Optimal. Leaf size=103 \[ 2 \cosh ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+3 \cosh ^{-1}(a x) \text {Li}_4\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \text {Li}_5\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {1}{5} \cosh ^{-1}(a x)^5+\cosh ^{-1}(a x)^4 \log \left (e^{2 \cosh ^{-1}(a x)}+1\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5660, 3718, 2190, 2531, 6609, 2282, 6589} \[ 2 \cosh ^{-1}(a x)^3 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (3,-e^{2 \cosh ^{-1}(a x)}\right )+3 \cosh ^{-1}(a x) \text {PolyLog}\left (4,-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \text {PolyLog}\left (5,-e^{2 \cosh ^{-1}(a x)}\right )-\frac {1}{5} \cosh ^{-1}(a x)^5+\cosh ^{-1}(a x)^4 \log \left (e^{2 \cosh ^{-1}(a x)}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3718
Rule 5660
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^4}{x} \, dx &=\operatorname {Subst}\left (\int x^4 \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{5} \cosh ^{-1}(a x)^5+2 \operatorname {Subst}\left (\int \frac {e^{2 x} x^4}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{5} \cosh ^{-1}(a x)^5+\cosh ^{-1}(a x)^4 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-4 \operatorname {Subst}\left (\int x^3 \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{5} \cosh ^{-1}(a x)^5+\cosh ^{-1}(a x)^4 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-6 \operatorname {Subst}\left (\int x^2 \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{5} \cosh ^{-1}(a x)^5+\cosh ^{-1}(a x)^4 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+6 \operatorname {Subst}\left (\int x \text {Li}_3\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{5} \cosh ^{-1}(a x)^5+\cosh ^{-1}(a x)^4 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+3 \cosh ^{-1}(a x) \text {Li}_4\left (-e^{2 \cosh ^{-1}(a x)}\right )-3 \operatorname {Subst}\left (\int \text {Li}_4\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{5} \cosh ^{-1}(a x)^5+\cosh ^{-1}(a x)^4 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+3 \cosh ^{-1}(a x) \text {Li}_4\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {\text {Li}_4(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=-\frac {1}{5} \cosh ^{-1}(a x)^5+\cosh ^{-1}(a x)^4 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+3 \cosh ^{-1}(a x) \text {Li}_4\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \text {Li}_5\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 103, normalized size = 1.00 \[ -2 \cosh ^{-1}(a x)^3 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \text {Li}_3\left (-e^{-2 \cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x) \text {Li}_4\left (-e^{-2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \text {Li}_5\left (-e^{-2 \cosh ^{-1}(a x)}\right )+\frac {1}{5} \cosh ^{-1}(a x)^5+\cosh ^{-1}(a x)^4 \log \left (e^{-2 \cosh ^{-1}(a x)}+1\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (a x\right )^{4}}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 165, normalized size = 1.60 \[ -\frac {\mathrm {arccosh}\left (a x \right )^{5}}{5}+\mathrm {arccosh}\left (a x \right )^{4} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+2 \mathrm {arccosh}\left (a x \right )^{3} \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (4, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {3 \polylog \left (5, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2} \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^4}{x} \,d x \]
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 \[ \int \frac {\operatorname {acosh}^{4}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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