Optimal. Leaf size=71 \[ -\frac{2 x}{b \sqrt{\sinh (a+b x)}}-\frac{4 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{b^2 \sqrt{\sinh (a+b x)}} \]
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Rubi [A] time = 0.0395053, antiderivative size = 71, normalized size of antiderivative = 1., 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 = {5372, 2642, 2641} \[ -\frac{2 x}{b \sqrt{\sinh (a+b x)}}-\frac{4 i \sqrt{i \sinh (a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b^2 \sqrt{\sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 5372
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{x \cosh (a+b x)}{\sinh ^{\frac{3}{2}}(a+b x)} \, dx &=-\frac{2 x}{b \sqrt{\sinh (a+b x)}}+\frac{2 \int \frac{1}{\sqrt{\sinh (a+b x)}} \, dx}{b}\\ &=-\frac{2 x}{b \sqrt{\sinh (a+b x)}}+\frac{\left (2 \sqrt{i \sinh (a+b x)}\right ) \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx}{b \sqrt{\sinh (a+b x)}}\\ &=-\frac{2 x}{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)}}{b^2 \sqrt{\sinh (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.164629, size = 56, normalized size = 0.79 \[ \frac{-2 b x+4 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )}{b^2 \sqrt{\sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}}\, 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 \frac{x \cosh \left (b x + a\right )}{\sinh \left (b x + a\right )^{\frac{3}{2}}}\,{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 \frac{x \cosh{\left (a + b x \right )}}{\sinh ^{\frac{3}{2}}{\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 \frac{x \cosh \left (b x + a\right )}{\sinh \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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