Integrand size = 15, antiderivative size = 87 \[ \int x^2 \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {3 i}{b n}\right ),\frac {3}{2} \left (1-\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+i b n} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4605, 4601, 371} \[ \int x^2 \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {3 i}{b n}\right ),\frac {3}{2} \left (1-\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+i b n} \]
[In]
[Out]
Rule 371
Rule 4601
Rule 4605
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^{-1+\frac {3}{n}} \sec (a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (2 e^{i a} x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^{-1+i b+\frac {3}{n}}}{1+e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n} \\ & = \frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {3 i}{b n}\right ),\frac {3}{2} \left (1-\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+i b n} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99 \[ \int x^2 \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {2 i e^{i a} x^3 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {3 i}{2 b n},\frac {3}{2}-\frac {3 i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{-3 i+b n} \]
[In]
[Out]
\[\int x^{2} \sec \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]
[In]
[Out]
\[ \int x^2 \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{2} \sec \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
[In]
[Out]
\[ \int x^2 \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^{2} \sec {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
[In]
[Out]
\[ \int x^2 \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{2} \sec \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
[In]
[Out]
\[ \int x^2 \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{2} \sec \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int x^2 \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {x^2}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
[In]
[Out]