Optimal. Leaf size=111 \[ \frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{b x \sqrt{d+e x^2} \text{EllipticF}\left (\tan ^{-1}(c x),1-\frac{e}{c^2 d}\right )}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}} \]
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Rubi [A] time = 0.075112, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {191, 6292, 12, 418} \[ \frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{b x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 6292
Rule 12
Rule 418
Rubi steps
\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{1}{d \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{1}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{d \sqrt{-c^2 x^2}}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{b x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d^2 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}\\ \end{align*}
Mathematica [A] time = 0.192663, size = 113, normalized size = 1.02 \[ \frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-c^2} x\right ),\frac{e}{c^2 d}\right )}{\sqrt{-c^2} d \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}+\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{d \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.451, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\rm arccsch} \left (cx\right )) \left ( e{x}^{2}+d \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*} b \int \frac{\log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + \frac{1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} + \frac{a x}{\sqrt{e x^{2} + d} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{b \operatorname{arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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