Optimal. Leaf size=42 \[ -\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}} \]
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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4643, 4641} \[ -\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4641
Rule 4643
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {1}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 42, normalized size = 1.00 \[ -\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 48, normalized size = 1.14 \[ \frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{3} c x^{2} - a c\right )} \sqrt {\arcsin \left (a x\right )}} \]
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 \[ \int \frac {1}{\sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 38, normalized size = 0.90 \[ -\frac {2 \sqrt {-a^{2} x^{2}+1}}{\sqrt {\arcsin \left (a x \right )}\, a \sqrt {-c \left (a^{2} x^{2}-1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^{3/2}\,\sqrt {c-a^2\,c\,x^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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