Optimal. Leaf size=475 \[ \frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {Li}_2\left (-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {Li}_2\left (-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{f-\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{f}-\frac {b \text {Li}_2\left (e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )}{f}-\frac {\log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f}-\frac {2 b^2 \text {Li}_3\left (-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{f}-\frac {2 b^2 \text {Li}_3\left (-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{f}+\frac {b^2 \text {Li}_3\left (e^{2 \text {csch}^{-1}(c+d x)}\right )}{2 f} \]
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Rubi [A] time = 1.07, 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 = {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 2190
Rule 2282
Rule 2531
Rule 3716
Rule 5561
Rule 5569
Rule 5596
Rule 6322
Rule 6589
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*}
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Mathematica [C] time = 2.98, 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 {Li}_2\left (e^{-2 \text {csch}^{-1}(c+d x)}\right )+2 \text {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (-e^{-\text {csch}^{-1}(c+d x)}\right ) \text {csch}^{-1}(c+d x)-12 \text {Li}_2\left (e^{\text {csch}^{-1}(c+d x)}\right ) \text {csch}^{-1}(c+d x)+12 \text {Li}_2\left (\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 {Li}_2\left (\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 {Li}_3\left (-e^{-\text {csch}^{-1}(c+d x)}\right )+12 \text {Li}_3\left (e^{\text {csch}^{-1}(c+d x)}\right )-12 \text {Li}_3\left (\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 {Li}_3\left (\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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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arcsch}\left (d x + c\right )^{2} + 2 \, a b \operatorname {arcsch}\left (d x + c\right ) + a^{2}}{f x + e}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, 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 \[ \frac {a^{2} \log \left (f x + e\right )}{f} + \int \frac {b^{2} \log \left (\sqrt {\frac {1}{{\left (d x + c\right )}^{2}} + 1} + \frac {1}{d x + c}\right )^{2}}{f x + e} + \frac {2 \, a b \log \left (\sqrt {\frac {1}{{\left (d x + c\right )}^{2}} + 1} + \frac {1}{d x + c}\right )}{f x + e}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2}{e+f\,x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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