3.543 \(\int \frac{\cos (\frac{3 x}{2})}{\sqrt [4]{3^{3 x}}} \, dx\)

Optimal. Leaf size=57 \[ \frac{8 \sin \left (\frac{3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}-\frac{4 \log (3) \cos \left (\frac{3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )} \]

[Out]

(-4*Cos[(3*x)/2]*Log[3])/(3*(3^(3*x))^(1/4)*(4 + Log[3]^2)) + (8*Sin[(3*x)/2])/(3*(3^(3*x))^(1/4)*(4 + Log[3]^
2))

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Rubi [A]  time = 0.0284353, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2281, 4433} \[ \frac{8 \sin \left (\frac{3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}-\frac{4 \log (3) \cos \left (\frac{3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )} \]

Antiderivative was successfully verified.

[In]

Int[Cos[(3*x)/2]/(3^(3*x))^(1/4),x]

[Out]

(-4*Cos[(3*x)/2]*Log[3])/(3*(3^(3*x))^(1/4)*(4 + Log[3]^2)) + (8*Sin[(3*x)/2])/(3*(3^(3*x))^(1/4)*(4 + Log[3]^
2))

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 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 \left (\frac{3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx &=\frac{3^{3 x/4} \int 3^{-3 x/4} \cos \left (\frac{3 x}{2}\right ) \, dx}{\sqrt [4]{3^{3 x}}}\\ &=-\frac{4 \cos \left (\frac{3 x}{2}\right ) \log (3)}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}+\frac{8 \sin \left (\frac{3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}\\ \end{align*}

Mathematica [A]  time = 0.066293, size = 37, normalized size = 0.65 \[ -\frac{4 \left (\log (3) \cos \left (\frac{3 x}{2}\right )-2 \sin \left (\frac{3 x}{2}\right )\right )}{3 \sqrt [4]{27^x} \left (4+\log ^2(3)\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[(3*x)/2]/(3^(3*x))^(1/4),x]

[Out]

(-4*(Cos[(3*x)/2]*Log[3] - 2*Sin[(3*x)/2]))/(3*(27^x)^(1/4)*(4 + Log[3]^2))

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Maple [C]  time = 0.05, size = 55, normalized size = 1. \begin{align*} -{\frac{-4\,i{{\rm e}^{-{\frac{3\,i}{2}}x}}+2\,{{\rm e}^{-3/2\,ix}}\ln \left ( 3 \right ) +4\,i{{\rm e}^{{\frac{3\,i}{2}}x}}+2\,\ln \left ( 3 \right ){{\rm e}^{3/2\,ix}}}{ \left ( 6\,i+3\,\ln \left ( 3 \right ) \right ) \left ( -2\,i+\ln \left ( 3 \right ) \right ) }{\frac{1}{\sqrt [4]{{27}^{x}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3/2*x)/(3^(3*x))^(1/4),x)

[Out]

-2/3/(2*I+ln(3))/(-2*I+ln(3))/(27^x)^(1/4)*(-2*I*exp(-3/2*I*x)+exp(-3/2*I*x)*ln(3)+2*I*exp(3/2*I*x)+ln(3)*exp(
3/2*I*x))

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Maxima [A]  time = 1.41806, size = 42, normalized size = 0.74 \begin{align*} -\frac{4 \,{\left (\cos \left (\frac{3}{2} \, x\right ) \log \left (3\right ) - 2 \, \sin \left (\frac{3}{2} \, x\right )\right )}}{3 \,{\left (\log \left (3\right )^{2} + 4\right )} 3^{\frac{3}{4} \, x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3/2*x)/(3^(3*x))^(1/4),x, algorithm="maxima")

[Out]

-4/3*(cos(3/2*x)*log(3) - 2*sin(3/2*x))/((log(3)^2 + 4)*3^(3/4*x))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3/2*x)/(3^(3*x))^(1/4),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 3.06312, size = 70, normalized size = 1.23 \begin{align*} \frac{8 \sin{\left (\frac{3 x}{2} \right )}}{3 \sqrt [4]{3^{3 x}} \log{\left (3 \right )}^{2} + 12 \sqrt [4]{3^{3 x}}} - \frac{4 \log{\left (3 \right )} \cos{\left (\frac{3 x}{2} \right )}}{3 \sqrt [4]{3^{3 x}} \log{\left (3 \right )}^{2} + 12 \sqrt [4]{3^{3 x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3/2*x)/(3**(3*x))**(1/4),x)

[Out]

8*sin(3*x/2)/(3*(3**(3*x))**(1/4)*log(3)**2 + 12*(3**(3*x))**(1/4)) - 4*log(3)*cos(3*x/2)/(3*(3**(3*x))**(1/4)
*log(3)**2 + 12*(3**(3*x))**(1/4))

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Giac [A]  time = 1.13985, size = 53, normalized size = 0.93 \begin{align*} -\frac{4 \,{\left (\frac{\cos \left (\frac{3}{2} \, x\right ) \log \left (3\right )}{\log \left (3\right )^{2} + 4} - \frac{2 \, \sin \left (\frac{3}{2} \, x\right )}{\log \left (3\right )^{2} + 4}\right )}}{3 \cdot 3^{\frac{3}{4} \, x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3/2*x)/(3^(3*x))^(1/4),x, algorithm="giac")

[Out]

-4/3*(cos(3/2*x)*log(3)/(log(3)^2 + 4) - 2*sin(3/2*x)/(log(3)^2 + 4))/3^(3/4*x)