Optimal. Leaf size=235 \[ -\frac{2 b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-i e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{2 b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,i e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{2 i b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{2 i b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac{a (e x)^{3 n}}{3 e n}-\frac{2 i b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n} \]
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Rubi [A] time = 0.18986, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {14, 4208, 4204, 4181, 2531, 2282, 6589} \[ -\frac{2 b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-i e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{2 b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,i e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{2 i b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{2 i b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac{a (e x)^{3 n}}{3 e n}-\frac{2 i b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 4208
Rule 4204
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+3 n}+b (e x)^{-1+3 n} \sec \left (c+d x^n\right )\right ) \, dx\\ &=\frac{a (e x)^{3 n}}{3 e n}+b \int (e x)^{-1+3 n} \sec \left (c+d x^n\right ) \, dx\\ &=\frac{a (e x)^{3 n}}{3 e n}+\frac{\left (b x^{-3 n} (e x)^{3 n}\right ) \int x^{-1+3 n} \sec \left (c+d x^n\right ) \, dx}{e}\\ &=\frac{a (e x)^{3 n}}{3 e n}+\frac{\left (b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x^2 \sec (c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac{a (e x)^{3 n}}{3 e n}-\frac{2 i b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac{\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}+\frac{\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}\\ &=\frac{a (e x)^{3 n}}{3 e n}-\frac{2 i b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac{2 i b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{2 i b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n}+\frac{\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n}\\ &=\frac{a (e x)^{3 n}}{3 e n}-\frac{2 i b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac{2 i b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{2 i b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^3 e n}\\ &=\frac{a (e x)^{3 n}}{3 e n}-\frac{2 i b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac{2 i b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{2 i b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{2 b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-i e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{2 b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (i e^{i \left (c+d x^n\right )}\right )}{d^3 e n}\\ \end{align*}
Mathematica [F] time = 1.17997, size = 0, normalized size = 0. \[ \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.378, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{-1+3\,n} \left ( a+b\sec \left ( c+d{x}^{n} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.19279, size = 1640, normalized size = 6.98 \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{\left (b \sec \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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