3.546 \(\int \frac{\cos ^3(\frac{x}{3})}{\sqrt{e^x}} \, dx\)

Optimal. Leaf size=79 \[ \frac{32 \sin \left (\frac{x}{3}\right )}{65 \sqrt{e^x}}-\frac{2 \cos ^3\left (\frac{x}{3}\right )}{5 \sqrt{e^x}}-\frac{48 \cos \left (\frac{x}{3}\right )}{65 \sqrt{e^x}}+\frac{4 \sin \left (\frac{x}{3}\right ) \cos ^2\left (\frac{x}{3}\right )}{5 \sqrt{e^x}} \]

[Out]

(-48*Cos[x/3])/(65*Sqrt[E^x]) - (2*Cos[x/3]^3)/(5*Sqrt[E^x]) + (32*Sin[x/3])/(65*Sqrt[E^x]) + (4*Cos[x/3]^2*Si
n[x/3])/(5*Sqrt[E^x])

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Rubi [A]  time = 0.0449323, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2281, 4435, 4433} \[ \frac{32 \sin \left (\frac{x}{3}\right )}{65 \sqrt{e^x}}-\frac{2 \cos ^3\left (\frac{x}{3}\right )}{5 \sqrt{e^x}}-\frac{48 \cos \left (\frac{x}{3}\right )}{65 \sqrt{e^x}}+\frac{4 \sin \left (\frac{x}{3}\right ) \cos ^2\left (\frac{x}{3}\right )}{5 \sqrt{e^x}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x/3]^3/Sqrt[E^x],x]

[Out]

(-48*Cos[x/3])/(65*Sqrt[E^x]) - (2*Cos[x/3]^3)/(5*Sqrt[E^x]) + (32*Sin[x/3])/(65*Sqrt[E^x]) + (4*Cos[x/3]^2*Si
n[x/3])/(5*Sqrt[E^x])

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 4435

Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*
x))*Cos[d + e*x]^m)/(e^2*m^2 + b^2*c^2*Log[F]^2), x] + (Dist[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^2*Log[F]^2), Int
[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x] + Simp[(e*m*F^(c*(a + b*x))*Sin[d + e*x]*Cos[d + e*x]^(m - 1))/(
e^2*m^2 + b^2*c^2*Log[F]^2), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[
m, 1]

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]

Rubi steps

\begin{align*} \int \frac{\cos ^3\left (\frac{x}{3}\right )}{\sqrt{e^x}} \, dx &=\frac{e^{x/2} \int e^{-x/2} \cos ^3\left (\frac{x}{3}\right ) \, dx}{\sqrt{e^x}}\\ &=-\frac{2 \cos ^3\left (\frac{x}{3}\right )}{5 \sqrt{e^x}}+\frac{4 \cos ^2\left (\frac{x}{3}\right ) \sin \left (\frac{x}{3}\right )}{5 \sqrt{e^x}}+\frac{\left (8 e^{x/2}\right ) \int e^{-x/2} \cos \left (\frac{x}{3}\right ) \, dx}{15 \sqrt{e^x}}\\ &=-\frac{48 \cos \left (\frac{x}{3}\right )}{65 \sqrt{e^x}}-\frac{2 \cos ^3\left (\frac{x}{3}\right )}{5 \sqrt{e^x}}+\frac{32 \sin \left (\frac{x}{3}\right )}{65 \sqrt{e^x}}+\frac{4 \cos ^2\left (\frac{x}{3}\right ) \sin \left (\frac{x}{3}\right )}{5 \sqrt{e^x}}\\ \end{align*}

Mathematica [A]  time = 0.0380714, size = 36, normalized size = 0.46 \[ \frac{90 \sin \left (\frac{x}{3}\right )+26 \sin (x)-135 \cos \left (\frac{x}{3}\right )-13 \cos (x)}{130 \sqrt{e^x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x/3]^3/Sqrt[E^x],x]

[Out]

(-135*Cos[x/3] - 13*Cos[x] + 90*Sin[x/3] + 26*Sin[x])/(130*Sqrt[E^x])

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Maple [A]  time = 0.039, size = 38, normalized size = 0.5 \begin{align*} -{\frac{\cos \left ( x \right ) }{10}{{\rm e}^{-{\frac{x}{2}}}}}+{\frac{\sin \left ( x \right ) }{5}{{\rm e}^{-{\frac{x}{2}}}}}-{\frac{27}{26}{{\rm e}^{-{\frac{x}{2}}}}\cos \left ({\frac{x}{3}} \right ) }+{\frac{9}{13}{{\rm e}^{-{\frac{x}{2}}}}\sin \left ({\frac{x}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*exp(-1/2*x)*cos(x)+1/5*exp(-1/2*x)*sin(x)-27/26*exp(-1/2*x)*cos(1/3*x)+9/13*exp(-1/2*x)*sin(1/3*x)

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Maxima [A]  time = 0.950952, size = 36, normalized size = 0.46 \begin{align*} -\frac{1}{130} \,{\left (135 \, \cos \left (\frac{1}{3} \, x\right ) + 13 \, \cos \left (x\right ) - 90 \, \sin \left (\frac{1}{3} \, x\right ) - 26 \, \sin \left (x\right )\right )} e^{\left (-\frac{1}{2} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/130*(135*cos(1/3*x) + 13*cos(x) - 90*sin(1/3*x) - 26*sin(x))*e^(-1/2*x)

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Fricas [A]  time = 1.90172, size = 138, normalized size = 1.75 \begin{align*} \frac{4}{65} \,{\left (13 \, \cos \left (\frac{1}{3} \, x\right )^{2} + 8\right )} e^{\left (-\frac{1}{2} \, x\right )} \sin \left (\frac{1}{3} \, x\right ) - \frac{2}{65} \,{\left (13 \, \cos \left (\frac{1}{3} \, x\right )^{3} + 24 \, \cos \left (\frac{1}{3} \, x\right )\right )} e^{\left (-\frac{1}{2} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/65*(13*cos(1/3*x)^2 + 8)*e^(-1/2*x)*sin(1/3*x) - 2/65*(13*cos(1/3*x)^3 + 24*cos(1/3*x))*e^(-1/2*x)

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Sympy [A]  time = 2.64452, size = 76, normalized size = 0.96 \begin{align*} \frac{32 \sin ^{3}{\left (\frac{x}{3} \right )}}{65 \sqrt{e^{x}}} - \frac{48 \sin ^{2}{\left (\frac{x}{3} \right )} \cos{\left (\frac{x}{3} \right )}}{65 \sqrt{e^{x}}} + \frac{84 \sin{\left (\frac{x}{3} \right )} \cos ^{2}{\left (\frac{x}{3} \right )}}{65 \sqrt{e^{x}}} - \frac{74 \cos ^{3}{\left (\frac{x}{3} \right )}}{65 \sqrt{e^{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32*sin(x/3)**3/(65*sqrt(exp(x))) - 48*sin(x/3)**2*cos(x/3)/(65*sqrt(exp(x))) + 84*sin(x/3)*cos(x/3)**2/(65*sqr
t(exp(x))) - 74*cos(x/3)**3/(65*sqrt(exp(x)))

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Giac [A]  time = 1.11083, size = 45, normalized size = 0.57 \begin{align*} -\frac{9}{26} \,{\left (3 \, \cos \left (\frac{1}{3} \, x\right ) - 2 \, \sin \left (\frac{1}{3} \, x\right )\right )} e^{\left (-\frac{1}{2} \, x\right )} - \frac{1}{10} \,{\left (\cos \left (x\right ) - 2 \, \sin \left (x\right )\right )} e^{\left (-\frac{1}{2} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/26*(3*cos(1/3*x) - 2*sin(1/3*x))*e^(-1/2*x) - 1/10*(cos(x) - 2*sin(x))*e^(-1/2*x)