Optimal. Leaf size=78 \[ \text{Unintegrable}\left (\frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )},x\right )+\frac{\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b}-\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b} \]
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Rubi [A] time = 0.398859, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt{1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx &=\int \left (\frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}-\frac{c^2 x}{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}\right ) \, dx\\ &=-\left (c^2 \int \frac{x}{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\right )+\int \frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ &=\int \frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx-\operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\left (\cos \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )\right )+\sin \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\int \frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ &=\frac{\text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{b}-\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b}+\int \frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ \end{align*}
Mathematica [A] time = 2.86417, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.267, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b\arcsin \left ( cx \right ) \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} x^{2} + 1}}{{\left (b \arcsin \left (c x\right ) + a\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} x^{2} + 1}}{b x \arcsin \left (c x\right ) + a x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}{x \left (a + b \operatorname{asin}{\left (c x \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} x^{2} + 1}}{{\left (b \arcsin \left (c x\right ) + a\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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