Optimal. Leaf size=64 \[ \frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b^2}+\frac {4 \sinh (a+b x)}{3 b^2 \sqrt {\cosh (a+b x)}}-\frac {2 x}{3 b \cosh ^{\frac {3}{2}}(a+b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 64, 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 = {5373, 2636, 2639} \[ \frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b^2}+\frac {4 \sinh (a+b x)}{3 b^2 \sqrt {\cosh (a+b x)}}-\frac {2 x}{3 b \cosh ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 5373
Rubi steps
\begin {align*} \int \frac {x \sinh (a+b x)}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx &=-\frac {2 x}{3 b \cosh ^{\frac {3}{2}}(a+b x)}+\frac {2 \int \frac {1}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx}{3 b}\\ &=-\frac {2 x}{3 b \cosh ^{\frac {3}{2}}(a+b x)}+\frac {4 \sinh (a+b x)}{3 b^2 \sqrt {\cosh (a+b x)}}-\frac {2 \int \sqrt {\cosh (a+b x)} \, dx}{3 b}\\ &=-\frac {2 x}{3 b \cosh ^{\frac {3}{2}}(a+b x)}+\frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b^2}+\frac {4 \sinh (a+b x)}{3 b^2 \sqrt {\cosh (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 57, normalized size = 0.89 \[ \frac {2 \left (\sinh (2 (a+b x))+2 i \cosh ^{\frac {3}{2}}(a+b x) E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )-b x\right )}{3 b^2 \cosh ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {x \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {x \sinh \left (b x +a \right )}{\cosh \left (b x +a \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {x \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x\,\mathrm {sinh}\left (a+b\,x\right )}{{\mathrm {cosh}\left (a+b\,x\right )}^{5/2}} \,d x \]
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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