Optimal. Leaf size=64 \[ \frac {12 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{25 b^2}-\frac {4 \sinh (a+b x) \cosh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \cosh ^{\frac {5}{2}}(a+b x)}{5 b} \]
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Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5373, 2635, 2639} \[ \frac {12 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{25 b^2}-\frac {4 \sinh (a+b x) \cosh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \cosh ^{\frac {5}{2}}(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 5373
Rubi steps
\begin {align*} \int x \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx &=\frac {2 x \cosh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \int \cosh ^{\frac {5}{2}}(a+b x) \, dx}{5 b}\\ &=\frac {2 x \cosh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {4 \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{25 b^2}-\frac {6 \int \sqrt {\cosh (a+b x)} \, dx}{25 b}\\ &=\frac {2 x \cosh ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {12 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{25 b^2}-\frac {4 \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{25 b^2}\\ \end {align*}
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Mathematica [C] time = 2.01, size = 142, normalized size = 2.22 \[ \frac {e^{-3 (a+b x)} \left (48 e^{2 (a+b x)} \sqrt {e^{2 (a+b x)}+1} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 (a+b x)}\right )+\left (e^{2 (a+b x)}+1\right ) \left (2 (5 b x-12) e^{2 (a+b x)}+(5 b x-2) e^{4 (a+b x)}+5 b x+2\right )\right )}{50 \sqrt {2} b^2 \sqrt {e^{-a-b x}+e^{a+b x}}} \]
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 \[ \int x \cosh \left (b x + a\right )^{\frac {3}{2}} \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x \left (\cosh ^{\frac {3}{2}}\left (b x +a \right )\right ) \sinh \left (b x +a \right )\, dx \]
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 \[ \int x \cosh \left (b x + a\right )^{\frac {3}{2}} \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\mathrm {cosh}\left (a+b\,x\right )}^{3/2}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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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