Optimal. Leaf size=31 \[ \frac{1}{3} e^{\cos (3 x+1)}-\frac{1}{3} e^{\cos (3 x+1)} \cos (3 x+1) \]
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Rubi [A] time = 0.0226498, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4335, 2176, 2194} \[ \frac{1}{3} e^{\cos (3 x+1)}-\frac{1}{3} e^{\cos (3 x+1)} \cos (3 x+1) \]
Antiderivative was successfully verified.
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Rule 4335
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{\cos (1+3 x)} \cos (1+3 x) \sin (1+3 x) \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int e^x x \, dx,x,\cos (1+3 x)\right )\right )\\ &=-\frac{1}{3} e^{\cos (1+3 x)} \cos (1+3 x)+\frac{1}{3} \operatorname{Subst}\left (\int e^x \, dx,x,\cos (1+3 x)\right )\\ &=\frac{1}{3} e^{\cos (1+3 x)}-\frac{1}{3} e^{\cos (1+3 x)} \cos (1+3 x)\\ \end{align*}
Mathematica [A] time = 0.121238, size = 24, normalized size = 0.77 \[ \frac{2}{3} \sin ^2\left (\frac{1}{2} (3 x+1)\right ) e^{\cos (3 x+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 26, normalized size = 0.8 \begin{align*}{\frac{{{\rm e}^{\cos \left ( 1+3\,x \right ) }}}{3}}-{\frac{{{\rm e}^{\cos \left ( 1+3\,x \right ) }}\cos \left ( 1+3\,x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976038, size = 23, normalized size = 0.74 \begin{align*} -\frac{1}{3} \,{\left (\cos \left (3 \, x + 1\right ) - 1\right )} e^{\left (\cos \left (3 \, x + 1\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90291, size = 57, normalized size = 1.84 \begin{align*} -\frac{1}{3} \,{\left (\cos \left (3 \, x + 1\right ) - 1\right )} e^{\left (\cos \left (3 \, x + 1\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.819937, size = 26, normalized size = 0.84 \begin{align*} - \frac{e^{\cos{\left (3 x + 1 \right )}} \cos{\left (3 x + 1 \right )}}{3} + \frac{e^{\cos{\left (3 x + 1 \right )}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10971, size = 23, normalized size = 0.74 \begin{align*} -\frac{1}{3} \,{\left (\cos \left (3 \, x + 1\right ) - 1\right )} e^{\left (\cos \left (3 \, x + 1\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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