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

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

[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)

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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 = {2319, 4518} \begin {gather*} \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 )} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.05, size = 37, normalized size = 0.65 \begin {gather*} -\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 )} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.05, size = 37, normalized size = 0.65

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 2.01, size = 31, normalized size = 0.54 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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Sympy [A]
time = 0.51, size = 70, normalized size = 1.23 \begin {gather*} \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 {gather*}

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 = 2.00, size = 39, normalized size = 0.68 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.04, size = 33, normalized size = 0.58 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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