How to solve $\int\tan^3(x)\,dx$? How to solve $\int\tan^3(x)\,dx$ ? Do I use substitution?
 A: One standard approach to integral involving powers of tangent is the following: Rewrite your power as something times $\tan^2(x)$. In this case: $\tan^3(x)=\tan(x)\cdot\tan^2(x)$.
Now, use that $\tan^2(x)=\sec^2(x)-1$, so
$$ \int\tan^3(x)dx=\int\tan(x)\sec^2(x)dx-\int\tan(x)dx.$$
The first integral on the right hand side can be solved using the substitution $u=\tan(x)$, so $du=\sec^2(x)dx$ and $$ \int\tan(x)\sec^2(x)dx=\int u du. $$ The second integral is $\ln|\sec x|$, and can be found writing $\tan(x)=\sin(x)/\cos(x)$ and using the substitution $v=\cos(x)$.

Note that the same idea allows you to solve $\displaystyle \int\tan^n(x) dx$ for any $n\ge 3$. For example:
$$\int\tan^5(x)dx=\int\tan^3(x)\tan^2(x)dx=\int\tan^3(x)(\sec^2(x)-1)dx,$$
as before we split this last integral into two:
$$\int\tan^3(x)\sec^2(x)dx-\int\tan^3(x)dx.$$
The first one is solved with the substitution $u=\tan(x)$. The second is the integral we solved before. 
This, by the way, is an example of a reduction method that transforms an integral that depends on an integer parameter $n$ (in this case, $\displaystyle \int\tan^n(x)dx$) into a similar integral, but that now depends on some $m<n$. This gives us an algorithm to solve the integral by repeatedly applying the reduction procedure.
A: A simple way out is to write $\tan^3(x) = \frac{\sin^3(x)}{\cos^3(x)}$. Now let $\cos(x) = t$ and rewrite $\sin^2(x) = 1-t^2$.
The integral now becomes $\int \frac{\sin^3(x)}{\cos^3(x)} dx$.
$\cos(x) = t \Rightarrow -\sin(x)dx = dt$ and $\sin^2(x) = 1-t^2$.
Note the numerator of the integral $\sin^3(x) dx$ can be written as $\sin^2(x) \times \sin(x) dx = (1-t^2) (-dt) = (t^2-1) dt$
So the integral becomes $\int \frac{t^2-1}{t^3} dt = \int \frac{dt}{t} - \int \frac{dt}{t^3} = \log(t) + \frac{1}{2t^2} = \log(\cos(x)) + \frac{1}{2 \cos^2(x)} = \log(\cos(x)) + \frac{\sec^2(x)}{2}$
A: HINT: Rewrite your integral in this form:
\begin{align*}
\int\tan^{3}{x} \ dx & = \int (\sec^{2}{x}-1) \cdot \tan{x} \ dx \\ &= \int \sec^{2}{x} \cdot \tan{x} \ dx - \int \tan{x} \ dx \\ &= \int \sec{x} \cdot (\sec{x}\tan{x}) \ dx - \int \tan{x} \ dx
\end{align*}
