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

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

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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]

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 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]

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 6742

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

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*}

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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]

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))

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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]

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

[Out]

Could not integrate

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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]

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)

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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]

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)

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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]

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

[Out]

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

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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]

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)

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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]

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

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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]

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))

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