Optimal. Leaf size=98 \[ -\frac{4 i \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{9 b^2}-\frac{4 \cosh (a+b x)}{9 b^2 \sqrt{\text{csch}(a+b x)}}+\frac{2 x}{3 b \text{csch}^{\frac{3}{2}}(a+b x)} \]
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Rubi [A] time = 0.0543573, 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 = {5445, 3769, 3771, 2641} \[ -\frac{4 \cosh (a+b x)}{9 b^2 \sqrt{\text{csch}(a+b x)}}-\frac{4 i \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{9 b^2}+\frac{2 x}{3 b \text{csch}^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 5445
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{x \cosh (a+b x)}{\sqrt{\text{csch}(a+b x)}} \, dx &=\frac{2 x}{3 b \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{2 \int \frac{1}{\text{csch}^{\frac{3}{2}}(a+b x)} \, dx}{3 b}\\ &=\frac{2 x}{3 b \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{9 b^2 \sqrt{\text{csch}(a+b x)}}+\frac{2 \int \sqrt{\text{csch}(a+b x)} \, dx}{9 b}\\ &=\frac{2 x}{3 b \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{9 b^2 \sqrt{\text{csch}(a+b x)}}+\frac{\left (2 \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}\right ) \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx}{9 b}\\ &=\frac{2 x}{3 b \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{9 b^2 \sqrt{\text{csch}(a+b x)}}-\frac{4 i \sqrt{\text{csch}(a+b x)} 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}\\ \end{align*}
Mathematica [A] time = 0.413625, size = 67, normalized size = 0.68 \[ \frac{-\frac{4 i \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )}{(i \sinh (a+b x))^{3/2}}-4 \coth (a+b x)+6 b x}{9 b^2 \text{csch}^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{x\cosh \left ( bx+a \right ){\frac{1}{\sqrt{{\rm csch} \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 \frac{x \cosh \left (b x + a\right )}{\sqrt{\operatorname{csch}\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 \frac{x \cosh{\left (a + b x \right )}}{\sqrt{\operatorname{csch}{\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 )}{\sqrt{\operatorname{csch}\left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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