Optimal. Leaf size=87 \[ -\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \cosh ^{-1}(a x) \text {PolyLog}\left (3,-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{4} \text {PolyLog}\left (4,-e^{2 \cosh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5882, 3799,
2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3}{2} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \cosh ^{-1}(a x) \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{4} \text {Li}_4\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (e^{2 \cosh ^{-1}(a x)}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 5882
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{x} \, dx &=\text {Subst}\left (\int x^3 \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+2 \text {Subst}\left (\int \frac {e^{2 x} x^3}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-3 \text {Subst}\left (\int x^2 \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-3 \text {Subst}\left (\int x \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \cosh ^{-1}(a x) \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \text {Subst}\left (\int \text {Li}_3\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \cosh ^{-1}(a x) \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \cosh ^{-1}(a x) \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{4} \text {Li}_4\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 82, normalized size = 0.94 \begin {gather*} \frac {1}{4} \left (\cosh ^{-1}(a x)^4+4 \cosh ^{-1}(a x)^3 \log \left (1+e^{-2 \cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-e^{-2 \cosh ^{-1}(a x)}\right )-3 \text {PolyLog}\left (4,-e^{-2 \cosh ^{-1}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.57, size = 132, normalized size = 1.52
method | result | size |
derivativedivides | \(-\frac {\mathrm {arccosh}\left (a x \right )^{4}}{4}+\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}+\frac {3 \polylog \left (4, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{4}\) | \(132\) |
default | \(-\frac {\mathrm {arccosh}\left (a x \right )^{4}}{4}+\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}+\frac {3 \polylog \left (4, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{4}\) | \(132\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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