Optimal. Leaf size=31 \[ \text {Int}\left (\frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )},x\right ) \]
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Rubi [A] time = 0.14, 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 = {} \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )} \, dx &=\int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ \end {align*}
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Mathematica [A] time = 6.52, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1}}{a c^{6} x^{8} - 3 \, a c^{4} x^{6} + 3 \, a c^{2} x^{4} - a x^{2} + {\left (b c^{6} x^{8} - 3 \, b c^{4} x^{6} + 3 \, b c^{2} x^{4} - b x^{2}\right )} \arcsin \left (c x\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 5.64, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (-c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (1-c^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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