Optimal. Leaf size=68 \[ x \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {4 b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c}+\frac {2 b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {2 b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right )}{c} \]
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Rubi [A] time = 0.07, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6280, 5452, 4182, 2279, 2391} \[ \frac {2 b^2 \text {PolyLog}\left (2,-e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {2 b^2 \text {PolyLog}\left (2,e^{\text {csch}^{-1}(c x)}\right )}{c}+x \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {4 b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4182
Rule 5452
Rule 6280
Rubi steps
\begin {align*} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \coth (x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {4 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {4 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {4 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {2 b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {2 b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 121, normalized size = 1.78 \[ \frac {a^2 c x+2 a b c x \text {csch}^{-1}(c x)-2 a b \log \left (\tanh \left (\frac {1}{2} \text {csch}^{-1}(c x)\right )\right )-2 b^2 \text {Li}_2\left (-e^{-\text {csch}^{-1}(c x)}\right )+2 b^2 \text {Li}_2\left (e^{-\text {csch}^{-1}(c x)}\right )+b^2 c x \text {csch}^{-1}(c x)^2-2 b^2 \text {csch}^{-1}(c x) \log \left (1-e^{-\text {csch}^{-1}(c x)}\right )+2 b^2 \text {csch}^{-1}(c x) \log \left (e^{-\text {csch}^{-1}(c x)}+1\right )}{c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.26, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \operatorname {arcsch}\left (c x\right )^{2} + 2 \, a b \operatorname {arcsch}\left (c x\right ) + a^{2}, x\right ) \]
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 \[ \int {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}\,{d x} \]
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 \[ \int \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}\, dx \]
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 \[ {\left (x \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{2} - \int -\frac {c^{2} x^{2} \log \relax (c)^{2} + {\left (c^{2} x^{2} + 1\right )} \log \relax (x)^{2} + \log \relax (c)^{2} + 2 \, {\left (c^{2} x^{2} \log \relax (c) + \log \relax (c)\right )} \log \relax (x) - 2 \, {\left (c^{2} x^{2} \log \relax (c) + {\left (c^{2} x^{2} + 1\right )} \log \relax (x) + {\left (c^{2} x^{2} {\left (\log \relax (c) + 1\right )} + {\left (c^{2} x^{2} + 1\right )} \log \relax (x) + \log \relax (c)\right )} \sqrt {c^{2} x^{2} + 1} + \log \relax (c)\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + {\left (c^{2} x^{2} \log \relax (c)^{2} + {\left (c^{2} x^{2} + 1\right )} \log \relax (x)^{2} + \log \relax (c)^{2} + 2 \, {\left (c^{2} x^{2} \log \relax (c) + \log \relax (c)\right )} \log \relax (x)\right )} \sqrt {c^{2} x^{2} + 1}}{c^{2} x^{2} + {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 1}\,{d x}\right )} b^{2} + a^{2} x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} a b}{c} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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