Optimal. Leaf size=39 \[ \frac {1}{2} x e^{\cos ^{-1}(a x)}-\frac {\sqrt {1-a^2 x^2} e^{\cos ^{-1}(a x)}}{2 a} \]
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Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4837, 4432} \[ \frac {1}{2} x e^{\cos ^{-1}(a x)}-\frac {\sqrt {1-a^2 x^2} e^{\cos ^{-1}(a x)}}{2 a} \]
Antiderivative was successfully verified.
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Rule 4432
Rule 4837
Rubi steps
\begin {align*} \int e^{\cos ^{-1}(a x)} \, dx &=-\frac {\operatorname {Subst}\left (\int e^x \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=\frac {1}{2} e^{\cos ^{-1}(a x)} x-\frac {e^{\cos ^{-1}(a x)} \sqrt {1-a^2 x^2}}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 0.82 \[ -\frac {\left (\sqrt {1-a^2 x^2}-a x\right ) e^{\cos ^{-1}(a x)}}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 28, normalized size = 0.72 \[ \frac {{\left (a x - \sqrt {-a^{2} x^{2} + 1}\right )} e^{\left (\arccos \left (a x\right )\right )}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 31, normalized size = 0.79 \[ \frac {1}{2} \, x e^{\left (\arccos \left (a x\right )\right )} - \frac {\sqrt {-a^{2} x^{2} + 1} e^{\left (\arccos \left (a x\right )\right )}}{2 \, a} \]
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 \[ \int {\mathrm e}^{\arccos \left (a x \right )}\, dx \]
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 \[ \int e^{\left (\arccos \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\mathrm {e}}^{\mathrm {acos}\left (a\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 37, normalized size = 0.95 \[ \begin {cases} \frac {x e^{\operatorname {acos}{\left (a x \right )}}}{2} - \frac {\sqrt {- a^{2} x^{2} + 1} e^{\operatorname {acos}{\left (a x \right )}}}{2 a} & \text {for}\: a \neq 0 \\x e^{\frac {\pi }{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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