Integration involving substitution, integration by part and partial fraction

Question

I am trying to design an integration question which involves three methods, namely substitution, integration by part and partial fraction.

However, I couldn't design such a question. The best I can do involves at most $2$ methods, but not $3$. For example,

$$\int x^8 e^{x^3} \ dx$$

The integral above involves only substitution $(u=x^3)$ and by part, but not partial fraction.

It would be good if someone can come out a question involving three methods.

Answer 1

An example could be $$ \int\frac{x^2\ln x}{\left(1+x^3\right)^2}\:dx. $$ By the change of variable $u=x^3$, $du=3x^2dx$, one gets $$ \int\frac{x^2\ln x}{\left(1+x^3\right)^2}\:dx=\frac19\int\frac{\ln u}{\left(1+u\right)^2}\:du, $$ then one may use an integration by parts $$ \frac19\int\frac{\ln u}{\left(1+u\right)^2}\:du=-\frac{\ln u}{9\left(1+u\right)}+\frac19\int\frac1{u\left(1+u\right)}\:du $$and conclude by partial fraction decomposition.

Answer 2

$$\int\left(xe^{x^2}+x\log x+{1\over x^2-1}\right)\,dx$$