Optimal. Leaf size=35 \[ \frac {6 \sin \left (\frac {x}{2}\right )}{13 \sqrt [3]{e^x}}-\frac {30 \cos \left (\frac {x}{2}\right )}{13 \sqrt [3]{e^x}} \]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2281, 6742, 4433, 4432} \[ \frac {6 \sin \left (\frac {x}{2}\right )}{13 \sqrt [3]{e^x}}-\frac {30 \cos \left (\frac {x}{2}\right )}{13 \sqrt [3]{e^x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2281
Rule 4432
Rule 4433
Rule 6742
Rubi steps
\begin {align*} \int \frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}{\sqrt [3]{e^x}} \, dx &=\frac {e^{x/3} \int e^{-x/3} \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \, dx}{\sqrt [3]{e^x}}\\ &=\frac {\left (6 e^{x/3}\right ) \operatorname {Subst}\left (\int e^{-2 x} (\cos (3 x)+\sin (3 x)) \, dx,x,\frac {x}{6}\right )}{\sqrt [3]{e^x}}\\ &=\frac {\left (6 e^{x/3}\right ) \operatorname {Subst}\left (\int \left (e^{-2 x} \cos (3 x)+e^{-2 x} \sin (3 x)\right ) \, dx,x,\frac {x}{6}\right )}{\sqrt [3]{e^x}}\\ &=\frac {\left (6 e^{x/3}\right ) \operatorname {Subst}\left (\int e^{-2 x} \cos (3 x) \, dx,x,\frac {x}{6}\right )}{\sqrt [3]{e^x}}+\frac {\left (6 e^{x/3}\right ) \operatorname {Subst}\left (\int e^{-2 x} \sin (3 x) \, dx,x,\frac {x}{6}\right )}{\sqrt [3]{e^x}}\\ &=-\frac {30 \cos \left (\frac {x}{2}\right )}{13 \sqrt [3]{e^x}}+\frac {6 \sin \left (\frac {x}{2}\right )}{13 \sqrt [3]{e^x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 26, normalized size = 0.74 \[ \frac {6 \left (\sin \left (\frac {x}{2}\right )-5 \cos \left (\frac {x}{2}\right )\right )}{13 \sqrt [3]{e^x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}{\sqrt [3]{e^x}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.35, size = 21, normalized size = 0.60 \[ -\frac {30}{13} \, \cos \left (\frac {1}{2} \, x\right ) e^{\left (-\frac {1}{3} \, x\right )} + \frac {6}{13} \, e^{\left (-\frac {1}{3} \, x\right )} \sin \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.60, size = 39, normalized size = 1.11 \[ -\frac {6}{13} \, {\left (3 \, \cos \left (\frac {1}{2} \, x\right ) + 2 \, \sin \left (\frac {1}{2} \, x\right )\right )} e^{\left (-\frac {1}{3} \, x\right )} - \frac {6}{13} \, {\left (2 \, \cos \left (\frac {1}{2} \, x\right ) - 3 \, \sin \left (\frac {1}{2} \, x\right )\right )} e^{\left (-\frac {1}{3} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 22, normalized size = 0.63
method | result | size |
default | \(-\frac {30 \,{\mathrm e}^{-\frac {x}{3}} \cos \left (\frac {x}{2}\right )}{13}+\frac {6 \,{\mathrm e}^{-\frac {x}{3}} \sin \left (\frac {x}{2}\right )}{13}\) | \(22\) |
risch | \(\frac {\left (-\frac {15}{169}-\frac {3 i}{169}\right ) \left (\left (25-5 i\right ) \cos \left (\frac {x}{2}\right )+\left (-5+i\right ) \sin \left (\frac {x}{2}\right )\right )}{\left ({\mathrm e}^{x}\right )^{\frac {1}{3}}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.66, size = 39, normalized size = 1.11 \[ -\frac {6}{13} \, {\left (3 \, \cos \left (\frac {1}{2} \, x\right ) + 2 \, \sin \left (\frac {1}{2} \, x\right )\right )} e^{\left (-\frac {1}{3} \, x\right )} - \frac {6}{13} \, {\left (2 \, \cos \left (\frac {1}{2} \, x\right ) - 3 \, \sin \left (\frac {1}{2} \, x\right )\right )} e^{\left (-\frac {1}{3} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 19, normalized size = 0.54 \[ -\frac {6\,{\mathrm {e}}^{-\frac {x}{3}}\,\left (5\,\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.85, size = 29, normalized size = 0.83 \[ \frac {6 \sin {\left (\frac {x}{2} \right )}}{13 \sqrt [3]{e^{x}}} - \frac {30 \cos {\left (\frac {x}{2} \right )}}{13 \sqrt [3]{e^{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________