Optimal. Leaf size=64 \[ -\frac{\text{Si}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac{1}{2 a^2 \sin ^{-1}(a x)}+\frac{x^2}{\sin ^{-1}(a x)} \]
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Rubi [A] time = 0.168357, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {4633, 4719, 4635, 4406, 12, 3299, 4641} \[ -\frac{\text{Si}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac{1}{2 a^2 \sin ^{-1}(a x)}+\frac{x^2}{\sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4633
Rule 4719
Rule 4635
Rule 4406
Rule 12
Rule 3299
Rule 4641
Rubi steps
\begin{align*} \int \frac{x}{\sin ^{-1}(a x)^3} \, dx &=-\frac{x \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2} \, dx}{2 a}-a \int \frac{x^2}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2} \, dx\\ &=-\frac{x \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac{1}{2 a^2 \sin ^{-1}(a x)}+\frac{x^2}{\sin ^{-1}(a x)}-2 \int \frac{x}{\sin ^{-1}(a x)} \, dx\\ &=-\frac{x \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac{1}{2 a^2 \sin ^{-1}(a x)}+\frac{x^2}{\sin ^{-1}(a x)}-\frac{2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=-\frac{x \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac{1}{2 a^2 \sin ^{-1}(a x)}+\frac{x^2}{\sin ^{-1}(a x)}-\frac{2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=-\frac{x \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac{1}{2 a^2 \sin ^{-1}(a x)}+\frac{x^2}{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=-\frac{x \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac{1}{2 a^2 \sin ^{-1}(a x)}+\frac{x^2}{\sin ^{-1}(a x)}-\frac{\text{Si}\left (2 \sin ^{-1}(a x)\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0596291, size = 61, normalized size = 0.95 \[ -\frac{a x \sqrt{1-a^2 x^2}+\left (1-2 a^2 x^2\right ) \sin ^{-1}(a x)+2 \sin ^{-1}(a x)^2 \text{Si}\left (2 \sin ^{-1}(a x)\right )}{2 a^2 \sin ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 45, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) }{4\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}-{\frac{\cos \left ( 2\,\arcsin \left ( ax \right ) \right ) }{2\,\arcsin \left ( ax \right ) }}-{\it Si} \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] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{4 \, a^{2} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2} \int \frac{x}{\arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )}\,{d x} + \sqrt{a x + 1} \sqrt{-a x + 1} a x -{\left (2 \, a^{2} x^{2} - 1\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )}{2 \, a^{2} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2}} \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 )^{3}}, 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}^{3}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41823, size = 90, normalized size = 1.41 \begin{align*} -\frac{\operatorname{Si}\left (2 \, \arcsin \left (a x\right )\right )}{a^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1} x}{2 \, a \arcsin \left (a x\right )^{2}} + \frac{a^{2} x^{2} - 1}{a^{2} \arcsin \left (a x\right )} + \frac{1}{2 \, a^{2} \arcsin \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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