Optimal. Leaf size=83 \[ \frac {2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac {2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}+\text {Int}\left (x^m \coth (a+b x),x\right ) \]
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Rubi [A]
time = 0.11, 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 = {}
\begin {gather*} \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx \end {gather*}
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*}
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Mathematica [A]
time = 17.06, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 2.27, size = 0, normalized size = 0.00 \[\int x^{m} \left (\cosh ^{3}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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