Optimal. Leaf size=31 \[ \frac {1}{3} e^{\cos (3 x+1)}-\frac {1}{3} e^{\cos (3 x+1)} \cos (3 x+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 2176
Rule 2194
Rule 4335
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 = 0.12, 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.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 17, normalized size = 0.55 \[ -\frac {1}{3} \, {\left (\cos \left (3 \, x + 1\right ) - 1\right )} e^{\left (\cos \left (3 \, x + 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 17, normalized size = 0.55 \[ -\frac {1}{3} \, {\left (\cos \left (3 \, x + 1\right ) - 1\right )} e^{\left (\cos \left (3 \, x + 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 26, normalized size = 0.84 \[ \frac {{\mathrm e}^{\cos \left (1+3 x \right )}}{3}-\frac {{\mathrm e}^{\cos \left (1+3 x \right )} \cos \left (1+3 x \right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 17, normalized size = 0.55 \[ -\frac {1}{3} \, {\left (\cos \left (3 \, x + 1\right ) - 1\right )} e^{\left (\cos \left (3 \, x + 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, size = 17, normalized size = 0.55 \[ -\frac {{\mathrm {e}}^{\cos \left (3\,x+1\right )}\,\left (\cos \left (3\,x+1\right )-1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.67, size = 26, normalized size = 0.84 \[ - \frac {e^{\cos {\left (3 x + 1 \right )}} \cos {\left (3 x + 1 \right )}}{3} + \frac {e^{\cos {\left (3 x + 1 \right )}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________