Optimal. Leaf size=1384 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.36653, antiderivative size = 1384, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {4208, 4204, 4191, 3324, 3321, 2264, 2190, 2531, 2282, 6589, 4522, 2279, 2391} \[ -\frac{2 i b^2 (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^n+c\right )}}{b-i \sqrt{a^2-b^2}}\right ) x^{-3 n}}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac{2 i b^2 (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^n+c\right )}}{b+i \sqrt{a^2-b^2}}\right ) x^{-3 n}}{a^2 \left (a^2-b^2\right ) d^3 e n}+\frac{4 i b (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{i \left (d x^n+c\right )}}{b-\sqrt{b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt{b^2-a^2} d^3 e n}-\frac{2 i b^3 (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{i \left (d x^n+c\right )}}{b-\sqrt{b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^3 e n}-\frac{4 i b (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{i \left (d x^n+c\right )}}{b+\sqrt{b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt{b^2-a^2} d^3 e n}+\frac{2 i b^3 (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{i \left (d x^n+c\right )}}{b+\sqrt{b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^3 e n}+\frac{2 b^2 (e x)^{3 n} \log \left (\frac{e^{i \left (d x^n+c\right )} a}{b-i \sqrt{a^2-b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac{2 b^2 (e x)^{3 n} \log \left (\frac{e^{i \left (d x^n+c\right )} a}{b+i \sqrt{a^2-b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac{4 b (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^n+c\right )}}{b-\sqrt{b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt{b^2-a^2} d^2 e n}-\frac{2 b^3 (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^n+c\right )}}{b-\sqrt{b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^2 e n}-\frac{4 b (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^n+c\right )}}{b+\sqrt{b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt{b^2-a^2} d^2 e n}+\frac{2 b^3 (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^n+c\right )}}{b+\sqrt{b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^2 e n}-\frac{i b^2 (e x)^{3 n} x^{-n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac{2 i b (e x)^{3 n} \log \left (\frac{e^{i \left (d x^n+c\right )} a}{b-\sqrt{b^2-a^2}}+1\right ) x^{-n}}{a^2 \sqrt{b^2-a^2} d e n}-\frac{i b^3 (e x)^{3 n} \log \left (\frac{e^{i \left (d x^n+c\right )} a}{b-\sqrt{b^2-a^2}}+1\right ) x^{-n}}{a^2 \left (b^2-a^2\right )^{3/2} d e n}-\frac{2 i b (e x)^{3 n} \log \left (\frac{e^{i \left (d x^n+c\right )} a}{b+\sqrt{b^2-a^2}}+1\right ) x^{-n}}{a^2 \sqrt{b^2-a^2} d e n}+\frac{i b^3 (e x)^{3 n} \log \left (\frac{e^{i \left (d x^n+c\right )} a}{b+\sqrt{b^2-a^2}}+1\right ) x^{-n}}{a^2 \left (b^2-a^2\right )^{3/2} d e n}+\frac{b^2 (e x)^{3 n} \sin \left (d x^n+c\right ) x^{-n}}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (d x^n+c\right )\right )}+\frac{(e x)^{3 n}}{3 a^2 e n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4208
Rule 4204
Rule 4191
Rule 3324
Rule 3321
Rule 2264
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 4522
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \int \frac{x^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx}{e}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{(a+b \sec (c+d x))^2} \, dx,x,x^n\right )}{e n}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \left (\frac{x^2}{a^2}+\frac{b^2 x^2}{a^2 (b+a \cos (c+d x))^2}-\frac{2 b x^2}{a^2 (b+a \cos (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+a \cos (c+d x)} \, dx,x,x^n\right )}{a^2 e n}+\frac{\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{(b+a \cos (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}-\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,x^n\right )}{a^2 e n}-\frac{\left (b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+a \cos (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x \sin (c+d x)}{b+a \cos (c+d x)} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) d e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac{b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}-\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}-\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \sqrt{-a^2+b^2} e n}+\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \sqrt{-a^2+b^2} e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{i b-\sqrt{a^2-b^2}+i a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) d e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{i b+\sqrt{a^2-b^2}+i a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) d e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac{2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}-\frac{2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}+\frac{b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}-\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) \sqrt{-a^2+b^2} e n}+\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) \sqrt{-a^2+b^2} e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{i a e^{i (c+d x)}}{i b-\sqrt{a^2-b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{i a e^{i (c+d x)}}{i b+\sqrt{a^2-b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac{\left (4 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt{-a^2+b^2} d e n}+\frac{\left (4 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt{-a^2+b^2} d e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac{i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac{2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}+\frac{i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac{2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}+\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}-\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}+\frac{\left (2 i b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i a x}{i b-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}+\frac{\left (2 i b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i a x}{i b+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}+\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}-\frac{\left (2 i b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt{-a^2+b^2} d e n}+\frac{\left (2 i b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt{-a^2+b^2} d e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac{i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac{2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}+\frac{i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac{2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}-\frac{2 i b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac{2 i b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}+\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}+\frac{\left (4 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \sqrt{-a^2+b^2} d^3 e n}-\frac{\left (4 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \sqrt{-a^2+b^2} d^3 e n}-\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt{-a^2+b^2} d^2 e n}+\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt{-a^2+b^2} d^2 e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac{i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac{2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}+\frac{i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac{2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}-\frac{2 i b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac{2 i b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}+\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}+\frac{4 i b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3 e n}-\frac{4 i b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}+\frac{\left (2 i b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \left (a^2-b^2\right ) \sqrt{-a^2+b^2} d^3 e n}-\frac{\left (2 i b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \left (a^2-b^2\right ) \sqrt{-a^2+b^2} d^3 e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac{i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac{2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}+\frac{i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac{2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}-\frac{2 i b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac{2 i b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}+\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}-\frac{2 i b^3 x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}+\frac{4 i b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3 e n}+\frac{2 i b^3 x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}-\frac{4 i b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}\\ \end{align*}
Mathematica [F] time = 10.0505, size = 0, normalized size = 0. \[ \int \frac{(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.961, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{-1+3\,n}}{ \left ( a+b\sec \left ( c+d{x}^{n} \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 4.06414, size = 8319, normalized size = 6.01 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{3 \, n - 1}}{{\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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