Optimal. Leaf size=98 \[ -\frac {4 \cosh (a+b x)}{3 b^2 \sqrt {\sinh (a+b x)}}-\frac {4 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{3 b^2 \sqrt {i \sinh (a+b x)}}-\frac {2 x}{3 b \sinh ^{\frac {3}{2}}(a+b x)} \]
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Rubi [A] time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5372, 2636, 2640, 2639} \[ -\frac {4 \cosh (a+b x)}{3 b^2 \sqrt {\sinh (a+b x)}}-\frac {4 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{3 b^2 \sqrt {i \sinh (a+b x)}}-\frac {2 x}{3 b \sinh ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2640
Rule 5372
Rubi steps
\begin {align*} \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {5}{2}}(a+b x)} \, dx &=-\frac {2 x}{3 b \sinh ^{\frac {3}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sinh ^{\frac {3}{2}}(a+b x)} \, dx}{3 b}\\ &=-\frac {2 x}{3 b \sinh ^{\frac {3}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{3 b^2 \sqrt {\sinh (a+b x)}}+\frac {2 \int \sqrt {\sinh (a+b x)} \, dx}{3 b}\\ &=-\frac {2 x}{3 b \sinh ^{\frac {3}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{3 b^2 \sqrt {\sinh (a+b x)}}+\frac {\left (2 \sqrt {\sinh (a+b x)}\right ) \int \sqrt {i \sinh (a+b x)} \, dx}{3 b \sqrt {i \sinh (a+b x)}}\\ &=-\frac {2 x}{3 b \sinh ^{\frac {3}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{3 b^2 \sqrt {\sinh (a+b x)}}-\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{3 b^2 \sqrt {i \sinh (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 66, normalized size = 0.67 \[ -\frac {2 \left (\sinh (2 (a+b x))+2 i (i \sinh (a+b x))^{3/2} E\left (\left .\frac {1}{4} (-2 i a-2 i b x+\pi )\right |2\right )+b x\right )}{3 b^2 \sinh ^{\frac {3}{2}}(a+b x)} \]
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 \[ \int \frac {x \cosh \left (b x + a\right )}{\sinh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {x \cosh \left (b x + a\right )}{\sinh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {sinh}\left (a+b\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cosh {\left (a + b x \right )}}{\sinh ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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