Optimal. Leaf size=98 \[ -\frac {4 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{3 b^2}-\frac {4 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{3 b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b} \]
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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 = {5445, 3768, 3771, 2639} \[ -\frac {4 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{3 b^2}-\frac {4 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{3 b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3768
Rule 3771
Rule 5445
Rubi steps
\begin {align*} \int x \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x) \, dx &=-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx}{3 b}\\ &=-\frac {4 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{3 b^2}-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx}{3 b}\\ &=-\frac {4 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{3 b^2}-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \int \sqrt {i \sinh (a+b x)} \, dx}{3 b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}}\\ &=-\frac {4 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{3 b^2}-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}-\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{3 b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 70, normalized size = 0.71 \[ -\frac {2 \sqrt {\text {csch}(a+b x)} \left (2 \cosh (a+b x)+b x \text {csch}(a+b x)-2 \sqrt {i \sinh (a+b x)} E\left (\left .\frac {1}{4} (-2 i a-2 i b x+\pi )\right |2\right )\right )}{3 b^2} \]
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 \[ \int x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int x \cosh \left (b x +a \right ) \mathrm {csch}\left (b x +a \right )^{\frac {5}{2}}\, dx \]
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 \[ \int x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{5/2} \,d x \]
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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