Optimal. Leaf size=10 \[ -\frac {1}{3} \cos ^3\left (e^x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2282, 2565, 30} \[ -\frac {1}{3} \cos ^3\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 30
Rule 2282
Rule 2565
Rubi steps
\begin {align*} \int e^x \cos ^2\left (e^x\right ) \sin \left (e^x\right ) \, dx &=\operatorname {Subst}\left (\int \cos ^2(x) \sin (x) \, dx,x,e^x\right )\\ &=-\operatorname {Subst}\left (\int x^2 \, dx,x,\cos \left (e^x\right )\right )\\ &=-\frac {1}{3} \cos ^3\left (e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 1.90 \[ -\frac {1}{4} \cos \left (e^x\right )-\frac {1}{12} \cos \left (3 e^x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 7, normalized size = 0.70 \[ -\frac {1}{3} \, \cos \left (e^{x}\right )^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 7, normalized size = 0.70 \[ -\frac {1}{3} \, \cos \left (e^{x}\right )^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 8, normalized size = 0.80 \[ -\frac {\left (\cos ^{3}\left ({\mathrm e}^{x}\right )\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 7, normalized size = 0.70 \[ -\frac {1}{3} \, \cos \left (e^{x}\right )^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 7, normalized size = 0.70 \[ -\frac {{\cos \left ({\mathrm {e}}^x\right )}^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.96, size = 8, normalized size = 0.80 \[ - \frac {\cos ^{3}{\left (e^{x} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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