Optimal. Leaf size=98 \[ -\frac {4 \sinh ^{\frac {3}{2}}(a+b x) \cosh (a+b x)}{25 b^2}-\frac {12 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt {i \sinh (a+b x)}}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b} \]
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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 = {5372, 2635, 2640, 2639} \[ -\frac {4 \sinh ^{\frac {3}{2}}(a+b x) \cosh (a+b x)}{25 b^2}-\frac {12 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt {i \sinh (a+b x)}}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rule 5372
Rubi steps
\begin {align*} \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx &=\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \int \sinh ^{\frac {5}{2}}(a+b x) \, dx}{5 b}\\ &=-\frac {4 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {6 \int \sqrt {\sinh (a+b x)} \, dx}{25 b}\\ &=-\frac {4 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {\left (6 \sqrt {\sinh (a+b x)}\right ) \int \sqrt {i \sinh (a+b x)} \, dx}{25 b \sqrt {i \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)}}{25 b^2 \sqrt {i \sinh (a+b x)}}-\frac {4 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}\\ \end {align*}
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Mathematica [C] time = 2.36, size = 143, normalized size = 1.46 \[ \frac {e^{-3 (a+b x)} \left (48 e^{2 (a+b x)} \sqrt {1-e^{2 (a+b x)}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};e^{2 (a+b x)}\right )+\left (e^{2 (a+b x)}-1\right ) \left ((24-10 b x) e^{2 (a+b x)}+(5 b x-2) e^{4 (a+b x)}+5 b x+2\right )\right )}{50 \sqrt {2} b^2 \sqrt {e^{a+b x}-e^{-a-b x}}} \]
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 \[ \int x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int x \cosh \left (b x +a \right ) \left (\sinh ^{\frac {3}{2}}\left (b x +a \right )\right )\, dx \]
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 \[ \int x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^{3/2} \,d x \]
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 \[ \int x \sinh ^{\frac {3}{2}}{\left (a + b x \right )} \cosh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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