Optimal. Leaf size=98 \[ -\frac{4 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{9 b^2 \sqrt{\sinh (a+b x)}}-\frac{4 \sqrt{\sinh (a+b x)} \cosh (a+b x)}{9 b^2}+\frac{2 x \sinh ^{\frac{3}{2}}(a+b x)}{3 b} \]
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Rubi [A] time = 0.0530854, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5372, 2635, 2642, 2641} \[ -\frac{4 \sqrt{\sinh (a+b x)} \cosh (a+b x)}{9 b^2}-\frac{4 i \sqrt{i \sinh (a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{9 b^2 \sqrt{\sinh (a+b x)}}+\frac{2 x \sinh ^{\frac{3}{2}}(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 5372
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int x \cosh (a+b x) \sqrt{\sinh (a+b x)} \, dx &=\frac{2 x \sinh ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{2 \int \sinh ^{\frac{3}{2}}(a+b x) \, dx}{3 b}\\ &=-\frac{4 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{9 b^2}+\frac{2 x \sinh ^{\frac{3}{2}}(a+b x)}{3 b}+\frac{2 \int \frac{1}{\sqrt{\sinh (a+b x)}} \, dx}{9 b}\\ &=-\frac{4 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{9 b^2}+\frac{2 x \sinh ^{\frac{3}{2}}(a+b x)}{3 b}+\frac{\left (2 \sqrt{i \sinh (a+b x)}\right ) \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx}{9 b \sqrt{\sinh (a+b x)}}\\ &=-\frac{4 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)}}{9 b^2 \sqrt{\sinh (a+b x)}}-\frac{4 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{9 b^2}+\frac{2 x \sinh ^{\frac{3}{2}}(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.199701, size = 77, normalized size = 0.79 \[ \frac{2 \left (2 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )+3 b x \sinh ^2(a+b x)-\sinh (2 (a+b x))\right )}{9 b^2 \sqrt{\sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int x\cosh \left ( bx+a \right ) \sqrt{\sinh \left ( bx+a \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (b x + a\right ) \sqrt{\sinh \left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\sinh{\left (a + b x \right )}} \cosh{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (b x + a\right ) \sqrt{\sinh \left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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