Integrand size = 20, antiderivative size = 448 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\frac {d \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 b d \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {2 b d \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {2 b^2 d \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {2 b^2 d \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}} \]
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Time = 0.77 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6457, 5577, 4276, 3403, 2296, 2221, 2317, 2438} \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=-\frac {2 b d \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{f-\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{(d e-c f) \sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac {2 b d \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{(d e-c f) \sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac {d \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 b^2 d \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{(d e-c f) \sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac {2 b^2 d \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{(d e-c f) \sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3403
Rule 4276
Rule 5577
Rule 6457
Rubi steps \begin{align*} \text {integral}& = -\left (d \text {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) \text {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) \text {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) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 \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 )}{(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 \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 )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 12.80 (sec) , antiderivative size = 2061, normalized size of antiderivative = 4.60 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}}{\left (f x +e \right )^{2}}d x\]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2}}{\left (e + f x\right )^{2}}\, dx \]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2}{{\left (e+f\,x\right )}^2} \,d x \]
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