\(\int \frac {\cos (\frac {x}{2})+\sin (\frac {x}{2})}{\sqrt [3]{e^x}} \, dx\) [542]

Optimal result

Integrand size = 21, antiderivative size = 35 \[ \int \frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}{\sqrt [3]{e^x}} \, dx=-\frac {30 \cos \left (\frac {x}{2}\right )}{13 \sqrt [3]{e^x}}+\frac {6 \sin \left (\frac {x}{2}\right )}{13 \sqrt [3]{e^x}} \]

[Out]

-30/13*cos(1/2*x)/exp(x)^(1/3)+6/13*sin(1/2*x)/exp(x)^(1/3)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2319, 6874, 4518, 4517} \[ \int \frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}{\sqrt [3]{e^x}} \, dx=\frac {6 \sin \left (\frac {x}{2}\right )}{13 \sqrt [3]{e^x}}-\frac {30 \cos \left (\frac {x}{2}\right )}{13 \sqrt [3]{e^x}} \]

[In]

Int[(Cos[x/2] + Sin[x/2])/(E^x)^(1/3),x]

[Out]

(-30*Cos[x/2])/(13*(E^x)^(1/3)) + (6*Sin[x/2])/(13*(E^x)^(1/3))

Rule 2319

Int[(u_.)*((a_.)*(F_)^(v_))^(n_), x_Symbol] :> Dist[(a*F^v)^n/F^(n*v), Int[u*F^(n*v), x], x] /; FreeQ[{F, a, n
}, x] &&  !IntegerQ[n]

Rule 4517

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(S
in[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] - Simp[e*F^(c*(a + b*x))*(Cos[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 4518

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(C
os[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] + Simp[e*F^(c*(a + b*x))*(Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}{\sqrt [3]{e^x}} \, dx=\frac {6 \left (-5 \cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )}{13 \sqrt [3]{e^x}} \]

[In]

Integrate[(Cos[x/2] + Sin[x/2])/(E^x)^(1/3),x]

[Out]

(6*(-5*Cos[x/2] + Sin[x/2]))/(13*(E^x)^(1/3))

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.51

method	result	size
parallelrisch	\(\frac {-\frac {30 \cos \left (\frac {x}{2}\right )}{13}+\frac {6 \sin \left (\frac {x}{2}\right )}{13}}{\left ({\mathrm e}^{x}\right )^{\frac {1}{3}}}\)	\(18\)
default	\(-\frac {30 \,{\mathrm e}^{-\frac {x}{3}} \cos \left (\frac {x}{2}\right )}{13}+\frac {6 \,{\mathrm e}^{-\frac {x}{3}} \sin \left (\frac {x}{2}\right )}{13}\)	\(22\)
parts	\(-\frac {30 \,{\mathrm e}^{-\frac {x}{3}} \cos \left (\frac {x}{2}\right )}{13}+\frac {6 \,{\mathrm e}^{-\frac {x}{3}} \sin \left (\frac {x}{2}\right )}{13}\)	\(22\)
risch	\(\frac {\left (-\frac {15}{169}-\frac {3 i}{169}\right ) \left (\left (25-5 i\right ) \cos \left (\frac {x}{2}\right )+\left (-5+i\right ) \sin \left (\frac {x}{2}\right )\right )}{\left ({\mathrm e}^{x}\right )^{\frac {1}{3}}}\)	\(26\)

[In]

int((cos(1/2*x)+sin(1/2*x))/exp(x)^(1/3),x,method=_RETURNVERBOSE)

[Out]

6/13/exp(x)^(1/3)*(-5*cos(1/2*x)+sin(1/2*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60 \[ \int \frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}{\sqrt [3]{e^x}} \, dx=-\frac {30}{13} \, \cos \left (\frac {1}{2} \, x\right ) e^{\left (-\frac {1}{3} \, x\right )} + \frac {6}{13} \, e^{\left (-\frac {1}{3} \, x\right )} \sin \left (\frac {1}{2} \, x\right ) \]

[In]

integrate((cos(1/2*x)+sin(1/2*x))/exp(x)^(1/3),x, algorithm="fricas")

[Out]

-30/13*cos(1/2*x)*e^(-1/3*x) + 6/13*e^(-1/3*x)*sin(1/2*x)

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}{\sqrt [3]{e^x}} \, dx=\frac {6 \sin {\left (\frac {x}{2} \right )}}{13 \sqrt [3]{e^{x}}} - \frac {30 \cos {\left (\frac {x}{2} \right )}}{13 \sqrt [3]{e^{x}}} \]

[In]

integrate((cos(1/2*x)+sin(1/2*x))/exp(x)**(1/3),x)

[Out]

6*sin(x/2)/(13*exp(x)**(1/3)) - 30*cos(x/2)/(13*exp(x)**(1/3))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11 \[ \int \frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}{\sqrt [3]{e^x}} \, dx=-\frac {6}{13} \, {\left (3 \, \cos \left (\frac {1}{2} \, x\right ) + 2 \, \sin \left (\frac {1}{2} \, x\right )\right )} e^{\left (-\frac {1}{3} \, x\right )} - \frac {6}{13} \, {\left (2 \, \cos \left (\frac {1}{2} \, x\right ) - 3 \, \sin \left (\frac {1}{2} \, x\right )\right )} e^{\left (-\frac {1}{3} \, x\right )} \]

[In]

integrate((cos(1/2*x)+sin(1/2*x))/exp(x)^(1/3),x, algorithm="maxima")

[Out]

-6/13*(3*cos(1/2*x) + 2*sin(1/2*x))*e^(-1/3*x) - 6/13*(2*cos(1/2*x) - 3*sin(1/2*x))*e^(-1/3*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11 \[ \int \frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}{\sqrt [3]{e^x}} \, dx=-\frac {6}{13} \, {\left (3 \, \cos \left (\frac {1}{2} \, x\right ) + 2 \, \sin \left (\frac {1}{2} \, x\right )\right )} e^{\left (-\frac {1}{3} \, x\right )} - \frac {6}{13} \, {\left (2 \, \cos \left (\frac {1}{2} \, x\right ) - 3 \, \sin \left (\frac {1}{2} \, x\right )\right )} e^{\left (-\frac {1}{3} \, x\right )} \]

[In]

integrate((cos(1/2*x)+sin(1/2*x))/exp(x)^(1/3),x, algorithm="giac")

[Out]

-6/13*(3*cos(1/2*x) + 2*sin(1/2*x))*e^(-1/3*x) - 6/13*(2*cos(1/2*x) - 3*sin(1/2*x))*e^(-1/3*x)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.54 \[ \int \frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}{\sqrt [3]{e^x}} \, dx=-\frac {6\,{\mathrm {e}}^{-\frac {x}{3}}\,\left (5\,\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{13} \]

[In]

int((cos(x/2) + sin(x/2))/exp(x)^(1/3),x)

[Out]

-(6*exp(-x/3)*(5*cos(x/2) - sin(x/2)))/13