Optimal. Leaf size=82 \[ \frac {(e x)^n}{a e n}+\frac {2 b x^{-n} (e x)^n \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d e n} \]
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Rubi [A]
time = 0.10, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5549, 5545,
3868, 2739, 632, 210} \begin {gather*} \frac {2 b x^{-n} (e x)^n \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2+b^2}}\right )}{a d e n \sqrt {a^2+b^2}}+\frac {(e x)^n}{a e n} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 3868
Rule 5545
Rule 5549
Rubi steps
\begin {align*} \int \frac {(e x)^{-1+n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx &=\frac {\left (x^{-n} (e x)^n\right ) \int \frac {x^{-1+n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx}{e}\\ &=\frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{a+b \text {csch}(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 ) \text {Subst}\left (\int \frac {1}{1+\frac {a \sinh (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 ) \text {Subst}\left (\int \frac {1}{1-\frac {2 i a x}{b}+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 {\left (4 i x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{-4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,-\frac {2 i a}{b}+2 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 \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d e n}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 84, normalized size = 1.02 \begin {gather*} \frac {(e x)^n \left (d+c x^{-n}-\frac {2 b x^{-n} \text {ArcTan}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}\right )}{a d e n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 4.32, size = 319, normalized size = 3.89
method | result | size |
risch | \(\frac {x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{a n}-\frac {2 b \,{\mathrm e}^{-\frac {i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{\frac {i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi n \mathrm {csgn}\left (i e x \right )^{3}}{2}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi \mathrm {csgn}\left (i e x \right )^{3}}{2}} e^{n} {\mathrm e}^{c} \arctan \left (\frac {2 a \,{\mathrm e}^{2 c +d \,x^{n}}+2 \,{\mathrm e}^{c} b}{2 \sqrt {-a^{2} {\mathrm e}^{2 c}-b^{2} {\mathrm e}^{2 c}}}\right )}{a n e d \sqrt {-a^{2} {\mathrm e}^{2 c}-b^{2} {\mathrm e}^{2 c}}}\) | \(319\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs.
\(2 (79) = 158\).
time = 0.51, size = 218, normalized size = 2.66 \begin {gather*} \frac {\sqrt {a^{2} + b^{2}} {\left (b \cosh \left (n - 1\right ) + b \sinh \left (n - 1\right )\right )} \log \left (\frac {a b + {\left (a^{2} + b^{2} + \sqrt {a^{2} + b^{2}} b\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - {\left (b^{2} + \sqrt {a^{2} + b^{2}} b\right )} \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sqrt {a^{2} + b^{2}} a}{a \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + b}\right ) + {\left ({\left (a^{2} + b^{2}\right )} d \cosh \left (n - 1\right ) + {\left (a^{2} + b^{2}\right )} d \sinh \left (n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left ({\left (a^{2} + b^{2}\right )} d \cosh \left (n - 1\right ) + {\left (a^{2} + b^{2}\right )} d \sinh \left (n - 1\right )\right )} \sinh \left (n \log \left (x\right )\right )}{{\left (a^{3} + a b^{2}\right )} d n} \end {gather*}
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 \begin {gather*} \int \frac {\left (e x\right )^{n - 1}}{a + b \operatorname {csch}{\left (c + d x^{n} \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.66, size = 410, normalized size = 5.00 \begin {gather*} \frac {x\,{\left (e\,x\right )}^{n-1}}{a\,n}-\frac {\left (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 )-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 )\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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