Optimal. Leaf size=98 \[ \frac {3}{2} a^2 \cosh ^{-1}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-3 a^2 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} a^2 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.22, antiderivative size = 98, 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 = {5883, 5918,
5882, 3799, 2221, 2317, 2438} \begin {gather*} -\frac {3}{2} a^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} a^2 \cosh ^{-1}(a x)^2-3 a^2 \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )-\frac {\cosh ^{-1}(a x)^3}{2 x^2}+\frac {3 a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5883
Rule 5918
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{x^3} \, dx &=-\frac {\cosh ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\cosh ^{-1}(a x)^2}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-\left (3 a^2\right ) \int \frac {\cosh ^{-1}(a x)}{x} \, dx\\ &=\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-\left (3 a^2\right ) \text {Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {3}{2} a^2 \cosh ^{-1}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-\left (6 a^2\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {3}{2} a^2 \cosh ^{-1}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-3 a^2 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\left (3 a^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {3}{2} a^2 \cosh ^{-1}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-3 a^2 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {1}{2} \left (3 a^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=\frac {3}{2} a^2 \cosh ^{-1}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-3 a^2 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} a^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 92, normalized size = 0.94 \begin {gather*} \frac {1}{2} \left (-\frac {\cosh ^{-1}(a x)^3}{x^2}+3 a^2 \left (\cosh ^{-1}(a x) \left (-\cosh ^{-1}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x)}{a x}-2 \log \left (1+e^{-2 \cosh ^{-1}(a x)}\right )\right )+\text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(a x)}\right )\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 3.60, size = 116, normalized size = 1.18
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\mathrm {arccosh}\left (a x \right )^{2} \left (-3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +3 a^{2} x^{2}+\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+3 \mathrm {arccosh}\left (a x \right )^{2}-3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {3 \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\right )\) | \(116\) |
default | \(a^{2} \left (-\frac {\mathrm {arccosh}\left (a x \right )^{2} \left (-3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +3 a^{2} x^{2}+\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+3 \mathrm {arccosh}\left (a x \right )^{2}-3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {3 \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\right )\) | \(116\) |
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^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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