Optimal. Leaf size=45 \[ \frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {14, 5441, 5437, 3770} \[ \frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 3770
Rule 5437
Rule 5441
Rubi steps
\begin {align*} \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+n}+b (e x)^{-1+n} \text {csch}\left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^n}{e n}+b \int (e x)^{-1+n} \text {csch}\left (c+d x^n\right ) \, dx\\ &=\frac {a (e x)^n}{e n}+\frac {\left (b x^{-n} (e x)^n\right ) \int x^{-1+n} \text {csch}\left (c+d x^n\right ) \, dx}{e}\\ &=\frac {a (e x)^n}{e n}+\frac {\left (b x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \text {csch}(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 45, normalized size = 1.00 \[ \frac {x^{-n} (e x)^n \left (a \left (c+d x^n\right )+b \log \left (\tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )\right )}{d e n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 180, normalized size = 4.00 \[ \frac {a d \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) \cosh \left (n \log \relax (x)\right ) + a d \cosh \left (n \log \relax (x)\right ) \sinh \left ({\left (n - 1\right )} \log \relax (e)\right ) - {\left (b \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) + b \sinh \left ({\left (n - 1\right )} \log \relax (e)\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right ) + 1\right ) + {\left (b \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) + b \sinh \left ({\left (n - 1\right )} \log \relax (e)\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right ) - 1\right ) + {\left (a d \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) + a d \sinh \left ({\left (n - 1\right )} \log \relax (e)\right )\right )} \sinh \left (n \log \relax (x)\right )}{d n} \]
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 \[ \int {\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.78, size = 155, normalized size = 3.44 \[ \frac {a x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \relax (x )+2 \ln \relax (e )\right )}{2}}}{n}-\frac {2 \arctanh \left ({\mathrm e}^{c +d \,x^{n}}\right ) e^{n} b \,{\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 75, normalized size = 1.67 \[ -b {\left (\frac {e^{n - 1} \log \left ({\left (e^{\left (d x^{n} + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{d n} - \frac {e^{n - 1} \log \left ({\left (e^{\left (d x^{n} + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{d n}\right )} + \frac {\left (e x\right )^{n} a}{e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.37, size = 112, normalized size = 2.49 \[ \frac {a\,x\,{\left (e\,x\right )}^{n-1}}{n}-\frac {2\,\mathrm {atan}\left (\frac {b\,x\,{\mathrm {e}}^{d\,x^n}\,{\mathrm {e}}^c\,{\left (e\,x\right )}^{n-1}\,\sqrt {-d^2\,n^2\,x^{2\,n}}}{d\,n\,x^n\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}\right )\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{\sqrt {-d^2\,n^2\,x^{2\,n}}} \]
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 \[ \int \left (e x\right )^{n - 1} \left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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