Optimal. Leaf size=64 \[ \frac {4 c \text {Int}\left (\frac {x}{\left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b}-\frac {1}{b c \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.11, 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 = {} \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {1}{b c \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}+\frac {(4 c) \int \frac {x}{\left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b}\\ \end {align*}
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Mathematica [A] time = 4.44, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1}}{a^{2} c^{6} x^{6} - 3 \, a^{2} c^{4} x^{4} + 3 \, a^{2} c^{2} x^{2} + {\left (b^{2} c^{6} x^{6} - 3 \, b^{2} c^{4} x^{4} + 3 \, b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - a^{2} + 2 \, {\left (a b c^{6} x^{6} - 3 \, a b c^{4} x^{4} + 3 \, a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]
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 \[ -\frac {\frac {4 \, {\left (a b c^{6} x^{4} - 2 \, a b c^{4} x^{2} + a b c^{2} + {\left (b^{2} c^{6} x^{4} - 2 \, b^{2} c^{4} x^{2} + b^{2} c^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \int \frac {x}{{\left (c x + 1\right )}^{3} {\left (c x - 1\right )}^{3} {\left (b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a\right )}}\,{d x}}{b} + 1}{a b c^{5} x^{4} - 2 \, a b c^{3} x^{2} + a b c + {\left (b^{2} c^{5} x^{4} - 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )} \]
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 \[ \int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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