Optimal. Leaf size=448 \[ -\frac{2 b^2 d \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{2 b^2 d \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}-\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)} \]
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Rubi [A] time = 1.11005, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6322, 5469, 4191, 3322, 2264, 2190, 2279, 2391} \[ -\frac{2 b^2 d \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{2 b^2 d \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}-\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)} \]
Antiderivative was successfully verified.
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Rule 6322
Rule 5469
Rule 4191
Rule 3322
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx &=-\left (d \operatorname{Subst}\left (\int \frac{(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 )\right )\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{a+b x}{d e-c f+f \text{csch}(x)} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac{(2 b d) \operatorname{Subst}\left (\int \left (\frac{a+b x}{d e-c f}+\frac{f (a+b x)}{(-d e+c f) \left (f+d e \left (1-\frac{c f}{d e}\right ) \sinh (x)\right )}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{(2 b d) \operatorname{Subst}\left (\int \frac{a+b x}{f+d e \left (1-\frac{c f}{d e}\right ) \sinh (x)} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d e-c f}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{(4 b d) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 e^x f-d e \left (1-\frac{c f}{d e}\right )+d e e^{2 x} \left (1-\frac{c f}{d e}\right )} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d e-c f}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{(4 b d) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 f+2 d e e^x \left (1-\frac{c f}{d e}\right )-2 \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 )}{\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{(4 b d) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 f+2 d e e^x \left (1-\frac{c f}{d e}\right )+2 \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 )}{\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{\left (2 b^2 d\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 d e e^x \left (1-\frac{c f}{d e}\right )}{2 f-2 \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{\left (2 b^2 d\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 d e e^x \left (1-\frac{c f}{d e}\right )}{2 f+2 \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{\left (2 b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 d e \left (1-\frac{c f}{d e}\right ) x}{2 f-2 \sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{\left (2 b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 d e \left (1-\frac{c f}{d e}\right ) x}{2 f+2 \sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{2 b^2 d \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{2 b^2 d \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ \end{align*}
Mathematica [C] time = 12.8638, size = 2061, normalized size = 4.6 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.543, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (dx+c\right ) \right ) ^{2}}{ \left ( fx+e \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*} -\frac{b^{2} \log \left (\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right )^{2}}{f^{2} x + e f} - \frac{a^{2}}{f^{2} x + e f} - \int -\frac{{\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} c d f x +{\left (c^{2} f + f\right )} b^{2}\right )} \log \left (d x + c\right )^{2} - 2 \,{\left (a b d^{2} f x^{2} + 2 \, a b c d f x +{\left (c^{2} f + f\right )} a b\right )} \log \left (d x + c\right ) + 2 \,{\left (a b d^{2} f x^{2} + 2 \, a b c d f x +{\left (c^{2} f + f\right )} a b -{\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} c d f x +{\left (c^{2} f + f\right )} b^{2}\right )} \log \left (d x + c\right ) +{\left (b^{2} c d e +{\left (c^{2} f + f\right )} a b +{\left (a b d^{2} f + b^{2} d^{2} f\right )} x^{2} +{\left (2 \, a b c d f +{\left (d^{2} e + c d f\right )} b^{2}\right )} x -{\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} c d f x +{\left (c^{2} f + f\right )} b^{2}\right )} \log \left (d x + c\right )\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right ) + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}{\left ({\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} c d f x +{\left (c^{2} f + f\right )} b^{2}\right )} \log \left (d x + c\right )^{2} - 2 \,{\left (a b d^{2} f x^{2} + 2 \, a b c d f x +{\left (c^{2} f + f\right )} a b\right )} \log \left (d x + c\right )\right )}}{d^{2} f^{3} x^{4} + c^{2} e^{2} f + 2 \,{\left (d^{2} e f^{2} + c d f^{3}\right )} x^{3} + e^{2} f +{\left (d^{2} e^{2} f + 4 \, c d e f^{2} + c^{2} f^{3} + f^{3}\right )} x^{2} + 2 \,{\left (c d e^{2} f + c^{2} e f^{2} + e f^{2}\right )} x +{\left (d^{2} f^{3} x^{4} + c^{2} e^{2} f + 2 \,{\left (d^{2} e f^{2} + c d f^{3}\right )} x^{3} + e^{2} f +{\left (d^{2} e^{2} f + 4 \, c d e f^{2} + c^{2} f^{3} + f^{3}\right )} x^{2} + 2 \,{\left (c d e^{2} f + c^{2} e f^{2} + e f^{2}\right )} x\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}\,{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^{2} x^{2} + 2 \, e f x + e^{2}}, 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}}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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