Optimal. Leaf size=82 \[ \frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{8 a^3}+\frac{9 \text{Si}\left (3 \cos ^{-1}(a x)\right )}{8 a^3}+\frac{x^2 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac{x}{a^2 \cos ^{-1}(a x)}+\frac{3 x^3}{2 \cos ^{-1}(a x)} \]
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Rubi [A] time = 0.24308, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4634, 4720, 4636, 4406, 3299, 4624} \[ \frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{8 a^3}+\frac{9 \text{Si}\left (3 \cos ^{-1}(a x)\right )}{8 a^3}+\frac{x^2 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac{x}{a^2 \cos ^{-1}(a x)}+\frac{3 x^3}{2 \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4634
Rule 4720
Rule 4636
Rule 4406
Rule 3299
Rule 4624
Rubi steps
\begin{align*} \int \frac{x^2}{\cos ^{-1}(a x)^3} \, dx &=\frac{x^2 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac{\int \frac{x}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx}{a}+\frac{1}{2} (3 a) \int \frac{x^3}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx\\ &=\frac{x^2 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac{x}{a^2 \cos ^{-1}(a x)}+\frac{3 x^3}{2 \cos ^{-1}(a x)}-\frac{9}{2} \int \frac{x^2}{\cos ^{-1}(a x)} \, dx+\frac{\int \frac{1}{\cos ^{-1}(a x)} \, dx}{a^2}\\ &=\frac{x^2 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac{x}{a^2 \cos ^{-1}(a x)}+\frac{3 x^3}{2 \cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^3}+\frac{9 \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^3}\\ &=\frac{x^2 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac{x}{a^2 \cos ^{-1}(a x)}+\frac{3 x^3}{2 \cos ^{-1}(a x)}-\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{a^3}+\frac{9 \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 x}+\frac{\sin (3 x)}{4 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{2 a^3}\\ &=\frac{x^2 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac{x}{a^2 \cos ^{-1}(a x)}+\frac{3 x^3}{2 \cos ^{-1}(a x)}-\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{a^3}+\frac{9 \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}+\frac{9 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}\\ &=\frac{x^2 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac{x}{a^2 \cos ^{-1}(a x)}+\frac{3 x^3}{2 \cos ^{-1}(a x)}+\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{8 a^3}+\frac{9 \text{Si}\left (3 \cos ^{-1}(a x)\right )}{8 a^3}\\ \end{align*}
Mathematica [A] time = 0.132357, size = 65, normalized size = 0.79 \[ \frac{\frac{4 a x \left (a x \sqrt{1-a^2 x^2}+\left (3 a^2 x^2-2\right ) \cos ^{-1}(a x)\right )}{\cos ^{-1}(a x)^2}+\text{Si}\left (\cos ^{-1}(a x)\right )+9 \text{Si}\left (3 \cos ^{-1}(a x)\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 82, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{\sin \left ( 3\,\arccos \left ( ax \right ) \right ) }{8\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}}+{\frac{3\,\cos \left ( 3\,\arccos \left ( ax \right ) \right ) }{8\,\arccos \left ( ax \right ) }}+{\frac{9\,{\it Si} \left ( 3\,\arccos \left ( ax \right ) \right ) }{8}}+{\frac{1}{8\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{ax}{8\,\arccos \left ( ax \right ) }}+{\frac{{\it Si} \left ( \arccos \left ( ax \right ) \right ) }{8}} \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{\sqrt{a x + 1} \sqrt{-a x + 1} a x^{2} - \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{2} \int \frac{9 \, a^{2} x^{2} - 2}{\arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )}\,{d x} +{\left (3 \, a^{2} x^{3} - 2 \, x\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )}{2 \, a^{2} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\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^{2}}{\arccos \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^{2}}{\operatorname{acos}^{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 = 1.21897, size = 97, normalized size = 1.18 \begin{align*} \frac{3 \, x^{3}}{2 \, \arccos \left (a x\right )} + \frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{2 \, a \arccos \left (a x\right )^{2}} - \frac{x}{a^{2} \arccos \left (a x\right )} + \frac{9 \, \operatorname{Si}\left (3 \, \arccos \left (a x\right )\right )}{8 \, a^{3}} + \frac{\operatorname{Si}\left (\arccos \left (a x\right )\right )}{8 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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