Optimal. Leaf size=72 \[ \text {Int}\left (x^m \text {csch}(a+b x),x\right )+\frac {e^a x^m (-b x)^{-m} \Gamma (m+1,-b x)}{2 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (m+1,b x)}{2 b} \]
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Rubi [A] time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \cosh (a+b x) \coth (a+b x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^m \cosh (a+b x) \coth (a+b x) \, dx &=\int x^m \text {csch}(a+b x) \, dx+\int x^m \sinh (a+b x) \, dx\\ &=\frac {1}{2} \int e^{-i (i a+i b x)} x^m \, dx-\frac {1}{2} \int e^{i (i a+i b x)} x^m \, dx+\int x^m \text {csch}(a+b x) \, dx\\ &=\frac {e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{2 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{2 b}+\int x^m \text {csch}(a+b x) \, dx\\ \end {align*}
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Mathematica [A] time = 19.59, size = 0, normalized size = 0.00 \[ \int x^m \cosh (a+b x) \coth (a+b x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\cosh ^{2}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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