Optimal. Leaf size=81 \[ \frac{e^{\cos ^{-1}(a x)} \cos \left (2 \cos ^{-1}(a x)\right )}{10 a^4}+\frac{e^{\cos ^{-1}(a x)} \cos \left (4 \cos ^{-1}(a x)\right )}{34 a^4}-\frac{e^{\cos ^{-1}(a x)} \sin \left (2 \cos ^{-1}(a x)\right )}{20 a^4}-\frac{e^{\cos ^{-1}(a x)} \sin \left (4 \cos ^{-1}(a x)\right )}{136 a^4} \]
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Rubi [A] time = 0.0643541, antiderivative size = 81, 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} \[ \frac{e^{\cos ^{-1}(a x)} \cos \left (2 \cos ^{-1}(a x)\right )}{10 a^4}+\frac{e^{\cos ^{-1}(a x)} \cos \left (4 \cos ^{-1}(a x)\right )}{34 a^4}-\frac{e^{\cos ^{-1}(a x)} \sin \left (2 \cos ^{-1}(a x)\right )}{20 a^4}-\frac{e^{\cos ^{-1}(a x)} \sin \left (4 \cos ^{-1}(a x)\right )}{136 a^4} \]
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^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{e^x \cos ^3(x) \sin (x)}{a^3} \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int e^x \cos ^3(x) \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} e^x \sin (2 x)+\frac{1}{8} e^x \sin (4 x)\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int e^x \sin (4 x) \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac{\operatorname{Subst}\left (\int e^x \sin (2 x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=\frac{e^{\cos ^{-1}(a x)} \cos \left (2 \cos ^{-1}(a x)\right )}{10 a^4}+\frac{e^{\cos ^{-1}(a x)} \cos \left (4 \cos ^{-1}(a x)\right )}{34 a^4}-\frac{e^{\cos ^{-1}(a x)} \sin \left (2 \cos ^{-1}(a x)\right )}{20 a^4}-\frac{e^{\cos ^{-1}(a x)} \sin \left (4 \cos ^{-1}(a x)\right )}{136 a^4}\\ \end{align*}
Mathematica [A] time = 0.139171, size = 50, normalized size = 0.62 \[ -\frac{e^{\cos ^{-1}(a x)} \left (-68 \cos \left (2 \cos ^{-1}(a x)\right )-20 \cos \left (4 \cos ^{-1}(a x)\right )+34 \sin \left (2 \cos ^{-1}(a x)\right )+5 \sin \left (4 \cos ^{-1}(a x)\right )\right )}{680 a^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.01, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\arccos \left ( ax \right ) }}{x}^{3}\, 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^{3} 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 = 2.61862, size = 130, normalized size = 1.6 \begin{align*} \frac{{\left (20 \, a^{4} x^{4} - 3 \, a^{2} x^{2} -{\left (5 \, a^{3} x^{3} + 6 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} - 6\right )} e^{\left (\arccos \left (a x\right )\right )}}{85 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.83622, size = 105, normalized size = 1.3 \begin{align*} \begin{cases} \frac{4 x^{4} e^{\operatorname{acos}{\left (a x \right )}}}{17} - \frac{x^{3} \sqrt{- a^{2} x^{2} + 1} e^{\operatorname{acos}{\left (a x \right )}}}{17 a} - \frac{3 x^{2} e^{\operatorname{acos}{\left (a x \right )}}}{85 a^{2}} - \frac{6 x \sqrt{- a^{2} x^{2} + 1} e^{\operatorname{acos}{\left (a x \right )}}}{85 a^{3}} - \frac{6 e^{\operatorname{acos}{\left (a x \right )}}}{85 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4} e^{\frac{\pi }{2}}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32475, size = 111, normalized size = 1.37 \begin{align*} \frac{4}{17} \, x^{4} e^{\left (\arccos \left (a x\right )\right )} - \frac{\sqrt{-a^{2} x^{2} + 1} x^{3} e^{\left (\arccos \left (a x\right )\right )}}{17 \, a} - \frac{3 \, x^{2} e^{\left (\arccos \left (a x\right )\right )}}{85 \, a^{2}} - \frac{6 \, \sqrt{-a^{2} x^{2} + 1} x e^{\left (\arccos \left (a x\right )\right )}}{85 \, a^{3}} - \frac{6 \, e^{\left (\arccos \left (a x\right )\right )}}{85 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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