Optimal. Leaf size=296 \[ -\frac {i a^2 \sqrt {a x-1} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {i a^2 \sqrt {a x-1} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {i a^2 \sqrt {a x-1} \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {i a^2 \sqrt {a x-1} \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {a^2 \sqrt {a x-1} \tan ^{-1}\left (\sqrt {a x-1} \sqrt {a x+1}\right )}{\sqrt {1-a x}}+\frac {a^2 \sqrt {a x-1} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {a \sqrt {a x-1} \cosh ^{-1}(a x)}{x \sqrt {1-a x}} \]
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Rubi [A] time = 0.72, antiderivative size = 398, normalized size of antiderivative = 1.34, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5798, 5748, 5761, 4180, 2531, 2282, 6589, 5662, 92, 205} \[ -\frac {i a^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {i a^2 \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {a^2 \sqrt {a x-1} \sqrt {a x+1} \tan ^{-1}\left (\sqrt {a x-1} \sqrt {a x+1}\right )}{\sqrt {1-a^2 x^2}}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (a x+1) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {a^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}} \]
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 5798
Rule 6589
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^2}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}-\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{x^2} \, dx}{\sqrt {1-a^2 x^2}}+\frac {\left (a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^2}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {\left (a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt {1-a^2 x^2}}-\frac {\left (a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (a^3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{a+a x^2} \, dx,x,\sqrt {-1+a x} \sqrt {1+a x}\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{\sqrt {1-a^2 x^2}}-\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{\sqrt {1-a^2 x^2}}-\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{\sqrt {1-a^2 x^2}}-\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.01, size = 233, normalized size = 0.79 \[ \frac {i a^2 \sqrt {-((a x-1) (a x+1))} \left (\frac {i \sqrt {\frac {a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^2}{a^2 x^2}+2 \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{-\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x) \text {Li}_2\left (i e^{-\cosh ^{-1}(a x)}\right )+2 \text {Li}_3\left (-i e^{-\cosh ^{-1}(a x)}\right )-2 \text {Li}_3\left (i e^{-\cosh ^{-1}(a x)}\right )+\frac {2 i \cosh ^{-1}(a x)}{a x}+\cosh ^{-1}(a x)^2 \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )\right )}{2 \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{2}}{a^{2} x^{5} - x^{3}}, x\right ) \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.67, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccosh}\left (a x \right )^{2}}{x^{3} \sqrt {-a^{2} x^{2}+1}}\, dx \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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