Optimal. Leaf size=87 \[ \frac {(e x)^n}{a e n}-\frac {2 b x^{-n} (e x)^n \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a d e n \sqrt {a-b} \sqrt {a+b}} \]
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Rubi [A] time = 0.15, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5440, 5436, 3783, 2659, 208} \[ \frac {(e x)^n}{a e n}-\frac {2 b x^{-n} (e x)^n \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a d e n \sqrt {a-b} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2659
Rule 3783
Rule 5436
Rule 5440
Rubi steps
\begin {align*} \int \frac {(e x)^{-1+n}}{a+b \text {sech}\left (c+d x^n\right )} \, dx &=\frac {\left (x^{-n} (e x)^n\right ) \int \frac {x^{-1+n}}{a+b \text {sech}\left (c+d x^n\right )} \, dx}{e}\\ &=\frac {\left (x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \frac {1}{a+b \text {sech}(c+d x)} \, dx,x,x^n\right )}{e n}\\ &=\frac {(e x)^n}{a e n}-\frac {\left (x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a \cosh (c+d x)}{b}} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^n}{a e n}+\frac {\left (2 i x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,i \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a d e n}\\ &=\frac {(e x)^n}{a e n}-\frac {2 b x^{-n} (e x)^n \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b} d e n}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 80, normalized size = 0.92 \[ \frac {(e x)^n \left (\frac {2 b x^{-n} \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+c x^{-n}+d\right )}{a d e n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 511, normalized size = 5.87 \[ \left [\frac {{\left (a^{2} - b^{2}\right )} d \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) \cosh \left (n \log \relax (x)\right ) + {\left (a^{2} - b^{2}\right )} d \cosh \left (n \log \relax (x)\right ) \sinh \left ({\left (n - 1\right )} \log \relax (e)\right ) - {\left (\sqrt {-a^{2} + b^{2}} b \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) + \sqrt {-a^{2} + b^{2}} b \sinh \left ({\left (n - 1\right )} \log \relax (e)\right )\right )} \log \left (\frac {a b + {\left (b^{2} + \sqrt {-a^{2} + b^{2}} b\right )} \cosh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right ) + {\left (a^{2} - b^{2} - \sqrt {-a^{2} + b^{2}} b\right )} \sinh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right ) + \sqrt {-a^{2} + b^{2}} a}{a \cosh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right ) + b}\right ) + {\left ({\left (a^{2} - b^{2}\right )} d \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) + {\left (a^{2} - b^{2}\right )} d \sinh \left ({\left (n - 1\right )} \log \relax (e)\right )\right )} \sinh \left (n \log \relax (x)\right )}{{\left (a^{3} - a b^{2}\right )} d n}, \frac {{\left (a^{2} - b^{2}\right )} d \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) \cosh \left (n \log \relax (x)\right ) + {\left (a^{2} - b^{2}\right )} d \cosh \left (n \log \relax (x)\right ) \sinh \left ({\left (n - 1\right )} \log \relax (e)\right ) + 2 \, {\left (\sqrt {a^{2} - b^{2}} b \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) + \sqrt {a^{2} - b^{2}} b \sinh \left ({\left (n - 1\right )} \log \relax (e)\right )\right )} \arctan \left (-\frac {\sqrt {a^{2} - b^{2}} a \cosh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right ) + \sqrt {a^{2} - b^{2}} a \sinh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right ) + \sqrt {a^{2} - b^{2}} b}{a^{2} - b^{2}}\right ) + {\left ({\left (a^{2} - b^{2}\right )} d \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) + {\left (a^{2} - b^{2}\right )} d \sinh \left ({\left (n - 1\right )} \log \relax (e)\right )\right )} \sinh \left (n \log \relax (x)\right )}{{\left (a^{3} - a b^{2}\right )} d n}\right ] \]
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 \[ \int \frac {\left (e x\right )^{n - 1}}{b \operatorname {sech}\left (d x^{n} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.56, size = 317, normalized size = 3.64 \[ \frac {x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \relax (x )+2 \ln \relax (e )\right )}{2}}}{a n}-\frac {2 b \,{\mathrm e}^{-\frac {i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{\frac {i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi n \mathrm {csgn}\left (i e x \right )^{3}}{2}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi \mathrm {csgn}\left (i e x \right )^{3}}{2}} e^{n} {\mathrm e}^{c} \arctan \left (\frac {2 a \,{\mathrm e}^{2 c +d \,x^{n}}+2 \,{\mathrm e}^{c} b}{2 \sqrt {a^{2} {\mathrm e}^{2 c}-b^{2} {\mathrm e}^{2 c}}}\right )}{a n e d \sqrt {a^{2} {\mathrm e}^{2 c}-b^{2} {\mathrm e}^{2 c}}} \]
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 \[ -2 \, b e^{n} \int \frac {e^{\left (d x^{n} + n \log \relax (x) + c\right )}}{a^{2} e x e^{\left (2 \, d x^{n} + 2 \, c\right )} + 2 \, a b e x e^{\left (d x^{n} + c\right )} + a^{2} e x}\,{d x} + \frac {e^{n - 1} x^{n}}{a n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.10, size = 409, normalized size = 4.70 \[ \frac {x\,{\left (e\,x\right )}^{n-1}}{a\,n}-\frac {\left (2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x^n}\,{\mathrm {e}}^c\,\left (\frac {2\,b\,x\,{\left (e\,x\right )}^{n-1}}{a^4\,d\,n\,x^n\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}+\frac {2\,b\,d\,n\,x^n\,{\left (e\,x\right )}^{1-n}\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{a^2\,x\,\sqrt {a^4\,d^2\,n^2\,x^{2\,n}-a^2\,b^2\,d^2\,n^2\,x^{2\,n}}\,\sqrt {a^2\,d^2\,n^2\,x^{2\,n}\,\left (a^2-b^2\right )}}\right )\,\sqrt {a^4\,d^2\,n^2\,x^{2\,n}-a^2\,b^2\,d^2\,n^2\,x^{2\,n}}}{2}+\frac {a\,d\,n\,x^n\,{\left (e\,x\right )}^{1-n}\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{x\,\sqrt {a^2\,d^2\,n^2\,x^{2\,n}\,\left (a^2-b^2\right )}}\right )+2\,\mathrm {atan}\left (\frac {x\,{\left (e\,x\right )}^{n-1}\,\sqrt {a^2\,d^2\,n^2\,x^{2\,n}\,\left (a^2-b^2\right )}}{a\,d\,n\,x^n\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}\right )\right )\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{\sqrt {a^4\,d^2\,n^2\,x^{2\,n}-a^2\,b^2\,d^2\,n^2\,x^{2\,n}}} \]
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 \[ \int \frac {\left (e x\right )^{n - 1}}{a + b \operatorname {sech}{\left (c + d x^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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