Optimal. Leaf size=98 \[ -\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {12 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}+\frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)} \]
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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 = {5445, 3769, 3771, 2639} \[ -\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {12 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}+\frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3769
Rule 3771
Rule 5445
Rubi steps
\begin {align*} \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx &=\frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)}-\frac {2 \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx}{5 b}\\ &=\frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(a+b x)}+\frac {6 \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx}{25 b}\\ &=\frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(a+b x)}+\frac {6 \int \sqrt {i \sinh (a+b x)} \, dx}{25 b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}}\\ &=\frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(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 )}{25 b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}}\\ \end {align*}
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Mathematica [C] time = 2.05, size = 111, normalized size = 1.13 \[ \frac {e^{-2 (a+b x)} \left (-\frac {48 e^{2 (a+b x)} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};e^{2 (a+b x)}\right )}{\sqrt {1-e^{2 (a+b x)}}}+(24-10 b x) e^{2 (a+b x)}+(5 b x-2) e^{4 (a+b x)}+5 b x+2\right )}{50 b^2 \sqrt {\text {csch}(a+b x)}} \]
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 \[ \int \frac {x \cosh \left (b x + a\right )}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {x \cosh \left (b x +a \right )}{\mathrm {csch}\left (b x +a \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x \cosh \left (b x + a\right )}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \cosh {\left (a + b x \right )}}{\operatorname {csch}^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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