Optimal. Leaf size=452 \[ \frac {2 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 e n \sqrt {b^2-a^2}}-\frac {2 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 e n \sqrt {b^2-a^2}}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}-\frac {b x^{-n} (e x)^{3 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )}{a d e n \sqrt {b^2-a^2}}+\frac {b x^{-n} (e x)^{3 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )}{a d e n \sqrt {b^2-a^2}}+\frac {(e x)^{3 n}}{3 a e n} \]
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Rubi [A] time = 0.86, antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5440, 5436, 4191, 3320, 2264, 2190, 2531, 2282, 6589} \[ \frac {2 b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,-\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 e n \sqrt {b^2-a^2}}-\frac {2 b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,-\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}\right )}{a d^3 e n \sqrt {b^2-a^2}}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}\right )}{a d^2 e n \sqrt {b^2-a^2}}-\frac {b x^{-n} (e x)^{3 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )}{a d e n \sqrt {b^2-a^2}}+\frac {b x^{-n} (e x)^{3 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )}{a d e n \sqrt {b^2-a^2}}+\frac {(e x)^{3 n}}{3 a e n} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3320
Rule 4191
Rule 5436
Rule 5440
Rule 6589
Rubi steps
\begin {align*} \int \frac {(e x)^{-1+3 n}}{a+b \text {sech}\left (c+d x^n\right )} \, dx &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \int \frac {x^{-1+3 n}}{a+b \text {sech}\left (c+d x^n\right )} \, dx}{e}\\ &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b \text {sech}(c+d x)} \, 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}-\frac {b x^2}{a (b+a \cosh (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {\left (b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{b+a \cosh (c+d x)} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{\sqrt {-a^2+b^2} e n}+\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{\sqrt {-a^2+b^2} e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} d e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} d^2 e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (2 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^{c+d x^n}\right )}{a \sqrt {-a^2+b^2} d^3 e n}-\frac {\left (2 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^{c+d x^n}\right )}{a \sqrt {-a^2+b^2} d^3 e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3 e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3 e n}\\ \end {align*}
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Mathematica [F] time = 7.22, size = 0, normalized size = 0.00 \[ \int \frac {(e x)^{-1+3 n}}{a+b \text {sech}\left (c+d x^n\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.45, size = 2005, normalized size = 4.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{3 \, n - 1}}{b \operatorname {sech}\left (d x^{n} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{3 n -1}}{a +b \,\mathrm {sech}\left (c +d \,x^{n}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, b e^{3 \, n} \int \frac {e^{\left (d x^{n} + 3 \, n \log \relax (x) + c\right )}}{a^{2} e x e^{\left (2 \, d x^{n} + 2 \, c\right )} + 2 \, a b e x e^{\left (d x^{n} + c\right )} + a^{2} e x}\,{d x} + \frac {e^{3 \, n - 1} x^{3 \, n}}{3 \, a n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{3\,n-1}}{a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{3 n - 1}}{a + b \operatorname {sech}{\left (c + d x^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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