Integrand size = 20, antiderivative size = 475 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac {b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c+d x)}\right )}{2 f}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f} \]
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Time = 0.76 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6457, 5715, 5688, 3797, 2221, 2611, 2320, 6724, 5680} \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{f-\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{f}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )}{f}-\frac {\log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{f}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{f}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c+d x)}\right )}{2 f} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5680
Rule 5688
Rule 5715
Rule 6457
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\text {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 ) \\ & = -\text {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 {\text {Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{f}+\frac {(d e-c f) \text {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 \text {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) \text {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) \text {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) \text {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) \text {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) \text {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 ) \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\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 ) \operatorname {PolyLog}\left (2,-\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 {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 x}\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{f}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\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 ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\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 ) \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\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 ) \operatorname {PolyLog}\left (2,-\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 {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \text {csch}^{-1}(c+d x)}\right )}{2 f}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\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 ) \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\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 ) \operatorname {PolyLog}\left (2,-\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 {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c+d x)}\right )}{2 f}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\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 \operatorname {PolyLog}\left (3,-\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*}
\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx \]
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\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}}{f x +e}d x\]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2}{e+f\,x} \,d x \]
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