Optimal. Leaf size=82 \[ \text{Unintegrable}\left (x^m \coth (a+b x),x\right )+\frac{e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}+\frac{e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b} \]
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Rubi [A] time = 0.131473, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx &=\int x^m \coth (a+b x) \, dx+\int x^m \cosh (a+b x) \sinh (a+b x) \, dx\\ &=\int x^m \coth (a+b x) \, dx+\int \frac{1}{2} x^m \sinh (2 a+2 b x) \, dx\\ &=\frac{1}{2} \int x^m \sinh (2 a+2 b x) \, dx+\int x^m \coth (a+b x) \, dx\\ &=\frac{1}{4} \int e^{-i (2 i a+2 i b x)} x^m \, dx-\frac{1}{4} \int e^{i (2 i a+2 i b x)} x^m \, dx+\int x^m \coth (a+b x) \, dx\\ &=\frac{2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac{2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}+\int x^m \coth (a+b x) \, dx\\ \end{align*}
Mathematica [A] time = 22.3679, size = 0, normalized size = 0. \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.074, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( bx+a \right ) \right ) ^{3}{\rm csch} \left (bx+a\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cosh \left (b x + a\right )^{3} \operatorname{csch}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \cosh \left (b x + a\right )^{3} \operatorname{csch}\left (b x + a\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cosh \left (b x + a\right )^{3} \operatorname{csch}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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