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I need to find \begin{align} \int\sqrt{1+9t^4}\:dt. \end{align} What I have so far: \begin{align} \int\sqrt{1+9t^4}\:dt & =\int\sqrt{1+\left(3t^2\right)^2}\:dt,\tag{1} \end{align} now let $3t^2=\tan\left(\theta\right)\implies \displaystyle t=\sqrt{\frac{\tan\left(\theta\right)}{3}},\:\:dt=\frac{1}{2}\left(\frac{\tan\left(\theta\right)}{3}\right)^{-1/2}\left(\frac{\sec^{2}\left(\theta\right)}{3}\right)$, which gives me \begin{align} \int\left[1+\tan^2\left(\theta\right)\right]^{1/2}\frac{\sqrt{3}\sec^2\left(\theta\right)}{6\sqrt{\tan\left(\theta\right)}}\:d\theta\tag{2}&=\frac{\sqrt{3}}{6}\int\frac{\sec^3\left(\theta\right)}{\sqrt{\tan\left(\theta\right)}}\:d\theta\\ & = \frac{\sqrt{3}}{6}\int\frac{\sec^2\left(\theta\right)\sec\left(\theta\right)\:d\theta}{\sqrt{\tan\left(\theta\right)}}\tag{3}, \end{align} now let $u=\tan\left(\theta\right),\:\:du=\sec^{2}\left(\theta\right)\:d\theta$, which gives us \begin{align} \frac{\sqrt{3}}{6}\int\frac{\sqrt{1+u^{\color{red}{2}}}\:du}{\sqrt{u}}\tag{4}, \end{align} and where do I go from here? Or perhaps I'm just going in circles and haven't made any progress with this result.

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    $\begingroup$ What are you trying to convey with "$\therefore$"? $\endgroup$
    – GFauxPas
    Feb 12, 2015 at 16:19
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    $\begingroup$ The result cannot be expressed in terms of elementary functions; you need elliptic integrals. $\endgroup$
    – user111187
    Feb 12, 2015 at 16:20
  • $\begingroup$ @GFauxPas Just me moving through the problem with "therefores," I suppose I could get rid of them. $\endgroup$
    – bjd2385
    Feb 12, 2015 at 16:20
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    $\begingroup$ @bd1251252: Doing it your way we end up with something like $\int \frac{\sqrt{1+u^2}}{\sqrt{u}}\,du$, crucially different from where you reached. $\endgroup$ Feb 12, 2015 at 17:00
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    $\begingroup$ If you are only interested in the result, it is $$\frac{3 (9 t^5+t)-2 \sqrt[4]{-1} \sqrt{27 t^4+3} F(i \sinh ^{-1}((1+i) \sqrt{\frac{3}{2}} t)|-1)}{9 \sqrt{9 t^4+1}}$$ $\endgroup$
    – dustin
    Feb 12, 2015 at 17:02

6 Answers 6

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Rescaling the integrand to $\sqrt{1+x^4}$ is a simple matter so let's start there. First, some trickery with integration by parts can reduce the desired integral to one closer to the form of an elliptic integral of the first kind:

$$\begin{align} \int_{0}^{a}\sqrt{1+x^4}\,\mathrm{d}x &=\int_{0}^{a}\frac{1+x^4}{\sqrt{1+x^4}}\,\mathrm{d}x\\ &=\int_{0}^{a}\frac{\mathrm{d}x}{\sqrt{1+x^4}}+\int_{0}^{a}\frac{x^4}{\sqrt{1+x^4}}\,\mathrm{d}x\\ &=\int_{0}^{a}\frac{\mathrm{d}x}{\sqrt{1+x^4}}+\int_{0}^{a}x\cdot\frac{x^3}{\sqrt{1+x^4}}\,\mathrm{d}x\\ &=\int_{0}^{a}\frac{\mathrm{d}x}{\sqrt{1+x^4}}+\left[\frac12x\sqrt{1+x^4}\right]_{0}^{a}-\frac12\int_{0}^{a}\sqrt{1+x^4}\,\mathrm{d}x\\ &=\int_{0}^{a}\frac{\mathrm{d}x}{\sqrt{1+x^4}}+\frac12a\sqrt{1+a^4}-\frac12\int_{0}^{a}\sqrt{1+x^4}\,\mathrm{d}x\\ \implies \frac32\int_{0}^{a}\sqrt{1+x^4}\,\mathrm{d}x &=\frac{a}{2}\sqrt{1+a^4}+\int_{0}^{a}\frac{\mathrm{d}x}{\sqrt{1+x^4}}\\ \implies \int_{0}^{a}\sqrt{1+x^4}\,\mathrm{d}x &=\frac{a}{3}\sqrt{1+a^4}+\frac23\int_{0}^{a}\frac{\mathrm{d}x}{\sqrt{1+x^4}}.\\ \end{align}$$

Focusing now on the integral $\int_{0}^{a}\frac{\mathrm{d}x}{\sqrt{1+x^4}}$, applying a Landen transformation of the form $x=\frac{1-y}{1+y}$ yields,

$$\begin{align} \int_{0}^{a}\frac{\mathrm{d}x}{\sqrt{1+x^4}} &=\int_{1}^{\frac{1-a}{1+a}}\frac{(1+y)^2}{\sqrt{2(1+6y^2+y^4)}}\cdot\frac{(-2)\,\mathrm{d}y}{(1+y)^2}\\ &=\sqrt{2}\int_{\frac{1-a}{1+a}}^{1}\frac{\mathrm{d}y}{\sqrt{1+6y^2+y^4}}\\ &=\sqrt{2}\int_{\frac{1-a}{1+a}}^{1}\frac{\mathrm{d}y}{\sqrt{\left(3+2\sqrt{2}+y^2\right)\left(3-2\sqrt{2}+y^2\right)}}.\\ \end{align}$$

Note that the constant terms in the last line above have the useful properties

$$(3+2\sqrt{2})^{-1}=3-2\sqrt{2};\\ \sqrt{3+2\sqrt{2}}=1+\sqrt{2}.$$

Scaling the integral by substituting $(\sqrt{2}+1)y=t$,

$$\begin{align} \int_{0}^{a}\frac{\mathrm{d}x}{\sqrt{1+x^4}} &=\sqrt{2}\int_{\frac{1-a}{1+a}}^{1}\frac{\mathrm{d}y}{\sqrt{\left[(1+\sqrt{2})^2+y^2\right]\left[(1-\sqrt{2})^2+y^2\right]}} \\ &=\sqrt{2}\int_{\frac{1-a}{1+a}}^{1}\frac{\mathrm{d}y}{(\sqrt{2}+1)(\sqrt{2}-1)\sqrt{\left[1+(\sqrt{2}-1)^2y^2\right]\left[1+(\sqrt{2}+1)^2y^2\right]}} \\ &=\sqrt{2}\int_{\frac{1-a}{1+a}}^{1}\frac{\mathrm{d}y}{\sqrt{\left[1+(\sqrt{2}-1)^2y^2\right]\left[1+(\sqrt{2}+1)^2y^2\right]}} \\ &=(2-\sqrt{2})\int_{(\sqrt{2}+1)\frac{1-a}{1+a}}^{\sqrt{2}+1}\frac{\mathrm{d}t}{\sqrt{\left[1+(\sqrt{2}-1)^4t^2\right]\left(1+t^2\right)}} \\ &=(2-\sqrt{2})\int_{(\sqrt{2}+1)\frac{1-a}{1+a}}^{\sqrt{2}+1}\frac{\mathrm{d}t}{\sqrt{1+t^2}\sqrt{1+(\sqrt{2}-1)^4t^2}}. \\ \end{align}$$

Now it's time for trigonometric substitution. For compactness of notation, write $(\sqrt{2}-1)^2=b$. Using $t=\tan{\theta}$,

$$\begin{align} \int_{0}^{a}\frac{\mathrm{d}x}{\sqrt{1+x^4}} &=(2-\sqrt{2})\int_{(\sqrt{2}+1)\frac{1-a}{1+a}}^{\sqrt{2}+1}\frac{\mathrm{d}t}{\sqrt{1+t^2}\sqrt{1+b^2t^2}} \\ &=(2-\sqrt{2})\int_{\tan^{-1}{\left[(\sqrt{2}+1)\frac{1-a}{1+a}\right]}}^{\frac{3\pi}{8}}\frac{\sec^2{\theta}\,\mathrm{d}\theta}{\sqrt{\sec^2{\theta}}\sqrt{1+b^2\tan^2}{\theta}} \\ &=(2-\sqrt{2})\int_{\tan^{-1}{\left[(\sqrt{2}+1)\frac{1-a}{1+a}\right]}}^{\frac{3\pi}{8}}\frac{\sec^2{\theta}\,\mathrm{d}\theta}{\sqrt{\sec^4{\theta}}\sqrt{\cos^2{\theta}+b^2\sin^2}{\theta}} \\ &=(2-\sqrt{2})\int_{\tan^{-1}{\left[(\sqrt{2}+1)\frac{1-a}{1+a}\right]}}^{\frac{3\pi}{8}}\frac{\mathrm{d}\theta}{\sqrt{\cos^2{\theta}+b^2\sin^2}{\theta}} \\ &=(2-\sqrt{2})\int_{\tan^{-1}{\left[(\sqrt{2}+1)\frac{1-a}{1+a}\right]}}^{\frac{3\pi}{8}}\frac{\mathrm{d}\theta}{\sqrt{1-(1-b^2)\sin^2}{\theta}} \\ &=(2-\sqrt{2})\int_{\tan^{-1}{\left[(\sqrt{2}+1)\frac{1-a}{1+a}\right]}}^{\frac{3\pi}{8}}\frac{\mathrm{d}\theta}{\sqrt{1-b^{\prime\,2}\sin^2}{\theta}}. \\ \end{align}$$

And presto chango, an elliptic integral of kind numero uno! I presume I can safely leave the remaining details to you, but let me know if should make anything clearer.

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Let's write $\sqrt{1+9t^4}$ as $$ \sqrt{1+9t^4} = \frac{3t^2\sqrt{1+9t^4}}{3t^2}=3t^2\sqrt{1/(9t^4)+1} $$ Now we can use the fractional power binomial expansion. \begin{align} (1/(9t^4)+1)^{1/2}&= 1 + \frac{1}{18t^4}-\frac{1}{5184t^8} +\cdots\\ 3t^2(1/(9t^4)+1)^{1/2}&= 3t^2 + \frac{1}{6t^2}-\frac{1}{1728t^6} +\cdots \end{align} Now let's integrate \begin{align} \int\sqrt{1+9t^4}dt &= t^3 - \frac{1}{6t} + \frac{1}{1080t^5} -\frac{1}{34992t^9} + \cdots\\ &= 3\sum_{n=0}^{\infty}\frac{(-1)^n(2n)!t^{3-4n}}{(4n-3)(2n-1)36^n(n!)^2} \end{align}

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Hint: for this part$$\int\frac{\sqrt{1+u}\:du}{\sqrt{u}}$$ Let $v=\sqrt\frac{1+u}{u}$ or $u=\frac{1}{v^2-1}$ and thus $du=-\frac{2vdv}{(v^2-1)^2}$
Then it comes to solving $$-\int \frac{2v^2dv}{(v^2-1)^2}$$ Gross as it looks, it is at least rational and therefore "doable".


However when I make Maple calculate $\int\sqrt{1+9t^4}\:dt$ it does not give a closed-form result. So I'm afraid there must be some mistake when you simplified it to $(4)$.

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  • $\begingroup$ It appears I made an error in step (4) and it would thus need to be $\sqrt{1+u^2}$... $\endgroup$
    – bjd2385
    Feb 12, 2015 at 17:16
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$$\int\sqrt{1+9t^4}dt = \int\sqrt{1+(3t^2)^2}dt$$ put $$3t^2 = \sinh x \implies 6tdt = \cosh xdx \implies dt = \frac{\cosh x}{2\sqrt{3}\sqrt{\sinh x}}dx$$ $$\int\sqrt{1+\sinh^2x}\frac{\cosh x}{2\sqrt{3}\sqrt{\sinh x}}dx$$ $$\frac{1}{2\sqrt{3}}\int\sqrt{\cosh^2x}\frac{\cosh x}{\sqrt{\sinh x}}dx$$ $$\frac{1}{2\sqrt{3}}\int \cosh x\frac{\cosh x}{\sqrt{\sinh x}}dx$$ Taking $\cosh x$ as 1st function and $\frac{\cosh x}{\sqrt{\sinh x}}$ as 2nd function then by parts intigeration we get $$\frac{1}{2\sqrt{3}}[\cosh x \frac{(\sinh x)^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}-\int \sinh x\frac{(\sinh x)^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}dx]$$ $$\frac{1}{2\sqrt{3}}[\cosh x\sinh x-\int \sinh^2xdx]$$ $$\frac{1}{2\sqrt{3}}[\cosh x\sinh x-\int \frac{\cosh 2x-1}{2}dx]$$ $$\frac{1}{2\sqrt{3}}[\cosh x\sinh x- \frac{1}{2} \int (\cosh 2x-1)dx]$$ $$\frac{1}{2\sqrt{3}}[\cosh x\sinh x- \frac{1}{2} (\frac{\sinh 2x}{2}-x)]$$ as $\sinh 2x = 2\sinh x\cosh x = 2(3t^2)\sqrt{1+9t^4} = 6t^2\sqrt{1+9t^4}$ $$\frac{1}{2\sqrt{3}}[\sqrt{1+9t^4}(3t^2)- \frac{1}{2} (\frac{6t^2\sqrt{1+9t^4}}{2}-\sinh^{-1}(3t^2))]$$

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  • $\begingroup$ You might want to take a look at your derivative of your solution. wolfram $\endgroup$
    – dustin
    Feb 12, 2015 at 18:28
  • $\begingroup$ what i have did wrong? $\endgroup$ Feb 12, 2015 at 18:42
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    $\begingroup$ I did not examine your steps but the derivative of your solution is $t\sqrt{3}\sqrt{1+9t^4}\neq\sqrt{1+9t^4}$ $\endgroup$
    – dustin
    Feb 12, 2015 at 18:44
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Substituting $u=t\sqrt3$ and $v=\dfrac1{1+u^4}$ we will be able to express this integral in terms of the

incomplete beta function of arguments $\dfrac14$ and $-\dfrac34$. Alternately, by expanding the integrand

into its binomial series, and switching the order of summation and integration, we can rewrite

the same integral in terms of the hypergeometric function.

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$$ \begin{aligned} \displaystyle \int \sqrt{1+9 t^4} \, \mathrm{d}t =&\frac{1}{3} t \sqrt{1+9 t^4}+\frac{2}{3} \displaystyle \int \frac{1}{\sqrt{1+9 t^4}} \, \mathrm{d}t\\ =&\frac{1}{3} t \sqrt{1+9 t^4}+\frac{\left(1+3 t^2\right) \sqrt{\frac{1+9 t^4}{\left(1+3 t^2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt{3} t\right)|\frac{1}{2}\right)}{3 \sqrt{3} \sqrt{1+9 t^4}}\\ \end{aligned} $$

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