Optimal. Leaf size=115 \[ -6 a^2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+3 a^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+2 a^2 \cosh ^{-1}(a x)^3-6 a^2 \cosh ^{-1}(a x)^2 \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )-\frac {\cosh ^{-1}(a x)^4}{2 x^2}+\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{x} \]
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Rubi [A] time = 0.36, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5662, 5724, 5660, 3718, 2190, 2531, 2282, 6589} \[ -6 a^2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )+3 a^2 \text {PolyLog}\left (3,-e^{2 \cosh ^{-1}(a x)}\right )+2 a^2 \cosh ^{-1}(a x)^3-6 a^2 \cosh ^{-1}(a x)^2 \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )-\frac {\cosh ^{-1}(a x)^4}{2 x^2}+\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{x} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3718
Rule 5660
Rule 5662
Rule 5724
Rule 6589
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^4}{x^3} \, dx &=-\frac {\cosh ^{-1}(a x)^4}{2 x^2}+(2 a) \int \frac {\cosh ^{-1}(a x)^3}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-\left (6 a^2\right ) \int \frac {\cosh ^{-1}(a x)^2}{x} \, dx\\ &=\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-\left (6 a^2\right ) \operatorname {Subst}\left (\int x^2 \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-\left (12 a^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-6 a^2 \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\left (12 a^2\right ) \operatorname {Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-6 a^2 \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-6 a^2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+\left (6 a^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-6 a^2 \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-6 a^2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-6 a^2 \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-6 a^2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+3 a^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 1.16, size = 112, normalized size = 0.97 \[ a^2 \left (6 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{-2 \cosh ^{-1}(a x)}\right )+3 \text {Li}_3\left (-e^{-2 \cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x)^2 \left (\frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)}{a x}-\cosh ^{-1}(a x)-3 \log \left (e^{-2 \cosh ^{-1}(a x)}+1\right )\right )\right )-\frac {\cosh ^{-1}(a x)^4}{2 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{3}}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 149, normalized size = 1.30 \[ 2 a^{2} \mathrm {arccosh}\left (a x \right )^{3}-\frac {\mathrm {arccosh}\left (a x \right )^{4}}{2 x^{2}}-6 a^{2} \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-6 a^{2} \mathrm {arccosh}\left (a x \right ) \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+3 a^{2} \polylog \left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {2 a \mathrm {arccosh}\left (a x \right )^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{x} \]
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 \[ -\frac {\log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{4}}{2 \, x^{2}} + \int \frac {2 \, {\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{3}}{a^{3} x^{5} - a x^{3} + {\left (a^{2} x^{4} - x^{2}\right )} \sqrt {a x + 1} \sqrt {a x - 1}}\,{d x} \]
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 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^4}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {acosh}^{4}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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