Optimal. Leaf size=81 \[ -b \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 b}-\log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {1}{2} b^2 \text {Li}_3\left (e^{2 \text {csch}^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6286, 3716, 2190, 2531, 2282, 6589} \[ -b \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{2} b^2 \text {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right )+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 b}-\log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 6286
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x} \, dx &=-\operatorname {Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 b}+2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)^2}{1-e^{2 x}} \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text {csch}^{-1}(c x)\right )^2 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+(2 b) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text {csch}^{-1}(c x)\right )^2 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-b \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+b^2 \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text {csch}^{-1}(c x)\right )^2 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-b \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \text {csch}^{-1}(c x)}\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text {csch}^{-1}(c x)\right )^2 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-b \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} b^2 \text {Li}_3\left (e^{2 \text {csch}^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 115, normalized size = 1.42 \[ a^2 \log (c x)+a b \left (\text {Li}_2\left (e^{-2 \text {csch}^{-1}(c x)}\right )-\text {csch}^{-1}(c x) \left (\text {csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )\right )+b^2 \left (-\text {csch}^{-1}(c x) \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} \text {Li}_3\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{3} \text {csch}^{-1}(c x)^3-\text {csch}^{-1}(c x)^2 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arcsch}\left (c x\right )^{2} + 2 \, a b \operatorname {arcsch}\left (c x\right ) + a^{2}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}}{x}\, dx \]
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 \[ b^{2} \log \relax (x) \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{2} + a^{2} \log \relax (x) - \int -\frac {b^{2} \log \relax (c)^{2} + {\left (b^{2} c^{2} \log \relax (c)^{2} - 2 \, a b c^{2} \log \relax (c)\right )} x^{2} - 2 \, a b \log \relax (c) + {\left (b^{2} c^{2} x^{2} + b^{2}\right )} \log \relax (x)^{2} + 2 \, {\left ({\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} + b^{2} \log \relax (c) - a b\right )} \log \relax (x) - 2 \, {\left ({\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} + b^{2} \log \relax (c) - a b + {\left (b^{2} c^{2} x^{2} + b^{2}\right )} \log \relax (x) + \sqrt {c^{2} x^{2} + 1} {\left ({\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} + b^{2} \log \relax (c) - a b + {\left (2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \relax (x)\right )}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + \sqrt {c^{2} x^{2} + 1} {\left (b^{2} \log \relax (c)^{2} + {\left (b^{2} c^{2} \log \relax (c)^{2} - 2 \, a b c^{2} \log \relax (c)\right )} x^{2} - 2 \, a b \log \relax (c) + {\left (b^{2} c^{2} x^{2} + b^{2}\right )} \log \relax (x)^{2} + 2 \, {\left ({\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} + b^{2} \log \relax (c) - a b\right )} \log \relax (x)\right )}}{c^{2} x^{3} + {\left (c^{2} x^{3} + x\right )} \sqrt {c^{2} x^{2} + 1} + x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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