Optimal. Leaf size=87 \[ \frac {20 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{147 b^2}-\frac {4 \sinh (a+b x) \cosh ^{\frac {5}{2}}(a+b x)}{49 b^2}-\frac {20 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{147 b^2}+\frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b} \]
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Rubi [A] time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5373, 2635, 2641} \[ \frac {20 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{147 b^2}-\frac {4 \sinh (a+b x) \cosh ^{\frac {5}{2}}(a+b x)}{49 b^2}-\frac {20 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{147 b^2}+\frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 5373
Rubi steps
\begin {align*} \int x \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx &=\frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \int \cosh ^{\frac {7}{2}}(a+b x) \, dx}{7 b}\\ &=\frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {4 \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x)}{49 b^2}-\frac {10 \int \cosh ^{\frac {3}{2}}(a+b x) \, dx}{49 b}\\ &=\frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {20 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{147 b^2}-\frac {4 \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x)}{49 b^2}-\frac {10 \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{147 b}\\ &=\frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b}+\frac {20 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{147 b^2}-\frac {20 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{147 b^2}-\frac {4 \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x)}{49 b^2}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 77, normalized size = 0.89 \[ \frac {\sqrt {\cosh (a+b x)} (-46 \sinh (a+b x)-6 \sinh (3 (a+b x))+63 b x \cosh (a+b x)+21 b x \cosh (3 (a+b x)))+40 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{294 b^2} \]
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 \[ \int x \cosh \left (b x + a\right )^{\frac {5}{2}} \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int x \left (\cosh ^{\frac {5}{2}}\left (b x +a \right )\right ) \sinh \left (b x +a \right )\, dx \]
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 \[ \int x \cosh \left (b x + a\right )^{\frac {5}{2}} \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {cosh}\left (a+b\,x\right )}^{5/2}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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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