Optimal. Leaf size=163 \[ -\frac{a+b \text{csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac{b c e \sqrt{\frac{1}{c^2 x^2}+1}}{2 d \left (c^2 d^2+e^2\right ) \left (\frac{d}{x}+e\right )}+\frac{b \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac{c^2 d-\frac{e}{x}}{c \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d^2+e^2}}\right )}{2 d^2 \left (c^2 d^2+e^2\right )^{3/2}}+\frac{b \text{csch}^{-1}(c x)}{2 d^2 e} \]
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Rubi [A] time = 0.29184, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6290, 1568, 1475, 1651, 844, 215, 725, 206} \[ -\frac{a+b \text{csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac{b c e \sqrt{\frac{1}{c^2 x^2}+1}}{2 d \left (c^2 d^2+e^2\right ) \left (\frac{d}{x}+e\right )}+\frac{b \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac{c^2 d-\frac{e}{x}}{c \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d^2+e^2}}\right )}{2 d^2 \left (c^2 d^2+e^2\right )^{3/2}}+\frac{b \text{csch}^{-1}(c x)}{2 d^2 e} \]
Antiderivative was successfully verified.
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Rule 6290
Rule 1568
Rule 1475
Rule 1651
Rule 844
Rule 215
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{(d+e x)^3} \, dx &=-\frac{a+b \text{csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac{b \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^2 (d+e x)^2} \, dx}{2 c e}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac{b \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} \left (e+\frac{d}{x}\right )^2 x^4} \, dx}{2 c e}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac{b \operatorname{Subst}\left (\int \frac{x^2}{(e+d x)^2 \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{2 c e}\\ &=-\frac{b c e \sqrt{1+\frac{1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac{d}{x}\right )}-\frac{a+b \text{csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac{(b c) \operatorname{Subst}\left (\int \frac{e-\left (d+\frac{e^2}{c^2 d}\right ) x}{(e+d x) \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{2 e \left (c^2 d^2+e^2\right )}\\ &=-\frac{b c e \sqrt{1+\frac{1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac{d}{x}\right )}-\frac{a+b \text{csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{2 c d^2 e}-\frac{\left (b c \left (2+\frac{e^2}{c^2 d^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{(e+d x) \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{2 \left (c^2 d^2+e^2\right )}\\ &=-\frac{b c e \sqrt{1+\frac{1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac{d}{x}\right )}+\frac{b \text{csch}^{-1}(c x)}{2 d^2 e}-\frac{a+b \text{csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac{\left (b c \left (2+\frac{e^2}{c^2 d^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{d^2+\frac{e^2}{c^2}-x^2} \, dx,x,\frac{d-\frac{e}{c^2 x}}{\sqrt{1+\frac{1}{c^2 x^2}}}\right )}{2 \left (c^2 d^2+e^2\right )}\\ &=-\frac{b c e \sqrt{1+\frac{1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac{d}{x}\right )}+\frac{b \text{csch}^{-1}(c x)}{2 d^2 e}-\frac{a+b \text{csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac{b \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac{c^2 d-\frac{e}{x}}{c \sqrt{c^2 d^2+e^2} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{2 d^2 \left (c^2 d^2+e^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.46198, size = 204, normalized size = 1.25 \[ \frac{1}{2} \left (-\frac{a}{e (d+e x)^2}-\frac{b c e x \sqrt{\frac{1}{c^2 x^2}+1}}{d \left (c^2 d^2+e^2\right ) (d+e x)}-\frac{b \left (2 c^2 d^2+e^2\right ) \log \left (c x \left (\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d^2+e^2}-c d\right )+e\right )}{d^2 \left (c^2 d^2+e^2\right )^{3/2}}+\frac{b \left (2 c^2 d^2+e^2\right ) \log (d+e x)}{d^2 \left (c^2 d^2+e^2\right )^{3/2}}+\frac{b \sinh ^{-1}\left (\frac{1}{c x}\right )}{d^2 e}-\frac{b \text{csch}^{-1}(c x)}{e (d+e x)^2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.273, size = 963, normalized size = 5.9 \begin{align*} \text{result too large to display} \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}{4} \,{\left (\frac{2 i \, c^{3} d{\left (\log \left (i \, c x + 1\right ) - \log \left (-i \, c x + 1\right )\right )}}{c^{4} d^{4} + 2 \, c^{2} d^{2} e^{2} + e^{4}} + 4 \, c^{2} \int \frac{x}{2 \,{\left (c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} + 2 \, d e^{2} x + d^{2} e +{\left (c^{2} d^{2} e + e^{3}\right )} x^{2} +{\left (c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} + 2 \, d e^{2} x + d^{2} e +{\left (c^{2} d^{2} e + e^{3}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} + 1}\right )}}\,{d x} - \frac{2 \,{\left (3 \, c^{2} d^{2} e + e^{3}\right )} \log \left (e x + d\right )}{c^{4} d^{6} + 2 \, c^{2} d^{4} e^{2} + d^{2} e^{4}} - \frac{2 \, c^{4} d^{6} \log \left (c\right ) + 2 \, d^{2} e^{4} \log \left (c\right ) - 2 \, d^{2} e^{4} + 2 \,{\left (2 \, d^{4} e^{2} \log \left (c\right ) - d^{4} e^{2}\right )} c^{2} - 2 \,{\left (c^{2} d^{3} e^{3} + d e^{5}\right )} x +{\left (c^{4} d^{6} - c^{2} d^{4} e^{2} +{\left (c^{4} d^{4} e^{2} - c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (c^{4} d^{5} e - c^{2} d^{3} e^{3}\right )} x\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \,{\left ({\left (c^{4} d^{4} e^{2} + 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + 2 \,{\left (c^{4} d^{5} e + 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x\right )} \log \left (x\right ) - 2 \,{\left (c^{4} d^{6} + 2 \, c^{2} d^{4} e^{2} + d^{2} e^{4}\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )}{c^{4} d^{8} e + 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} +{\left (c^{4} d^{6} e^{3} + 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \,{\left (c^{4} d^{7} e^{2} + 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x}\right )} b - \frac{a}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 5.63123, size = 1524, normalized size = 9.35 \begin{align*} -\frac{a c^{4} d^{6} + b c^{3} d^{5} e + 2 \, a c^{2} d^{4} e^{2} + b c d^{3} e^{3} + a d^{2} e^{4} +{\left (b c^{3} d^{3} e^{3} + b c d e^{5}\right )} x^{2} -{\left (2 \, b c^{2} d^{4} e + b d^{2} e^{3} +{\left (2 \, b c^{2} d^{2} e^{3} + b e^{5}\right )} x^{2} + 2 \,{\left (2 \, b c^{2} d^{3} e^{2} + b d e^{4}\right )} x\right )} \sqrt{c^{2} d^{2} + e^{2}} \log \left (-\frac{c^{3} d^{2} x - c d e +{\left (c^{3} d^{2} + c e^{2}\right )} x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} +{\left (c^{2} d x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + c^{2} d x - e\right )} \sqrt{c^{2} d^{2} + e^{2}}}{e x + d}\right ) + 2 \,{\left (b c^{3} d^{4} e^{2} + b c d^{2} e^{4}\right )} x -{\left (b c^{4} d^{6} + 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} +{\left (b c^{4} d^{4} e^{2} + 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \,{\left (b c^{4} d^{5} e + 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) +{\left (b c^{4} d^{6} + 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} +{\left (b c^{4} d^{4} e^{2} + 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \,{\left (b c^{4} d^{5} e + 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) +{\left (b c^{4} d^{6} + 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left ({\left (b c^{3} d^{3} e^{3} + b c d e^{5}\right )} x^{2} +{\left (b c^{3} d^{4} e^{2} + b c d^{2} e^{4}\right )} x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \,{\left (c^{4} d^{8} e + 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} +{\left (c^{4} d^{6} e^{3} + 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \,{\left (c^{4} d^{7} e^{2} + 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x\right )}} \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{a + b \operatorname{acsch}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, 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{b \operatorname{arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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