Optimal. Leaf size=42 \[ \frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4643, 4641} \[ \frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 4641
Rule 4643
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^2}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {\sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 42, normalized size = 1.00 \[ \frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{2}}{a^{2} c x^{2} - c}, x\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} c x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 52, normalized size = 1.24 \[ -\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{3}}{3 a c \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 14, normalized size = 0.33 \[ \frac {\arcsin \left (a x\right )^{3}}{3 \, a \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{\sqrt {c-a^2\,c\,x^2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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