Optimal. Leaf size=47 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b c}-\frac {1}{b c x \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.15, 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 = {} \[ \int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {1}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {\int \frac {1}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ \end {align*}
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Mathematica [A] time = 8.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1}}{a^{2} c^{2} x^{3} - a^{2} x + {\left (b^{2} c^{2} x^{3} - b^{2} x\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} x^{3} - a b x\right )} \arcsin \left (c x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +b \arcsin \left (c x \right )\right )^{2} \sqrt {-c^{2} x^{2}+1}}\, dx \]
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 \[ -\frac {\frac {{\left (b^{2} c x \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x\right )} \int \frac {1}{{\left (b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a\right )} x^{2}}\,{d x}}{b c} + 1}{b^{2} c x \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]
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 \[ \int \frac {1}{x \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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