Optimal. Leaf size=501 \[ -\frac{b^2 f^2 (d e-c f) \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{b^2 f^2 (d e-c f) \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{2 b^2 (d e-c f)^3 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{2 b^2 (d e-c f)^3 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{b f^2 (c+d x)^2 \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{2 b f^2 (d e-c f) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{3 b f (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{4 b (d e-c f)^3 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{b f^3 (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}+\frac{b^2 f^2 x (d e-c f)}{d^3}+\frac{3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}-\frac{b^2 f^3 \log (c+d x)}{3 d^4} \]
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Rubi [A] time = 0.886769, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6322, 5469, 4190, 4182, 2279, 2391, 4184, 3475, 4185} \[ -\frac{b^2 f^2 (d e-c f) \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{b^2 f^2 (d e-c f) \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{2 b^2 (d e-c f)^3 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{2 b^2 (d e-c f)^3 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{b f^2 (c+d x)^2 \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{2 b f^2 (d e-c f) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{3 b f (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{4 b (d e-c f)^3 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{b f^3 (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}+\frac{b^2 f^2 x (d e-c f)}{d^3}+\frac{3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}-\frac{b^2 f^3 \log (c+d x)}{3 d^4} \]
Antiderivative was successfully verified.
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Rule 6322
Rule 5469
Rule 4190
Rule 4182
Rule 2279
Rule 2391
Rule 4184
Rule 3475
Rule 4185
Rubi steps
\begin{align*} \int (e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \text{csch}(x) (d e-c f+f \text{csch}(x))^3 \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}\\ &=\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{b \operatorname{Subst}\left (\int (a+b x) (d e-c f+f \text{csch}(x))^4 \, dx,x,\text{csch}^{-1}(c+d x)\right )}{2 d^4 f}\\ &=\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{b \operatorname{Subst}\left (\int \left (d^4 e^4 \left (1+\frac{c f \left (-4 d^3 e^3+6 c d^2 e^2 f-4 c^2 d e f^2+c^3 f^3\right )}{d^4 e^4}\right ) (a+b x)+4 d^3 e^3 f \left (1-\frac{c f \left (3 d^2 e^2-3 c d e f+c^2 f^2\right )}{d^3 e^3}\right ) (a+b x) \text{csch}(x)+6 d^2 e^2 f^2 \left (1+\frac{c f (-2 d e+c f)}{d^2 e^2}\right ) (a+b x) \text{csch}^2(x)+4 d e f^3 \left (1-\frac{c f}{d e}\right ) (a+b x) \text{csch}^3(x)+f^4 (a+b x) \text{csch}^4(x)\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{2 d^4 f}\\ &=-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{\left (b f^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}^4(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{2 d^4}-\frac{\left (2 b f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}^3(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{\left (3 b f (d e-c f)^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{\left (2 b (d e-c f)^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}\\ &=\frac{b^2 f^2 (d e-c f) x}{d^3}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}+\frac{3 b f (d e-c f)^2 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^2 (d e-c f) (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}+\frac{4 b (d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{\left (b f^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{\left (b f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{\left (3 b^2 f (d e-c f)^2\right ) \operatorname{Subst}\left (\int \coth (x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{\left (2 b^2 (d e-c f)^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{\left (2 b^2 (d e-c f)^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}\\ &=\frac{b^2 f^2 (d e-c f) x}{d^3}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}-\frac{b f^3 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{3 b f (d e-c f)^2 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^2 (d e-c f) (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{2 b f^2 (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{4 b (d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac{\left (b^2 f^3\right ) \operatorname{Subst}\left (\int \coth (x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^4}-\frac{\left (b^2 f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{\left (b^2 f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{\left (2 b^2 (d e-c f)^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{\left (2 b^2 (d e-c f)^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}\\ &=\frac{b^2 f^2 (d e-c f) x}{d^3}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}-\frac{b f^3 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{3 b f (d e-c f)^2 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^2 (d e-c f) (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{2 b f^2 (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{4 b (d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{b^2 f^3 \log (c+d x)}{3 d^4}+\frac{3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac{2 b^2 (d e-c f)^3 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{2 b^2 (d e-c f)^3 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{\left (b^2 f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{\left (b^2 f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}\\ &=\frac{b^2 f^2 (d e-c f) x}{d^3}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}-\frac{b f^3 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{3 b f (d e-c f)^2 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^2 (d e-c f) (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{2 b f^2 (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{4 b (d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{b^2 f^3 \log (c+d x)}{3 d^4}+\frac{3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}-\frac{b^2 f^2 (d e-c f) \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{2 b^2 (d e-c f)^3 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{b^2 f^2 (d e-c f) \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{2 b^2 (d e-c f)^3 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}\\ \end{align*}
Mathematica [C] time = 12.6609, size = 1429, normalized size = 2.85 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.425, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) ^{3} \left ( a+b{\rm arccsch} \left (dx+c\right ) \right ) ^{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*} \text{result too large to display} \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 (a^{2} f^{3} x^{3} + 3 \, a^{2} e f^{2} x^{2} + 3 \, a^{2} e^{2} f x + a^{2} e^{3} +{\left (b^{2} f^{3} x^{3} + 3 \, b^{2} e f^{2} x^{2} + 3 \, b^{2} e^{2} f x + b^{2} e^{3}\right )} \operatorname{arcsch}\left (d x + c\right )^{2} + 2 \,{\left (a b f^{3} x^{3} + 3 \, a b e f^{2} x^{2} + 3 \, a b e^{2} f x + a b e^{3}\right )} \operatorname{arcsch}\left (d x + c\right ), 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{\left (f x + e\right )}^{3}{\left (b \operatorname{arcsch}\left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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