Optimal. Leaf size=307 \[ -\frac {b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}+\frac {b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}-\frac {b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )}{a d e n \sqrt {b^2-a^2}}+\frac {b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )}{a d e n \sqrt {b^2-a^2}}+\frac {(e x)^{2 n}}{2 a e n} \]
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Rubi [A] time = 0.56, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5440, 5436, 4191, 3320, 2264, 2190, 2279, 2391} \[ -\frac {b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}+\frac {b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}\right )}{a d^2 e n \sqrt {b^2-a^2}}-\frac {b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )}{a d e n \sqrt {b^2-a^2}}+\frac {b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )}{a d e n \sqrt {b^2-a^2}}+\frac {(e x)^{2 n}}{2 a e n} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3320
Rule 4191
Rule 5436
Rule 5440
Rubi steps
\begin {align*} \int \frac {(e x)^{-1+2 n}}{a+b \text {sech}\left (c+d x^n\right )} \, dx &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int \frac {x^{-1+2 n}}{a+b \text {sech}\left (c+d x^n\right )} \, dx}{e}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {x}{a+b \text {sech}(c+d x)} \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \left (\frac {x}{a}-\frac {b x}{a (b+a \cosh (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {(e x)^{2 n}}{2 a e n}-\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {x}{b+a \cosh (c+d x)} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^{2 n}}{2 a e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^{2 n}}{2 a e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x}{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^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x}{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)^{2 n}}{2 a e n}-\frac {b x^{-n} (e x)^{2 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)^{2 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 (b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \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 (b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \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)^{2 n}}{2 a e n}-\frac {b x^{-n} (e x)^{2 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)^{2 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 (b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a \sqrt {-a^2+b^2} d^2 e n}\\ &=\frac {(e x)^{2 n}}{2 a e n}-\frac {b x^{-n} (e x)^{2 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)^{2 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^{-2 n} (e x)^{2 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 {b x^{-2 n} (e x)^{2 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}\\ \end {align*}
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Mathematica [C] time = 2.10, size = 859, normalized size = 2.80 \[ \frac {(e x)^{2 n} \left (b+a \cosh \left (d x^n+c\right )\right ) \left (\frac {2 b \left (2 \left (d x^n+c\right ) \tan ^{-1}\left (\frac {(a+b) \coth \left (\frac {1}{2} \left (d x^n+c\right )\right )}{\sqrt {a^2-b^2}}\right )+2 \left (c-i \cos ^{-1}\left (-\frac {b}{a}\right )\right ) \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )}{\sqrt {a^2-b^2}}\right )+\left (\cos ^{-1}\left (-\frac {b}{a}\right )+2 \left (\tan ^{-1}\left (\frac {(a+b) \coth \left (\frac {1}{2} \left (d x^n+c\right )\right )}{\sqrt {a^2-b^2}}\right )+\tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )}{\sqrt {a^2-b^2}}\right )\right )\right ) \log \left (\frac {\sqrt {a^2-b^2} e^{-\frac {d x^n}{2}-\frac {c}{2}}}{\sqrt {2} \sqrt {a} \sqrt {b+a \cosh \left (d x^n+c\right )}}\right )+\left (\cos ^{-1}\left (-\frac {b}{a}\right )-2 \left (\tan ^{-1}\left (\frac {(a+b) \coth \left (\frac {1}{2} \left (d x^n+c\right )\right )}{\sqrt {a^2-b^2}}\right )+\tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )}{\sqrt {a^2-b^2}}\right )\right )\right ) \log \left (\frac {\sqrt {a^2-b^2} e^{\frac {1}{2} \left (d x^n+c\right )}}{\sqrt {2} \sqrt {a} \sqrt {b+a \cosh \left (d x^n+c\right )}}\right )-\left (\cos ^{-1}\left (-\frac {b}{a}\right )+2 \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (\frac {(a+b) \left (-a+b+i \sqrt {a^2-b^2}\right ) \left (\tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )-1\right )}{a \left (a+b+i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac {b}{a}\right )-2 \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (\frac {(a+b) \left (a-b+i \sqrt {a^2-b^2}\right ) \left (\tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )+1\right )}{a \left (a+b+i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )\right )}\right )+i \left (\text {Li}_2\left (\frac {\left (b-i \sqrt {a^2-b^2}\right ) \left (a+b-i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )\right )}{a \left (a+b+i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )\right )}\right )-\text {Li}_2\left (\frac {\left (b+i \sqrt {a^2-b^2}\right ) \left (a+b-i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )\right )}{a \left (a+b+i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )\right )}\right )\right )\right ) x^{-2 n}}{\sqrt {a^2-b^2} d^2}+1\right ) \text {sech}\left (d x^n+c\right )}{2 a e n \left (a+b \text {sech}\left (d x^n+c\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 1286, normalized size = 4.19 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (e x\right )^{2 \, n - 1}}{b \operatorname {sech}\left (d x^{n} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.66, size = 587, normalized size = 1.91 \[ \frac {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 a 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 {x^{n} d \left (\ln \left (\frac {a \,{\mathrm e}^{2 c +d \,x^{n}}+{\mathrm e}^{c} b -\sqrt {b^{2} {\mathrm e}^{2 c}-a^{2} {\mathrm e}^{2 c}}}{{\mathrm e}^{c} b -\sqrt {b^{2} {\mathrm e}^{2 c}-a^{2} {\mathrm e}^{2 c}}}\right )-\ln \left (\frac {a \,{\mathrm e}^{2 c +d \,x^{n}}+{\mathrm e}^{c} b +\sqrt {b^{2} {\mathrm e}^{2 c}-a^{2} {\mathrm e}^{2 c}}}{{\mathrm e}^{c} b +\sqrt {b^{2} {\mathrm e}^{2 c}-a^{2} {\mathrm e}^{2 c}}}\right )\right )}{2 \sqrt {b^{2} {\mathrm e}^{2 c}-a^{2} {\mathrm e}^{2 c}}}+\frac {\dilog \left (\frac {a \,{\mathrm e}^{2 c +d \,x^{n}}+{\mathrm e}^{c} b -\sqrt {b^{2} {\mathrm e}^{2 c}-a^{2} {\mathrm e}^{2 c}}}{{\mathrm e}^{c} b -\sqrt {b^{2} {\mathrm e}^{2 c}-a^{2} {\mathrm e}^{2 c}}}\right )-\dilog \left (\frac {a \,{\mathrm e}^{2 c +d \,x^{n}}+{\mathrm e}^{c} b +\sqrt {b^{2} {\mathrm e}^{2 c}-a^{2} {\mathrm e}^{2 c}}}{{\mathrm e}^{c} b +\sqrt {b^{2} {\mathrm e}^{2 c}-a^{2} {\mathrm e}^{2 c}}}\right )}{2 \sqrt {b^{2} {\mathrm e}^{2 c}-a^{2} {\mathrm e}^{2 c}}}\right )}{a 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 \[ -2 \, b e^{2 \, n} \int \frac {e^{\left (d x^{n} + 2 \, n \log \relax (x) + c\right )}}{a^{2} e x e^{\left (2 \, d x^{n} + 2 \, c\right )} + 2 \, a b e x e^{\left (d x^{n} + c\right )} + a^{2} e x}\,{d x} + \frac {e^{2 \, n - 1} x^{2 \, n}}{2 \, a n} \]
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 \[ \int \frac {{\left (e\,x\right )}^{2\,n-1}}{a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}} \,d x \]
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 \[ \int \frac {\left (e x\right )^{2 n - 1}}{a + b \operatorname {sech}{\left (c + d x^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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