Optimal. Leaf size=179 \[ -\frac {i \sqrt {1-a^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4653, 4675, 3719, 2190, 2279, 2391} \[ -\frac {i \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4653
Rule 4675
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {\left (2 a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {\left (2 \sqrt {1-a^2 x^2}\right ) \operatorname {Subst}\left (\int x \tan (x) \, dx,x,\sin ^{-1}(a x)\right )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (4 i \sqrt {1-a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {\left (2 \sqrt {1-a^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (i \sqrt {1-a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 108, normalized size = 0.60 \[ \frac {\sin ^{-1}(a x) \left (a x \sin ^{-1}(a x)+\sqrt {1-a^2 x^2} \left (2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )-i \sin ^{-1}(a x)\right )\right )-i \sqrt {1-a^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c \left (1-a^2 x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{2}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 169, normalized size = 0.94 \[ -\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}+a x \right ) \arcsin \left (a x \right )^{2}}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (a x \right ) \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (a x \right )^{2}+\polylog \left (2, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )\right )}{c^{2} a \left (a^{2} x^{2}-1\right )} \]
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 \[ \int \frac {\arcsin \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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