Optimal. Leaf size=278 \[ -\frac{b x \left (3 c^2 d-2 e\right ) \sqrt{d+e x^2} \text{EllipticF}\left (\tan ^{-1}(c x),1-\frac{e}{c^2 d}\right )}{3 d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left (c^2 d-e\right ) \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac{2 x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{b c \sqrt{e} x \sqrt{-c^2 x^2-1} E\left (\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )|1-\frac{c^2 d}{e}\right )}{3 d^{3/2} \sqrt{-c^2 x^2} \left (c^2 d-e\right ) \sqrt{d+e x^2} \sqrt{\frac{d \left (c^2 x^2+1\right )}{d+e x^2}}} \]
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Rubi [A] time = 0.176579, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {192, 191, 6292, 12, 525, 418, 411} \[ \frac{2 x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{b x \left (3 c^2 d-2 e\right ) \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3 d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left (c^2 d-e\right ) \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac{b c \sqrt{e} x \sqrt{-c^2 x^2-1} E\left (\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )|1-\frac{c^2 d}{e}\right )}{3 d^{3/2} \sqrt{-c^2 x^2} \left (c^2 d-e\right ) \sqrt{d+e x^2} \sqrt{\frac{d \left (c^2 x^2+1\right )}{d+e x^2}}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rule 6292
Rule 12
Rule 525
Rule 418
Rule 411
Rubi steps
\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{3 d+2 e x^2}{3 d^2 \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{3 d+2 e x^2}{\sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{-c^2 x^2}}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{\left (b c \left (3 c^2 d-2 e\right ) x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d^2 \left (c^2 d-e\right ) \sqrt{-c^2 x^2}}-\frac{(b c e x) \int \frac{\sqrt{-1-c^2 x^2}}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d \left (c^2 d-e\right ) \sqrt{-c^2 x^2}}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{csch}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{b c \sqrt{e} x \sqrt{-1-c^2 x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )|1-\frac{c^2 d}{e}\right )}{3 d^{3/2} \left (c^2 d-e\right ) \sqrt{-c^2 x^2} \sqrt{\frac{d \left (1+c^2 x^2\right )}{d+e x^2}} \sqrt{d+e x^2}}-\frac{b \left (3 c^2 d-2 e\right ) x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3 d^3 \left (c^2 d-e\right ) \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 [C] time = 0.532478, size = 248, normalized size = 0.89 \[ \frac{x \left (a \left (c^2 d-e\right ) \left (3 d+2 e x^2\right )-b c e x \sqrt{\frac{1}{c^2 x^2}+1} \left (d+e x^2\right )+b \left (c^2 d-e\right ) \text{csch}^{-1}(c x) \left (3 d+2 e x^2\right )\right )}{3 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}-\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left (2 \left (c^2 d-e\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{c^2} x\right ),\frac{e}{c^2 d}\right )+c^2 d E\left (i \sinh ^{-1}\left (\sqrt{c^2} x\right )|\frac{e}{c^2 d}\right )\right )}{3 \sqrt{c^2} d^2 \sqrt{c^2 x^2+1} \left (c^2 d-e\right ) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.483, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\rm arccsch} \left (cx\right )) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{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*} \frac{1}{3} \, a{\left (\frac{2 \, x}{\sqrt{e x^{2} + d} d^{2}} + \frac{x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} d}\right )} + 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{5}{2}}}\,{d x} \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^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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