Optimal. Leaf size=265 \[ -\frac {3 i \sqrt {a x-1} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i \sqrt {a x-1} \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {6 i \sqrt {a x-1} \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {6 i \sqrt {a x-1} \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {6 i \sqrt {a x-1} \text {Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {6 i \sqrt {a x-1} \text {Li}_4\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 \sqrt {a x-1} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}} \]
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Rubi [A] time = 0.48, antiderivative size = 356, normalized size of antiderivative = 1.34, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5798, 5761, 4180, 2531, 6609, 2282, 6589} \[ -\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {6 i \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {6 i \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {6 i \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {6 i \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4180
Rule 5761
Rule 5798
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^3}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (3 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (3 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_4\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 488, normalized size = 1.84 \[ \frac {i \sqrt {-((a x-1) (a x+1))} \left (192 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+384 \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{-\cosh ^{-1}(a x)}\right )-384 \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-48 \left (\pi -2 i \cosh ^{-1}(a x)\right )^2 \text {Li}_2\left (-i e^{-\cosh ^{-1}(a x)}\right )-48 \pi ^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \text {Li}_3\left (-i e^{-\cosh ^{-1}(a x)}\right )-192 i \pi \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )+384 \text {Li}_4\left (-i e^{-\cosh ^{-1}(a x)}\right )+384 \text {Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )-16 \cosh ^{-1}(a x)^4-32 i \pi \cosh ^{-1}(a x)^3+24 \pi ^2 \cosh ^{-1}(a x)^2+8 i \pi ^3 \cosh ^{-1}(a x)-64 \cosh ^{-1}(a x)^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+64 \cosh ^{-1}(a x)^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )-96 i \pi \cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+96 i \pi \cosh ^{-1}(a x)^2 \log \left (1-i e^{\cosh ^{-1}(a x)}\right )+48 \pi ^2 \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-48 \pi ^2 \cosh ^{-1}(a x) \log \left (1-i e^{\cosh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \cosh ^{-1}(a x)\right )\right )\right )+7 \pi ^4\right )}{64 \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{3}}{a^{2} x^{3} - x}, x\right ) \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccosh}\left (a x \right )^{3}}{x \sqrt {-a^{2} x^{2}+1}}\, dx \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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