Optimal. Leaf size=41 \[ \frac {e^{\cos ^{-1}(a x)} \cos \left (2 \cos ^{-1}(a x)\right )}{5 a^2}-\frac {e^{\cos ^{-1}(a x)} \sin \left (2 \cos ^{-1}(a x)\right )}{10 a^2} \]
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Rubi [A] time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4837, 12, 4469, 4432} \[ \frac {e^{\cos ^{-1}(a x)} \cos \left (2 \cos ^{-1}(a x)\right )}{5 a^2}-\frac {e^{\cos ^{-1}(a x)} \sin \left (2 \cos ^{-1}(a x)\right )}{10 a^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 4432
Rule 4469
Rule 4837
Rubi steps
\begin {align*} \int e^{\cos ^{-1}(a x)} x \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {e^x \cos (x) \sin (x)}{a} \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=-\frac {\operatorname {Subst}\left (\int e^x \cos (x) \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{2} e^x \sin (2 x) \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int e^x \sin (2 x) \, dx,x,\cos ^{-1}(a x)\right )}{2 a^2}\\ &=\frac {e^{\cos ^{-1}(a x)} \cos \left (2 \cos ^{-1}(a x)\right )}{5 a^2}-\frac {e^{\cos ^{-1}(a x)} \sin \left (2 \cos ^{-1}(a x)\right )}{10 a^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 30, normalized size = 0.73 \[ -\frac {e^{\cos ^{-1}(a x)} \left (\sin \left (2 \cos ^{-1}(a x)\right )-2 \cos \left (2 \cos ^{-1}(a x)\right )\right )}{10 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 36, normalized size = 0.88 \[ \frac {{\left (2 \, a^{2} x^{2} - \sqrt {-a^{2} x^{2} + 1} a x - 1\right )} e^{\left (\arccos \left (a x\right )\right )}}{5 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 44, normalized size = 1.07 \[ \frac {2}{5} \, x^{2} e^{\left (\arccos \left (a x\right )\right )} - \frac {\sqrt {-a^{2} x^{2} + 1} x e^{\left (\arccos \left (a x\right )\right )}}{5 \, a} - \frac {e^{\left (\arccos \left (a x\right )\right )}}{5 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arccos \left (a x \right )} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x e^{\left (\arccos \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\mathrm {e}}^{\mathrm {acos}\left (a\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 58, normalized size = 1.41 \[ \begin {cases} \frac {2 x^{2} e^{\operatorname {acos}{\left (a x \right )}}}{5} - \frac {x \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {acos}{\left (a x \right )}}}{5 a} - \frac {e^{\operatorname {acos}{\left (a x \right )}}}{5 a^{2}} & \text {for}\: a \neq 0 \\\frac {x^{2} e^{\frac {\pi }{2}}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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