Optimal. Leaf size=76 \[ -\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}-i a \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )-i a \sin ^{-1}(a x)^2+2 a \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4681, 4625, 3717, 2190, 2279, 2391} \[ -i a \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}-i a \sin ^{-1}(a x)^2+2 a \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rule 4681
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}+(2 a) \int \frac {\sin ^{-1}(a x)}{x} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}+(2 a) \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^2-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}-(4 i a) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^2-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}+2 a \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-(2 a) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^2-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}+2 a \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )+(i a) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )\\ &=-i a \sin ^{-1}(a x)^2-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}+2 a \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-i a \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.30, size = 72, normalized size = 0.95 \[ \sin ^{-1}(a x) \left (2 a \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac {\left (\sqrt {1-a^2 x^2}+i a x\right ) \sin ^{-1}(a x)}{x}\right )-i a \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{2} x^{4} - x^{2}}, x\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 148, normalized size = 1.95 \[ \frac {\left (i a x -\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right )^{2}}{x}+2 a \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 a \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right )^{2} a -2 i \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right ) a -2 i \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right ) a \]
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 \[ -\frac {\sqrt {a x + 1} \sqrt {-a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2} - 2 \, a x \int \frac {\arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}{x}\,{d x}}{x} \]
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 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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