Optimal. Leaf size=120 \[ \frac {6 b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c}-\frac {6 b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c}+x \left (a+b \text {csch}^{-1}(c x)\right )^3+\frac {6 b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2}{c}-\frac {6 b^3 \text {Li}_3\left (-e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {6 b^3 \text {Li}_3\left (e^{\text {csch}^{-1}(c x)}\right )}{c} \]
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Rubi [A] time = 0.12, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6280, 5452, 4182, 2531, 2282, 6589} \[ \frac {6 b^2 \text {PolyLog}\left (2,-e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c}-\frac {6 b^2 \text {PolyLog}\left (2,e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c}-\frac {6 b^3 \text {PolyLog}\left (3,-e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {6 b^3 \text {PolyLog}\left (3,e^{\text {csch}^{-1}(c x)}\right )}{c}+x \left (a+b \text {csch}^{-1}(c x)\right )^3+\frac {6 b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2}{c} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4182
Rule 5452
Rule 6280
Rule 6589
Rubi steps
\begin {align*} \int \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^3 \coth (x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {(3 b) \operatorname {Subst}\left (\int (a+b x)^2 \text {csch}(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^3+\frac {6 b \left (a+b \text {csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int (a+b x) \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 )^3+\frac {6 b \left (a+b \text {csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {6 b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {6 b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {\left (6 b^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}+\frac {\left (6 b^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^3+\frac {6 b \left (a+b \text {csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {6 b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {6 b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {\left (6 b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {\left (6 b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^3+\frac {6 b \left (a+b \text {csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {6 b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {6 b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {6 b^3 \text {Li}_3\left (-e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {6 b^3 \text {Li}_3\left (e^{\text {csch}^{-1}(c x)}\right )}{c}\\ \end {align*}
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Mathematica [B] time = 0.34, size = 246, normalized size = 2.05 \[ a^3 x+\frac {3 a^2 b \log \left (c x \left (\sqrt {\frac {c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{c}+3 a^2 b x \text {csch}^{-1}(c x)+\frac {3 a b^2 \left (-2 \text {Li}_2\left (-e^{-\text {csch}^{-1}(c x)}\right )+2 \text {Li}_2\left (e^{-\text {csch}^{-1}(c x)}\right )+\text {csch}^{-1}(c x) \left (c x \text {csch}^{-1}(c x)-2 \log \left (1-e^{-\text {csch}^{-1}(c x)}\right )+2 \log \left (e^{-\text {csch}^{-1}(c x)}+1\right )\right )\right )}{c}+\frac {b^3 \left (-6 \text {csch}^{-1}(c x) \text {Li}_2\left (-e^{-\text {csch}^{-1}(c x)}\right )+6 \text {csch}^{-1}(c x) \text {Li}_2\left (e^{-\text {csch}^{-1}(c x)}\right )-6 \text {Li}_3\left (-e^{-\text {csch}^{-1}(c x)}\right )+6 \text {Li}_3\left (e^{-\text {csch}^{-1}(c x)}\right )+c x \text {csch}^{-1}(c x)^3-3 \text {csch}^{-1}(c x)^2 \log \left (1-e^{-\text {csch}^{-1}(c x)}\right )+3 \text {csch}^{-1}(c x)^2 \log \left (e^{-\text {csch}^{-1}(c x)}+1\right )\right )}{c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} \operatorname {arcsch}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {arcsch}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {arcsch}\left (c x\right ) + a^{3}, x\right ) \]
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 \[ \int {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3}\,{d x} \]
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 \[ \int \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{3}\, dx \]
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^{3} x \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{3} + a^{3} x + \frac {3 \, {\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^{2} b}{2 \, c} - \int \frac {b^{3} \log \relax (c)^{3} - 3 \, a b^{2} \log \relax (c)^{2} + {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x)^{3} + {\left (b^{3} c^{2} \log \relax (c)^{3} - 3 \, a b^{2} c^{2} \log \relax (c)^{2}\right )} x^{2} + 3 \, {\left (b^{3} \log \relax (c) - a b^{2} + {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)^{2} + 3 \, {\left (b^{3} \log \relax (c) - a b^{2} + {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2} + {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x) + \sqrt {c^{2} x^{2} + 1} {\left (b^{3} \log \relax (c) - a b^{2} + {\left (b^{3} c^{2} {\left (\log \relax (c) + 1\right )} - a b^{2} c^{2}\right )} x^{2} + {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x)\right )}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{2} + 3 \, {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2}\right )} \log \relax (x) - 3 \, {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2} + {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x)^{2} + 2 \, {\left (b^{3} \log \relax (c) - a b^{2} + {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x) + {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2} + {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x)^{2} + 2 \, {\left (b^{3} \log \relax (c) - a b^{2} + {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + {\left (b^{3} \log \relax (c)^{3} - 3 \, a b^{2} \log \relax (c)^{2} + {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x)^{3} + {\left (b^{3} c^{2} \log \relax (c)^{3} - 3 \, a b^{2} c^{2} \log \relax (c)^{2}\right )} x^{2} + 3 \, {\left (b^{3} \log \relax (c) - a b^{2} + {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)^{2} + 3 \, {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2}\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} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]
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 \[ \int \left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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