Optimal. Leaf size=64 \[ \frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{9 b^2}-\frac {4 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{9 b^2}+\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b} \]
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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, 2641} \[ \frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{9 b^2}-\frac {4 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{9 b^2}+\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 5373
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 = 0.16, size = 56, normalized size = 0.88 \[ \frac {2 \sqrt {\cosh (a+b x)} (3 b x \cosh (a+b x)-2 \sinh (a+b x))+4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{9 b^2} \]
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 \[ \int x \sqrt {\cosh \left (b x + a\right )} \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, 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 \[ \int x \sqrt {\cosh \left (b x + a\right )} \sinh \left (b x + a\right )\,{d x} \]
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 \[ \int x\,\sqrt {\mathrm {cosh}\left (a+b\,x\right )}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sinh {\left (a + b x \right )} \sqrt {\cosh {\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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