Optimal. Leaf size=38 \[ \frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
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Rubi [A] time = 0.0247518, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4631, 3302} \[ \frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4631
Rule 3302
Rubi steps
\begin{align*} \int \frac{x}{\sin ^{-1}(a x)^2} \, dx &=-\frac{x \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=-\frac{x \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac{\text{Ci}\left (2 \sin ^{-1}(a x)\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0032364, size = 32, normalized size = 0.84 \[ \frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac{\sin \left (2 \sin ^{-1}(a x)\right )}{2 a^2 \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 28, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) }{2\,\arcsin \left ( ax \right ) }}+{\it Ci} \left ( 2\,\arcsin \left ( ax \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{\arcsin \left (a x\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\operatorname{asin}^{2}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37361, size = 49, normalized size = 1.29 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x}{a \arcsin \left (a x\right )} + \frac{\operatorname{Ci}\left (2 \, \arcsin \left (a x\right )\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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