Optimal. Leaf size=217 \[ \frac {a (e x)^{3 n}}{3 e n}+\frac {2 i b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-i e^{d x^n+c}\right )}{d^3 e n}-\frac {2 i b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (i e^{d x^n+c}\right )}{d^3 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-i e^{d x^n+c}\right )}{d^2 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (i e^{d x^n+c}\right )}{d^2 e n}+\frac {2 b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n} \]
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Rubi [A] time = 0.18, antiderivative size = 217, normalized size of antiderivative = 1.00, 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, 5440, 5436, 4180, 2531, 2282, 6589} \[ \frac {2 i b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,-i e^{c+d x^n}\right )}{d^3 e n}-\frac {2 i b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,i e^{c+d x^n}\right )}{d^3 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}+\frac {a (e x)^{3 n}}{3 e n}+\frac {2 b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2282
Rule 2531
Rule 4180
Rule 5436
Rule 5440
Rule 6589
Rubi steps
\begin {align*} \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+3 n}+b (e x)^{-1+3 n} \text {sech}\left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^{3 n}}{3 e n}+b \int (e x)^{-1+3 n} \text {sech}\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} \text {sech}\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 \text {sech}(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a (e x)^{3 n}}{3 e n}+\frac {2 b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \log \left (1-i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}\\ &=\frac {a (e x)^{3 n}}{3 e n}+\frac {2 b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-i e^{c+d x^n}\right )}{d^2 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (i e^{c+d 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^{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^{c+d x}\right ) \, dx,x,x^n\right )}{d^2 e n}\\ &=\frac {a (e x)^{3 n}}{3 e n}+\frac {2 b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-i e^{c+d x^n}\right )}{d^2 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (i e^{c+d x^n}\right )}{d^2 e n}+\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^3 e n}-\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^3 e n}\\ &=\frac {a (e x)^{3 n}}{3 e n}+\frac {2 b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-i e^{c+d x^n}\right )}{d^2 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (i e^{c+d x^n}\right )}{d^2 e n}+\frac {2 i b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-i e^{c+d x^n}\right )}{d^3 e n}-\frac {2 i b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (i e^{c+d x^n}\right )}{d^3 e n}\\ \end {align*}
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Mathematica [F] time = 10.53, size = 0, normalized size = 0.00 \[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.49, size = 1072, normalized size = 4.94 \[ \text {result too large to display} \]
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 \[ \int {\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.16, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{3 n -1} \left (a +b \,\mathrm {sech}\left (c +d \,x^{n}\right )\right )\, dx \]
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 \[ 2 \, b \int \frac {\left (e x\right )^{3 \, n - 1}}{e^{\left (d x^{n} + c\right )} + e^{\left (-d x^{n} - c\right )}}\,{d x} + \frac {\left (e x\right )^{3 \, n} a}{3 \, e n} \]
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 \[ \int \left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{3\,n-1} \,d x \]
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 \[ \int \left (e x\right )^{3 n - 1} \left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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