\(\int \frac {\cos (\frac {3 x}{2})}{\sqrt [4]{3^{3 x}}} \, dx\) [543]

Optimal result

Integrand size = 16, antiderivative size = 57 \[ \int \frac {\cos \left (\frac {3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx=-\frac {4 \cos \left (\frac {3 x}{2}\right ) \log (3)}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}+\frac {8 \sin \left (\frac {3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )} \]

[Out]

-4/3*cos(3/2*x)*ln(3)/(3^(3*x))^(1/4)/(4+ln(3)^2)+8/3*sin(3/2*x)/(3^(3*x))^(1/4)/(4+ln(3)^2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2319, 4518} \[ \int \frac {\cos \left (\frac {3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx=\frac {8 \sin \left (\frac {3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}-\frac {4 \log (3) \cos \left (\frac {3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )} \]

[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 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 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]

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

Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.65 \[ \int \frac {\cos \left (\frac {3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx=-\frac {4 \left (\cos \left (\frac {3 x}{2}\right ) \log (3)-2 \sin \left (\frac {3 x}{2}\right )\right )}{3 \sqrt [4]{27^x} \left (4+\log ^2(3)\right )} \]

[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))

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.56

method	result	size
parallelrisch	\(-\frac {4 \left (\cos \left (\frac {3 x}{2}\right ) \ln \left (3\right )-2 \sin \left (\frac {3 x}{2}\right )\right )}{\left (27^{x}\right )^{\frac {1}{4}} \left (3 \ln \left (3\right )^{2}+12\right )}\)	\(32\)
risch	\(-\frac {2 \left (2 \cos \left (\frac {3 x}{2}\right ) \ln \left (3\right )-4 \sin \left (\frac {3 x}{2}\right )\right )}{3 \left (2 i+\ln \left (3\right )\right ) \left (-2 i+\ln \left (3\right )\right ) \left (27^{x}\right )^{\frac {1}{4}}}\)	\(37\)

[In]

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

[Out]

-4*(cos(3/2*x)*ln(3)-2*sin(3/2*x))/(27^x)^(1/4)/(3*ln(3)^2+12)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\cos \left (\frac {3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.23 \[ \int \frac {\cos \left (\frac {3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx=\frac {8 \sin {\left (\frac {3 x}{2} \right )}}{3 \sqrt [4]{3^{3 x}} \log {\left (3 \right )}^{2} + 12 \sqrt [4]{3^{3 x}}} - \frac {4 \log {\left (3 \right )} \cos {\left (\frac {3 x}{2} \right )}}{3 \sqrt [4]{3^{3 x}} \log {\left (3 \right )}^{2} + 12 \sqrt [4]{3^{3 x}}} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.54 \[ \int \frac {\cos \left (\frac {3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx=-\frac {4 \, {\left (\cos \left (\frac {3}{2} \, x\right ) \log \left (3\right ) - 2 \, \sin \left (\frac {3}{2} \, x\right )\right )}}{3 \, {\left (\log \left (3\right )^{2} + 4\right )} 3^{\frac {3}{4} \, x}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.68 \[ \int \frac {\cos \left (\frac {3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx=-\frac {4 \, {\left (\frac {\cos \left (\frac {3}{2} \, x\right ) \log \left (3\right )}{\log \left (3\right )^{2} + 4} - \frac {2 \, \sin \left (\frac {3}{2} \, x\right )}{\log \left (3\right )^{2} + 4}\right )}}{3 \cdot 3^{\frac {3}{4} \, x}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.58 \[ \int \frac {\cos \left (\frac {3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx=\frac {\frac {3\,\sin \left (\frac {3\,x}{2}\right )}{2}-\frac {3\,\cos \left (\frac {3\,x}{2}\right )\,\ln \left (3\right )}{4}}{3^{\frac {3\,x}{4}}\,\left (\frac {9\,{\ln \left (3\right )}^2}{16}+\frac {9}{4}\right )} \]

[In]

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

[Out]

((3*sin((3*x)/2))/2 - (3*cos((3*x)/2)*log(3))/4)/(3^((3*x)/4)*((9*log(3)^2)/16 + 9/4))