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

Optimal. Leaf size=35 \[ \frac{6 \sin \left (\frac{x}{2}\right )}{13 \sqrt [3]{e^x}}-\frac{30 \cos \left (\frac{x}{2}\right )}{13 \sqrt [3]{e^x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.111145, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, 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]

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 2281

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 6742

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

Rule 4433

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 4432

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]

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 = 0.0585615, 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]

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]  time = 0.014, size = 22, normalized size = 0.6 \begin{align*} -{\frac{30}{13}{{\rm e}^{-{\frac{x}{3}}}}\cos \left ({\frac{x}{2}} \right ) }+{\frac{6}{13}{{\rm e}^{-{\frac{x}{3}}}}\sin \left ({\frac{x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.929995, size = 53, normalized size = 1.51 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 2.17471, size = 80, normalized size = 2.29 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0.829124, size = 29, normalized size = 0.83 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [A]  time = 1.0828, size = 53, normalized size = 1.51 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)