Optimal. Leaf size=110 \[ \frac {3}{2} b^2 \text {Li}_3\left (e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )-\frac {3}{2} b \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}-\log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {3}{4} b^3 \text {Li}_4\left (e^{2 \text {csch}^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6286, 3716, 2190, 2531, 6609, 2282, 6589} \[ \frac {3}{2} b^2 \text {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )-\frac {3}{2} b \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {3}{4} b^3 \text {PolyLog}\left (4,e^{2 \text {csch}^{-1}(c x)}\right )+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}-\log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^3 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 6286
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x} \, dx &=-\operatorname {Subst}\left (\int (a+b x)^3 \coth (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}+2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)^3}{1-e^{2 x}} \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}-\left (a+b \text {csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+(3 b) \operatorname {Subst}\left (\int (a+b x)^2 \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 )^4}{4 b}-\left (a+b \text {csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-\frac {3}{2} b \left (a+b \text {csch}^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\left (3 b^2\right ) \operatorname {Subst}\left (\int (a+b x) \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 )^4}{4 b}-\left (a+b \text {csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-\frac {3}{2} b \left (a+b \text {csch}^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_3\left (e^{2 \text {csch}^{-1}(c x)}\right )-\frac {1}{2} \left (3 b^3\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (e^{2 x}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}-\left (a+b \text {csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-\frac {3}{2} b \left (a+b \text {csch}^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_3\left (e^{2 \text {csch}^{-1}(c x)}\right )-\frac {1}{4} \left (3 b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 \text {csch}^{-1}(c x)}\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}-\left (a+b \text {csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-\frac {3}{2} b \left (a+b \text {csch}^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \text {Li}_3\left (e^{2 \text {csch}^{-1}(c x)}\right )-\frac {3}{4} b^3 \text {Li}_4\left (e^{2 \text {csch}^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.23, size = 198, normalized size = 1.80 \[ \frac {1}{4} \left (4 a^3 \log (c x)+6 a^2 b \left (\text {Li}_2\left (e^{-2 \text {csch}^{-1}(c x)}\right )-\text {csch}^{-1}(c x) \left (\text {csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )\right )+2 a b^2 \left (-6 \text {csch}^{-1}(c x) \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+3 \text {Li}_3\left (e^{2 \text {csch}^{-1}(c x)}\right )+2 \text {csch}^{-1}(c x)^2 \left (\text {csch}^{-1}(c x)-3 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )\right )\right )+b^3 \left (-6 \text {csch}^{-1}(c x)^2 \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+6 \text {csch}^{-1}(c x) \text {Li}_3\left (e^{2 \text {csch}^{-1}(c x)}\right )-3 \text {Li}_4\left (e^{2 \text {csch}^{-1}(c x)}\right )+\text {csch}^{-1}(c x)^4-4 \text {csch}^{-1}(c x)^3 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {arcsch}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {arcsch}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {arcsch}\left (c x\right ) + a^{3}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{3}}{x}\, dx \]
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^{3} \log \relax (x) \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{3} + a^{3} \log \relax (x) - \int \frac {b^{3} \log \relax (c)^{3} - 3 \, a b^{2} \log \relax (c)^{2} + 3 \, a^{2} b \log \relax (c) + {\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} + 3 \, a^{2} b c^{2} \log \relax (c)\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} \log \relax (c) - a b^{2} c^{2}\right )} x^{2} + {\left (2 \, 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) + a^{2} b + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c) + a^{2} b c^{2}\right )} x^{2}\right )} \log \relax (x) - 3 \, {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) + a^{2} b + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c) + a^{2} b c^{2}\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) + a^{2} b + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c) + a^{2} b c^{2}\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} + 3 \, a^{2} b \log \relax (c) + {\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} + 3 \, a^{2} b c^{2} \log \relax (c)\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) + a^{2} b + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c) + a^{2} b c^{2}\right )} x^{2}\right )} \log \relax (x)\right )} \sqrt {c^{2} x^{2} + 1}}{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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{3}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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