Optimal. Leaf size=59 \[ \frac {a \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)},x\right )}{c}-\frac {1}{a c \sqrt {1-a^2 x^2} \sin ^{-1}(a x)} \]
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Rubi [A] time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \sin ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (c-a^2 c x^2\right ) \sin ^{-1}(a x)^2} \, dx &=-\frac {1}{a c \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}+\frac {a \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)} \, dx}{c}\\ \end {align*}
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Mathematica [A] time = 4.07, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \sin ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (a^{2} c x^{2} - c\right )} \arcsin \left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (a^{2} c x^{2} - c\right )} \arcsin \left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-a^{2} c \,x^{2}+c \right ) \arcsin \left (a x \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^2\,\left (c-a^2\,c\,x^2\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {1}{a^{2} x^{2} \operatorname {asin}^{2}{\left (a x \right )} - \operatorname {asin}^{2}{\left (a x \right )}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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