Optimal. Leaf size=85 \[ -\text {Int}\left (x^m \tanh (a+b x),x\right )+\frac {e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \Gamma (m+1,-2 b x)}{b}+\frac {e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \Gamma (m+1,2 b x)}{b} \]
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Rubi [A] time = 0.14, 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 = {} \[ \int x^m \sinh ^2(a+b x) \tanh (a+b x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^m \sinh ^2(a+b x) \tanh (a+b x) \, dx &=\int x^m \cosh (a+b x) \sinh (a+b x) \, dx-\int x^m \tanh (a+b x) \, dx\\ &=\int \frac {1}{2} x^m \sinh (2 a+2 b x) \, dx-\int x^m \tanh (a+b x) \, dx\\ &=\frac {1}{2} \int x^m \sinh (2 a+2 b x) \, dx-\int x^m \tanh (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 \tanh (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 \tanh (a+b x) \, dx\\ \end {align*}
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Mathematica [A] time = 22.60, size = 0, normalized size = 0.00 \[ \int x^m \sinh ^2(a+b x) \tanh (a+b x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}, x\right ) \]
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 \[ \int x^{m} \operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}\,{d x} \]
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 \[ \int x^{m} \mathrm {sech}\left (b x +a \right ) \left (\sinh ^{3}\left (b x +a \right )\right )\, dx \]
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 \[ \int x^{m} \operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}\,{d x} \]
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 \[ \int \frac {x^m\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{\mathrm {cosh}\left (a+b\,x\right )} \,d x \]
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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