Optimal. Leaf size=135 \[ \frac {a (e x)^{2 n}}{2 e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-i e^{d x^n+c}\right )}{d^2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (i e^{d x^n+c}\right )}{d^2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n} \]
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Rubi [A] time = 0.11, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {14, 5440, 5436, 4180, 2279, 2391} \[ -\frac {i b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}+\frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2279
Rule 2391
Rule 4180
Rule 5436
Rule 5440
Rubi steps
\begin {align*} \int (e x)^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+2 n}+b (e x)^{-1+2 n} \text {sech}\left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^{2 n}}{2 e n}+b \int (e x)^{-1+2 n} \text {sech}\left (c+d x^n\right ) \, dx\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \text {sech}\left (c+d x^n\right ) \, dx}{e}\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int x \text {sech}(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^2 e n}+\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^2 e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-i e^{c+d x^n}\right )}{d^2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (i e^{c+d x^n}\right )}{d^2 e n}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 260, normalized size = 1.93 \[ \frac {x^{-2 n} (e x)^{2 n} \left (a d^2 x^{2 n}-2 i b \text {Li}_2\left (-i e^{d x^n+c}\right )+2 i b \text {Li}_2\left (i e^{d x^n+c}\right )+2 i b d x^n \log \left (1-i e^{c+d x^n}\right )-2 i b d x^n \log \left (1+i e^{c+d x^n}\right )+2 i b c \log \left (1-i e^{c+d x^n}\right )-\pi b \log \left (1-i e^{c+d x^n}\right )-2 i b c \log \left (1+i e^{c+d x^n}\right )+\pi b \log \left (1+i e^{c+d x^n}\right )-2 i b c \log \left (\cot \left (\frac {1}{4} \left (2 i c+2 i d x^n+\pi \right )\right )\right )+\pi b \log \left (\cot \left (\frac {1}{4} \left (2 i c+2 i d x^n+\pi \right )\right )\right )\right )}{2 d^2 e n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 658, normalized size = 4.87 \[ \text {result too large to display} \]
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 \[ \int {\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.81, size = 368, normalized size = 2.73 \[ \frac {a x \,{\mathrm e}^{\frac {\left (-1+2 n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \relax (x )+2 \ln \relax (e )\right )}{2}}}{2 n}+\frac {2 b \,{\mathrm e}^{-i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )} {\mathrm e}^{i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}} {\mathrm e}^{i \pi n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}} {\mathrm e}^{-i \pi n \mathrm {csgn}\left (i e x \right )^{3}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi \mathrm {csgn}\left (i e x \right )^{3}}{2}} e^{2 n} {\mathrm e}^{c} \left (-\frac {\sqrt {-{\mathrm e}^{2 c}}\, x^{n} d \left (\ln \left (1+{\mathrm e}^{d \,x^{n}} \sqrt {-{\mathrm e}^{2 c}}\right )-\ln \left (1-{\mathrm e}^{d \,x^{n}} \sqrt {-{\mathrm e}^{2 c}}\right )\right ) {\mathrm e}^{-2 c}}{2}-\frac {\sqrt {-{\mathrm e}^{2 c}}\, \left (\dilog \left (1+{\mathrm e}^{d \,x^{n}} \sqrt {-{\mathrm e}^{2 c}}\right )-\dilog \left (1-{\mathrm e}^{d \,x^{n}} \sqrt {-{\mathrm e}^{2 c}}\right )\right ) {\mathrm e}^{-2 c}}{2}\right )}{e n \,d^{2}} \]
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 \[ 2 \, b \int \frac {\left (e x\right )^{2 \, n - 1}}{e^{\left (d x^{n} + c\right )} + e^{\left (-d x^{n} - c\right )}}\,{d x} + \frac {\left (e x\right )^{2 \, n} a}{2 \, e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{2\,n-1} \,d x \]
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 \[ \int \left (e x\right )^{2 n - 1} \left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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