Optimal. Leaf size=121 \[ -\frac {4 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{49 b^2}+\frac {20 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{147 b^2}+\frac {20 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{147 b^2 \sqrt {\sinh (a+b x)}}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b} \]
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Rubi [A] time = 0.07, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5372, 2635, 2642, 2641} \[ -\frac {4 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{49 b^2}+\frac {20 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{147 b^2}+\frac {20 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{147 b^2 \sqrt {\sinh (a+b x)}}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 5372
Rubi steps
\begin {align*} \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx &=\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \int \sinh ^{\frac {7}{2}}(a+b x) \, dx}{7 b}\\ &=-\frac {4 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}+\frac {10 \int \sinh ^{\frac {3}{2}}(a+b x) \, dx}{49 b}\\ &=\frac {20 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{147 b^2}-\frac {4 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {10 \int \frac {1}{\sqrt {\sinh (a+b x)}} \, dx}{147 b}\\ &=\frac {20 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{147 b^2}-\frac {4 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {\left (10 \sqrt {i \sinh (a+b x)}\right ) \int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx}{147 b \sqrt {\sinh (a+b x)}}\\ &=\frac {20 i F\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {i \sinh (a+b x)}}{147 b^2 \sqrt {\sinh (a+b x)}}+\frac {20 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{147 b^2}-\frac {4 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 103, normalized size = 0.85 \[ \frac {52 \sinh (2 (a+b x))-6 \sinh (4 (a+b x))-84 b x \cosh (2 (a+b x))+21 b x \cosh (4 (a+b x))-80 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{4} (-2 i a-2 i b x+\pi )\right |2\right )+63 b x}{588 b^2 \sqrt {\sinh (a+b x)}} \]
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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int x \cosh \left (b x +a \right ) \left (\sinh ^{\frac {5}{2}}\left (b x +a \right )\right )\, dx \]
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 \[ \int x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^{5/2} \,d x \]
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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