Optimal. Leaf size=64 \[ \frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{9 b^2}-\frac {4 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{9 b^2} \]
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Rubi [A]
time = 0.03, 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 = {5481, 2715,
2720} \begin {gather*} \frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{9 b^2}-\frac {4 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{9 b^2}+\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 5481
Rubi steps
\begin {align*} \int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx &=\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}-\frac {2 \int \cosh ^{\frac {3}{2}}(a+b x) \, dx}{3 b}\\ &=\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}-\frac {4 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{9 b^2}-\frac {2 \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{9 b}\\ &=\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{9 b^2}-\frac {4 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{9 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 56, normalized size = 0.88 \begin {gather*} \frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )+2 \sqrt {\cosh (a+b x)} (3 b x \cosh (a+b x)-2 \sinh (a+b x))}{9 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.37, size = 0, normalized size = 0.00 \[\int x \sinh \left (b x +a \right ) \left (\sqrt {\cosh }\left (b x +a \right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int x \sinh {\left (a + b x \right )} \sqrt {\cosh {\left (a + b x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x\,\sqrt {\mathrm {cosh}\left (a+b\,x\right )}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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