Optimal. Leaf size=76 \[ -i a \text {ArcSin}(a x)^2-\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{x}+2 a \text {ArcSin}(a x) \log \left (1-e^{2 i \text {ArcSin}(a x)}\right )-i a \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(a x)}\right ) \]
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Rubi [A]
time = 0.11, 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 = {4771, 4721,
3798, 2221, 2317, 2438} \begin {gather*} -\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{x}-i a \text {Li}_2\left (e^{2 i \text {ArcSin}(a x)}\right )-i a \text {ArcSin}(a x)^2+2 a \text {ArcSin}(a x) \log \left (1-e^{2 i \text {ArcSin}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 4771
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) \text {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) \text {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) \text {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) \text {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 = 0.18, size = 72, normalized size = 0.95 \begin {gather*} \text {ArcSin}(a x) \left (-\frac {\left (i a x+\sqrt {1-a^2 x^2}\right ) \text {ArcSin}(a x)}{x}+2 a \log \left (1-e^{2 i \text {ArcSin}(a x)}\right )\right )-i a \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 141, normalized size = 1.86
method | result | size |
default | \(\frac {\left (i a x -\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right )^{2}}{x}-2 i a \left (i \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+i \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{2}+\polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+\polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(141\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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