Optimal. Leaf size=122 \[ -\frac {2 b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^3}+\frac {b x^2 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 x}{3 c^2} \]
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Rubi [A] time = 0.13, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6286, 5452, 4185, 4182, 2279, 2391} \[ -\frac {b^2 \text {PolyLog}\left (2,-e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 \text {PolyLog}\left (2,e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}+\frac {b x^2 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{3 c}-\frac {2 b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^3}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b^2 x}{3 c^2} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4182
Rule 4185
Rule 5452
Rule 6286
Rubi steps
\begin {align*} \int x^2 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \coth (x) \text {csch}^3(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{c^3}\\ &=\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \text {csch}^3(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{3 c^3}\\ &=\frac {b^2 x}{3 c^2}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{3 c^3}\\ &=\frac {b^2 x}{3 c^2}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {2 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}-\frac {b^2 \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{3 c^3}+\frac {b^2 \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{3 c^3}\\ &=\frac {b^2 x}{3 c^2}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {2 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}\\ &=\frac {b^2 x}{3 c^2}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {2 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}-\frac {b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}\\ \end {align*}
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Mathematica [A] time = 1.43, size = 211, normalized size = 1.73 \[ \frac {a^2 c^3 x^3+2 a b c^3 x^3 \text {csch}^{-1}(c x)+a b c^2 x^2 \sqrt {\frac {1}{c^2 x^2}+1}-\frac {a b c x \sqrt {\frac {1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}+b^2 c^3 x^3 \text {csch}^{-1}(c x)^2+b^2 c^2 x^2 \sqrt {\frac {1}{c^2 x^2}+1} \text {csch}^{-1}(c x)+b^2 \text {Li}_2\left (-e^{-\text {csch}^{-1}(c x)}\right )-b^2 \text {Li}_2\left (e^{-\text {csch}^{-1}(c x)}\right )+b^2 c x+b^2 \text {csch}^{-1}(c x) \log \left (1-e^{-\text {csch}^{-1}(c x)}\right )-b^2 \text {csch}^{-1}(c x) \log \left (e^{-\text {csch}^{-1}(c x)}+1\right )}{3 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.37, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{2} \operatorname {arcsch}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname {arcsch}\left (c x\right ) + a^{2} x^{2}, x\right ) \]
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 \[ \int {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \]
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 \[ \int x^{2} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}\, dx \]
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 \[ \frac {1}{3} \, a^{2} x^{3} + \frac {1}{6} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} a b + \frac {1}{3} \, {\left (x^{3} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{2} - 3 \, \int -\frac {3 \, c^{2} x^{4} \log \relax (c)^{2} + 3 \, x^{2} \log \relax (c)^{2} + 3 \, {\left (c^{2} x^{4} + x^{2}\right )} \log \relax (x)^{2} + 6 \, {\left (c^{2} x^{4} \log \relax (c) + x^{2} \log \relax (c)\right )} \log \relax (x) - 2 \, {\left (3 \, c^{2} x^{4} \log \relax (c) + 3 \, x^{2} \log \relax (c) + 3 \, {\left (c^{2} x^{4} + x^{2}\right )} \log \relax (x) + {\left (c^{2} x^{4} {\left (3 \, \log \relax (c) + 1\right )} + 3 \, x^{2} \log \relax (c) + 3 \, {\left (c^{2} x^{4} + x^{2}\right )} \log \relax (x)\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + 3 \, {\left (c^{2} x^{4} \log \relax (c)^{2} + x^{2} \log \relax (c)^{2} + {\left (c^{2} x^{4} + x^{2}\right )} \log \relax (x)^{2} + 2 \, {\left (c^{2} x^{4} \log \relax (c) + x^{2} \log \relax (c)\right )} \log \relax (x)\right )} \sqrt {c^{2} x^{2} + 1}}{3 \, {\left (c^{2} x^{2} + {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 1\right )}}\,{d x}\right )} b^{2} \]
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 \[ \int x^2\,{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
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 \[ \int x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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