Optimal. Leaf size=121 \[ -\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}+\frac {12 \cosh (a+b x)}{35 b^2 \sqrt {\sinh (a+b x)}}+\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{35 b^2 \sqrt {i \sinh (a+b x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5480, 2716,
2721, 2719} \begin {gather*} -\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}+\frac {12 \cosh (a+b x)}{35 b^2 \sqrt {\sinh (a+b x)}}+\frac {12 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 )}{35 b^2 \sqrt {i \sinh (a+b x)}}-\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2719
Rule 2721
Rule 5480
Rubi steps
\begin {align*} \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {9}{2}}(a+b x)} \, dx &=-\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx}{7 b}\\ &=-\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}-\frac {6 \int \frac {1}{\sinh ^{\frac {3}{2}}(a+b x)} \, dx}{35 b}\\ &=-\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}+\frac {12 \cosh (a+b x)}{35 b^2 \sqrt {\sinh (a+b x)}}-\frac {6 \int \sqrt {\sinh (a+b x)} \, dx}{35 b}\\ &=-\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}+\frac {12 \cosh (a+b x)}{35 b^2 \sqrt {\sinh (a+b x)}}-\frac {\left (6 \sqrt {\sinh (a+b x)}\right ) \int \sqrt {i \sinh (a+b x)} \, dx}{35 b \sqrt {i \sinh (a+b x)}}\\ &=-\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}+\frac {12 \cosh (a+b x)}{35 b^2 \sqrt {\sinh (a+b x)}}+\frac {12 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)}}{35 b^2 \sqrt {i \sinh (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 89, normalized size = 0.74 \begin {gather*} -\frac {2 \left (5 b x-6 \cosh (a+b x) \sinh ^3(a+b x)+6 E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {i \sinh (a+b x)} \sinh ^3(a+b x)+\sinh (2 (a+b x))\right )}{35 b^2 \sinh ^{\frac {7}{2}}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.40, size = 0, normalized size = 0.00 \[\int \frac {x \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{\frac {9}{2}}}\, dx\]
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 \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 = -0.01 \begin {gather*} \int \frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {sinh}\left (a+b\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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