Optimal. Leaf size=57 \[ -\frac {2 x \sqrt {\text {sech}(a+b x)}}{b}-\frac {4 i \sqrt {\cosh (a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{b^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5552, 3856,
2720} \begin {gather*} -\frac {2 x \sqrt {\text {sech}(a+b x)}}{b}-\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3856
Rule 5552
Rubi steps
\begin {align*} \int x \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx &=-\frac {2 x \sqrt {\text {sech}(a+b x)}}{b}+\frac {2 \int \sqrt {\text {sech}(a+b x)} \, dx}{b}\\ &=-\frac {2 x \sqrt {\text {sech}(a+b x)}}{b}+\frac {\left (2 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{b}\\ &=-\frac {2 x \sqrt {\text {sech}(a+b x)}}{b}-\frac {4 i \sqrt {\cosh (a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 46, normalized size = 0.81 \begin {gather*} -\frac {2 \left (b x+2 i \sqrt {\cosh (a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )\right ) \sqrt {\text {sech}(a+b x)}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.60, size = 0, normalized size = 0.00 \[\int x \mathrm {sech}\left (b x +a \right )^{\frac {3}{2}} \sinh \left (b x +a \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 [F]
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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,\mathrm {sinh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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