Optimal. Leaf size=47 \[ -\frac {c \log (a+b \text {ArcSin}(c x))}{b}+\text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))},x\right ) \]
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Rubi [A]
time = 0.21, 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 = {}
\begin {gather*} \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {ArcSin}(c x))} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1-c^2 x^2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx &=\int \left (-\frac {c^2}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}+\frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}\right ) \, dx\\ &=-\left (c^2 \int \frac {1}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\right )+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ &=-\frac {c \log \left (a+b \sin ^{-1}(c x)\right )}{b}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {ArcSin}(c x))} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.44, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-c^{2} x^{2}+1}}{x^{2} \left (a +b \arcsin \left (c x \right )\right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {1-c^2\,x^2}}{x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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