Optimal. Leaf size=475 \[ -\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac {b \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {b^2 \text {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c+d x)}\right )}{2 f}-\frac {2 b^2 \text {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac {2 b^2 \text {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f} \]
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Rubi [A]
time = 0.74, antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6457, 5715,
5688, 3797, 2221, 2611, 2320, 6724, 5680} \begin {gather*} \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 f e+\left (c^2+1\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 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 \text {Li}_2\left (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 \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 f e+\left (c^2+1\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 f e+\left (c^2+1\right ) f^2}}\right )}{f}+\frac {b^2 \text {Li}_3\left (e^{2 \text {csch}^{-1}(c+d x)}\right )}{2 f} \end {gather*}
Antiderivative was successfully verified.
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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*} \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx &=-\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 ) \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 {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 ) \text {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 ) \text {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 \text {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 ) \text {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 ) \text {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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.04, size = 1008, normalized size = 2.12 \begin {gather*} \frac {6 a^2 \log (e+f x)+6 a 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 \text {ArcSin}\left (\sqrt {\frac {d e+i f-c f}{2 d e-2 c f}}\right ) \text {ArcTan}\left (\frac {(i d e+f-i c f) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\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 (\pi +4 \text {ArcSin}\left (\sqrt {\frac {d e+i f-c f}{2 d e-2 c f}}\right )\right )\right ) \log \left (\frac {d e-c f-e^{\text {csch}^{-1}(c+d x)} 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 \text {ArcSin}\left (\sqrt {\frac {d e+i f-c f}{2 d e-2 c f}}\right )\right )\right ) \log \left (-\frac {-d e+c f+e^{\text {csch}^{-1}(c+d x)} 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 (i \pi +2 \text {csch}^{-1}(c+d x)\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 )+b^2 \left (-i \pi ^3-2 \text {csch}^{-1}(c+d x)^3-6 \text {csch}^{-1}(c+d x)^2 \log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )-6 \text {csch}^{-1}(c+d x)^2 \log \left (1-e^{\text {csch}^{-1}(c+d x)}\right )+6 \text {csch}^{-1}(c+d x)^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 )+6 \text {csch}^{-1}(c+d x)^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 )+12 \text {csch}^{-1}(c+d x) \text {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-12 \text {csch}^{-1}(c+d x) \text {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )+12 \text {csch}^{-1}(c+d x) \text {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 )+12 \text {csch}^{-1}(c+d x) \text {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 )+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)}{-f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )-12 \text {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 )\right )}{6 f} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccsch}\left (d x +c \right )\right )^{2}}{f x +e}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2}{e+f\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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