Optimal. Leaf size=351 \[ \frac{2 b^2 (d e-c f)^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{2 b^2 (d e-c f)^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{b^2 f^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{b^2 f^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}-\frac{(d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{2 b f (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{4 b (d e-c f)^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{b f^2 (c+d x)^2 \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{2 b f^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}+\frac{2 b^2 f (d e-c f) \log (c+d x)}{d^3}+\frac{b^2 f^2 x}{3 d^2} \]
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Rubi [A] time = 0.50625, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 17, 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} \[ \frac{2 b^2 (d e-c f)^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{2 b^2 (d e-c f)^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{b^2 f^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{b^2 f^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}-\frac{(d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{2 b f (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{4 b (d e-c f)^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{b f^2 (c+d x)^2 \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{2 b f^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}+\frac{2 b^2 f (d e-c f) \log (c+d x)}{d^3}+\frac{b^2 f^2 x}{3 d^2} \]
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)^2 \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))^2 \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}\\ &=\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) (d e-c f+f \text{csch}(x))^3 \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3 f}\\ &=\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}-\frac{(2 b) \operatorname{Subst}\left (\int \left (d^3 e^3 \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)+3 d^2 e^2 f \left (1+\frac{c f (-2 d e+c f)}{d^2 e^2}\right ) (a+b x) \text{csch}(x)+3 d e f^2 \left (1-\frac{c f}{d e}\right ) (a+b x) \text{csch}^2(x)+f^3 (a+b x) \text{csch}^3(x)\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3 f}\\ &=-\frac{(d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}-\frac{\left (2 b f^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}^3(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{(2 b f (d e-c f)) \operatorname{Subst}\left (\int (a+b x) \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}-\frac{\left (2 b (d e-c f)^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 b f (d e-c f) (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{b f^2 (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}+\frac{4 b (d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}+\frac{\left (b f^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{\left (2 b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \coth (x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{\left (2 b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}-\frac{\left (2 b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 b f (d e-c f) (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{b f^2 (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}-\frac{2 b f^2 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{4 b (d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}+\frac{2 b^2 f (d e-c f) \log (c+d x)}{d^3}-\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3}+\frac{\left (2 b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{\left (2 b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 b f (d e-c f) (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{b f^2 (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}-\frac{2 b f^2 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{4 b (d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}+\frac{2 b^2 f (d e-c f) \log (c+d x)}{d^3}+\frac{2 b^2 (d e-c f)^2 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{2 b^2 (d e-c f)^2 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 b f (d e-c f) (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{b f^2 (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}-\frac{2 b f^2 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{4 b (d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}+\frac{2 b^2 f (d e-c f) \log (c+d x)}{d^3}-\frac{b^2 f^2 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{2 b^2 (d e-c f)^2 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}+\frac{b^2 f^2 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}-\frac{2 b^2 (d e-c f)^2 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}\\ \end{align*}
Mathematica [C] time = 9.14219, size = 864, normalized size = 2.46 \[ \frac{1}{3} a^2 f^2 x^3+a^2 e f x^2+a^2 e^2 x-\frac{2 b^2 d e f \left (\frac{(c+d x)^2 \text{csch}^{-1}(c+d x)^2}{2 d^2}-\frac{c \coth \left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right ) \text{csch}^{-1}(c+d x)^2}{2 d^2}+\frac{c \tanh \left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right ) \text{csch}^{-1}(c+d x)^2}{2 d^2}+\frac{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \text{csch}^{-1}(c+d x)}{d^2}-\frac{\log \left (\frac{1}{c+d x}\right )}{d^2}-\frac{2 i c \left (i \text{csch}^{-1}(c+d x) \left (\log \left (1-e^{-\text{csch}^{-1}(c+d x)}\right )-\log \left (1+e^{-\text{csch}^{-1}(c+d x)}\right )\right )+i \left (\text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right )-\text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c+d x)}\right )\right )\right )}{d^2}\right ) x}{(c+d x) \left (\frac{c}{c+d x}-1\right )}+\frac{1}{3} a b \left (2 x \left (3 e^2+3 f x e+f^2 x^2\right ) \text{csch}^{-1}(c+d x)+\frac{-f (c+d x) \sqrt{\frac{c^2+2 d x c+d^2 x^2+1}{(c+d x)^2}} (5 c f-d (6 e+f x))+2 c \left (3 d^2 e^2-3 c d f e+c^2 f^2\right ) \sinh ^{-1}\left (\frac{1}{c+d x}\right )+\left (6 d^2 e^2-12 c d f e+\left (6 c^2-1\right ) f^2\right ) \log \left ((c+d x) \left (\sqrt{\frac{c^2+2 d x c+d^2 x^2+1}{(c+d x)^2}}+1\right )\right )}{d^3}\right )-\frac{b^2 e^2 \left (-\text{csch}^{-1}(c+d x) \left ((c+d x) \text{csch}^{-1}(c+d x)-2 \log \left (1-e^{-\text{csch}^{-1}(c+d x)}\right )+2 \log \left (1+e^{-\text{csch}^{-1}(c+d x)}\right )\right )+2 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right )-2 \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c+d x)}\right )\right )}{d}-\frac{b^2 f^2 \left (-\frac{\text{csch}^{-1}(c+d x)^2 \text{csch}^4\left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )}{2 (c+d x)}+2 \text{csch}^{-1}(c+d x) \left (3 c \text{csch}^{-1}(c+d x)-1\right ) \text{csch}^2\left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )-8 (c+d x)^3 \text{csch}^{-1}(c+d x)^2 \sinh ^4\left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )-2 \text{csch}^{-1}(c+d x) \left (3 c \text{csch}^{-1}(c+d x)+1\right ) \text{sech}^2\left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )+2 \left (-6 c^2 \text{csch}^{-1}(c+d x)^2+\text{csch}^{-1}(c+d x)^2+12 c \text{csch}^{-1}(c+d x)-2\right ) \coth \left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )-48 c \log \left (\frac{1}{c+d x}\right )+8 \left (6 c^2-1\right ) \left (\text{csch}^{-1}(c+d x) \left (\log \left (1-e^{-\text{csch}^{-1}(c+d x)}\right )-\log \left (1+e^{-\text{csch}^{-1}(c+d x)}\right )\right )+\text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right )-\text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c+d x)}\right )\right )+2 \left (6 c^2 \text{csch}^{-1}(c+d x)^2-\text{csch}^{-1}(c+d x)^2+12 c \text{csch}^{-1}(c+d x)+2\right ) \tanh \left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )\right )}{24 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.403, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) ^{2} \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^{2} x^{2} + 2 \, a^{2} e f x + a^{2} e^{2} +{\left (b^{2} f^{2} x^{2} + 2 \, b^{2} e f x + b^{2} e^{2}\right )} \operatorname{arcsch}\left (d x + c\right )^{2} + 2 \,{\left (a b f^{2} x^{2} + 2 \, a b e f x + a b e^{2}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acsch}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{2}\, 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{\left (f x + e\right )}^{2}{\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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