Optimal. Leaf size=475 \[ \frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{f}-\frac{b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{f}-\frac{2 b^2 \text{PolyLog}\left (3,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{f}-\frac{2 b^2 \text{PolyLog}\left (3,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{f}+\frac{b^2 \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c+d x)}\right )}{2 f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{f}-\frac{\log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f} \]
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Rubi [A] time = 1.07282, antiderivative size = 475, 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, 5596, 5569, 3716, 2190, 2531, 2282, 6589, 5561} \[ \frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{f}-\frac{b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{f}-\frac{2 b^2 \text{PolyLog}\left (3,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{f}-\frac{2 b^2 \text{PolyLog}\left (3,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{f}+\frac{b^2 \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c+d x)}\right )}{2 f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{f}-\frac{\log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f} \]
Antiderivative was successfully verified.
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Rule 6322
Rule 5596
Rule 5569
Rule 3716
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 5561
Rubi steps
\begin{align*} \int \frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^2 \coth (x) \text{csch}(x)}{d e-c f+f \text{csch}(x)} \, dx,x,\text{csch}^{-1}(c+d x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{(a+b x)^2 \coth (x)}{f+(d e-c f) \sinh (x)} \, dx,x,\text{csch}^{-1}(c+d x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}+\frac{(d e-c f) \operatorname{Subst}\left (\int \frac{(a+b x)^2 \cosh (x)}{f+(d e-c f) \sinh (x)} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^2}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}+\frac{(d e-c f) \operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{f+e^x (d e-c f)-\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}+\frac{(d e-c f) \operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{f+e^x (d e-c f)+\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+\frac{e^x (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+\frac{e^x (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac{b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{e^x (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{e^x (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac{b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \text{csch}^{-1}(c+d x)}\right )}{2 f}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{(d e-c f) x}{-f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{f}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{(d e-c f) x}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac{b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{b^2 \text{Li}_3\left (e^{2 \text{csch}^{-1}(c+d x)}\right )}{2 f}-\frac{2 b^2 \text{Li}_3\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac{2 b^2 \text{Li}_3\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}\\ \end{align*}
Mathematica [C] time = 2.78343, size = 1008, normalized size = 2.12 \[ \frac{6 \log (e+f x) a^2+6 b \left (\frac{1}{4} \left (\pi -2 i \text{csch}^{-1}(c+d x)\right )^2-\text{csch}^{-1}(c+d x)^2-8 \sin ^{-1}\left (\sqrt{\frac{d e-c f+i f}{2 d e-2 c f}}\right ) \tan ^{-1}\left (\frac{(i d e-i c f+f) \cot \left (\frac{1}{4} \left (2 i \text{csch}^{-1}(c+d x)+\pi \right )\right )}{\sqrt{f^2+(d e-c f)^2}}\right )-2 \text{csch}^{-1}(c+d x) \log \left (1-e^{-2 \text{csch}^{-1}(c+d x)}\right )+\left (2 \text{csch}^{-1}(c+d x)+i \left (4 \sin ^{-1}\left (\sqrt{\frac{d e-c f+i f}{2 d e-2 c f}}\right )+\pi \right )\right ) \log \left (\frac{d e-e^{\text{csch}^{-1}(c+d x)} f-c f+e^{\text{csch}^{-1}(c+d x)} \sqrt{f^2+(d e-c f)^2}}{d e-c f}\right )+\left (2 \text{csch}^{-1}(c+d x)+i \left (\pi -4 \sin ^{-1}\left (\sqrt{\frac{d e-c f+i f}{2 d e-2 c f}}\right )\right )\right ) \log \left (-\frac{-d e+e^{\text{csch}^{-1}(c+d x)} f+c f+e^{\text{csch}^{-1}(c+d x)} \sqrt{f^2+(d e-c f)^2}}{d e-c f}\right )+2 \text{csch}^{-1}(c+d x) \log \left (\frac{d (e+f x)}{c+d x}\right )-\left (2 \text{csch}^{-1}(c+d x)+i \pi \right ) \log \left (\frac{d (e+f x)}{c+d x}\right )+\text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c+d x)}\right )+2 \text{PolyLog}\left (2,\frac{e^{\text{csch}^{-1}(c+d x)} \left (f-\sqrt{f^2+(d e-c f)^2}\right )}{d e-c f}\right )+2 \text{PolyLog}\left (2,\frac{e^{\text{csch}^{-1}(c+d x)} \left (f+\sqrt{f^2+(d e-c f)^2}\right )}{d e-c f}\right )\right ) a+b^2 \left (-2 \text{csch}^{-1}(c+d x)^3-6 \log \left (1+e^{-\text{csch}^{-1}(c+d x)}\right ) \text{csch}^{-1}(c+d x)^2-6 \log \left (1-e^{\text{csch}^{-1}(c+d x)}\right ) \text{csch}^{-1}(c+d x)^2+6 \log \left (\frac{e^{\text{csch}^{-1}(c+d x)} (c f-d e)}{\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-f}+1\right ) \text{csch}^{-1}(c+d x)^2+6 \log \left (\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+1\right ) \text{csch}^{-1}(c+d x)^2+12 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right ) \text{csch}^{-1}(c+d x)-12 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right ) \text{csch}^{-1}(c+d x)+12 \text{PolyLog}\left (2,\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-f}\right ) \text{csch}^{-1}(c+d x)+12 \text{PolyLog}\left (2,\frac{e^{\text{csch}^{-1}(c+d x)} (c f-d e)}{f+\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right ) \text{csch}^{-1}(c+d x)+12 \text{PolyLog}\left (3,-e^{-\text{csch}^{-1}(c+d x)}\right )+12 \text{PolyLog}\left (3,e^{\text{csch}^{-1}(c+d x)}\right )-12 \text{PolyLog}\left (3,\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-f}\right )-12 \text{PolyLog}\left (3,\frac{e^{\text{csch}^{-1}(c+d x)} (c f-d e)}{f+\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )-i \pi ^3\right )}{6 f} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.611, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (dx+c\right ) \right ) ^{2}}{fx+e}}\, 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{a^{2} \log \left (f x + e\right )}{f} + \int \frac{b^{2} \log \left (\sqrt{\frac{1}{{\left (d x + c\right )}^{2}} + 1} + \frac{1}{d x + c}\right )^{2}}{f x + e} + \frac{2 \, a b \log \left (\sqrt{\frac{1}{{\left (d x + c\right )}^{2}} + 1} + \frac{1}{d x + c}\right )}{f x + e}\,{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{b^{2} \operatorname{arcsch}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcsch}\left (d x + c\right ) + a^{2}}{f x + e}, 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{{\left (b \operatorname{arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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