Optimal. Leaf size=57 \[ \frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{2 a^4}-\frac{\text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )}{2 a^4}-\frac{x^3 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
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Rubi [A] time = 0.0486051, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4631, 3302} \[ \frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{2 a^4}-\frac{\text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )}{2 a^4}-\frac{x^3 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4631
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^3}{\sin ^{-1}(a x)^2} \, dx &=-\frac{x^3 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (\frac{\cos (2 x)}{2 x}-\frac{\cos (4 x)}{2 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=-\frac{x^3 \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 )}{2 a^4}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{2 a^4}\\ &=-\frac{x^3 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac{\text{Ci}\left (2 \sin ^{-1}(a x)\right )}{2 a^4}-\frac{\text{Ci}\left (4 \sin ^{-1}(a x)\right )}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.0168856, size = 56, normalized size = 0.98 \[ \frac{4 \sin ^{-1}(a x) \text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )-4 \sin ^{-1}(a x) \text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )-2 \sin \left (2 \sin ^{-1}(a x)\right )+\sin \left (4 \sin ^{-1}(a x)\right )}{8 a^4 \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 54, normalized size = 1. \begin{align*}{\frac{1}{{a}^{4}} \left ( -{\frac{\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) }{4\,\arcsin \left ( ax \right ) }}+{\frac{{\it Ci} \left ( 2\,\arcsin \left ( ax \right ) \right ) }{2}}+{\frac{\sin \left ( 4\,\arcsin \left ( ax \right ) \right ) }{8\,\arcsin \left ( ax \right ) }}-{\frac{{\it Ci} \left ( 4\,\arcsin \left ( ax \right ) \right ) }{2}} \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^{3}}{\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^{3}}{\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 = 1.38474, size = 97, normalized size = 1.7 \begin{align*} \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{a^{3} \arcsin \left (a x\right )} - \frac{\sqrt{-a^{2} x^{2} + 1} x}{a^{3} \arcsin \left (a x\right )} - \frac{\operatorname{Ci}\left (4 \, \arcsin \left (a x\right )\right )}{2 \, a^{4}} + \frac{\operatorname{Ci}\left (2 \, \arcsin \left (a x\right )\right )}{2 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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