Optimal. Leaf size=183 \[ -i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+i a^3 \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-i a^3 \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )-a^3 \tan ^{-1}\left (\sqrt {a x-1} \sqrt {a x+1}\right )+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac {a^2 \cosh ^{-1}(a x)}{x}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 x^2} \]
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Rubi [A] time = 0.58, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5662, 5748, 5761, 4180, 2531, 2282, 6589, 92, 205} \[ -i a^3 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+i a^3 \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-i a^3 \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )-a^3 \tan ^{-1}\left (\sqrt {a x-1} \sqrt {a x+1}\right )+\frac {a^2 \cosh ^{-1}(a x)}{x}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 92
Rule 205
Rule 2282
Rule 2531
Rule 4180
Rule 5662
Rule 5748
Rule 5761
Rule 6589
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{x^4} \, dx &=-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a \int \frac {\cosh ^{-1}(a x)^2}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}-a^2 \int \frac {\cosh ^{-1}(a x)}{x^2} \, dx+\frac {1}{2} a^3 \int \frac {\cosh ^{-1}(a x)^2}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+\frac {1}{2} a^3 \operatorname {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )-a^3 \int \frac {1}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\left (i a^3\right ) \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\left (i a^3\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-a^4 \operatorname {Subst}\left (\int \frac {1}{a+a x^2} \, dx,x,\sqrt {-1+a x} \sqrt {1+a x}\right )\\ &=\frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-a^3 \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+\left (i a^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-\left (i a^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-a^3 \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+\left (i a^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )-\left (i a^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )\\ &=\frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-a^3 \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+i a^3 \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-i a^3 \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.58, size = 201, normalized size = 1.10 \[ \frac {1}{6} \left (-3 i a^3 \left (2 \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{-\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x) \text {Li}_2\left (i e^{-\cosh ^{-1}(a x)}\right )+2 \text {Li}_3\left (-i e^{-\cosh ^{-1}(a x)}\right )-2 \text {Li}_3\left (i e^{-\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x)^2 \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )\right )+\frac {6 a^2 \cosh ^{-1}(a x)}{x}-\frac {2 \cosh ^{-1}(a x)^3}{x^3}+\frac {3 a \sqrt {\frac {a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^2}{x^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccosh}\left (a x \right )^{3}}{x^{4}}\, dx \]
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 \[ -\frac {\log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{3}}{3 \, x^{3}} + \int \frac {{\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{a^{3} x^{6} - a x^{4} + {\left (a^{2} x^{5} - x^{3}\right )} \sqrt {a x + 1} \sqrt {a x - 1}}\,{d x} \]
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 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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