Optimal. Leaf size=82 \[ -\frac{\sqrt{1-a^2 x^2} e^{\cos ^{-1}(a x)}}{8 a^3}+\frac{x e^{\cos ^{-1}(a x)}}{8 a^2}+\frac{3 e^{\cos ^{-1}(a x)} \cos \left (3 \cos ^{-1}(a x)\right )}{40 a^3}-\frac{e^{\cos ^{-1}(a x)} \sin \left (3 \cos ^{-1}(a x)\right )}{40 a^3} \]
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Rubi [A] time = 0.0613399, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4837, 12, 4469, 4432} \[ -\frac{\sqrt{1-a^2 x^2} e^{\cos ^{-1}(a x)}}{8 a^3}+\frac{x e^{\cos ^{-1}(a x)}}{8 a^2}+\frac{3 e^{\cos ^{-1}(a x)} \cos \left (3 \cos ^{-1}(a x)\right )}{40 a^3}-\frac{e^{\cos ^{-1}(a x)} \sin \left (3 \cos ^{-1}(a x)\right )}{40 a^3} \]
Antiderivative was successfully verified.
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Rule 4837
Rule 12
Rule 4469
Rule 4432
Rubi steps
\begin{align*} \int e^{\cos ^{-1}(a x)} x^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{e^x \cos ^2(x) \sin (x)}{a^2} \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int e^x \cos ^2(x) \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} e^x \sin (x)+\frac{1}{4} e^x \sin (3 x)\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int e^x \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}-\frac{\operatorname{Subst}\left (\int e^x \sin (3 x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}\\ &=\frac{e^{\cos ^{-1}(a x)} x}{8 a^2}-\frac{e^{\cos ^{-1}(a x)} \sqrt{1-a^2 x^2}}{8 a^3}+\frac{3 e^{\cos ^{-1}(a x)} \cos \left (3 \cos ^{-1}(a x)\right )}{40 a^3}-\frac{e^{\cos ^{-1}(a x)} \sin \left (3 \cos ^{-1}(a x)\right )}{40 a^3}\\ \end{align*}
Mathematica [A] time = 0.106531, size = 50, normalized size = 0.61 \[ -\frac{e^{\cos ^{-1}(a x)} \left (5 \sqrt{1-a^2 x^2}-5 a x-3 \cos \left (3 \cos ^{-1}(a x)\right )+\sin \left (3 \cos ^{-1}(a x)\right )\right )}{40 a^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.009, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\arccos \left ( ax \right ) }}{x}^{2}\, dx \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*} \int x^{2} e^{\left (\arccos \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.5661, size = 107, normalized size = 1.3 \begin{align*} \frac{{\left (3 \, a^{3} x^{3} - a x -{\left (a^{2} x^{2} + 1\right )} \sqrt{-a^{2} x^{2} + 1}\right )} e^{\left (\arccos \left (a x\right )\right )}}{10 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.7158, size = 85, normalized size = 1.04 \begin{align*} \begin{cases} \frac{3 x^{3} e^{\operatorname{acos}{\left (a x \right )}}}{10} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1} e^{\operatorname{acos}{\left (a x \right )}}}{10 a} - \frac{x e^{\operatorname{acos}{\left (a x \right )}}}{10 a^{2}} - \frac{\sqrt{- a^{2} x^{2} + 1} e^{\operatorname{acos}{\left (a x \right )}}}{10 a^{3}} & \text{for}\: a \neq 0 \\\frac{x^{3} e^{\frac{\pi }{2}}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37588, size = 93, normalized size = 1.13 \begin{align*} \frac{3}{10} \, x^{3} e^{\left (\arccos \left (a x\right )\right )} - \frac{\sqrt{-a^{2} x^{2} + 1} x^{2} e^{\left (\arccos \left (a x\right )\right )}}{10 \, a} - \frac{x e^{\left (\arccos \left (a x\right )\right )}}{10 \, a^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1} e^{\left (\arccos \left (a x\right )\right )}}{10 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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