3.103 \(\int \cos (\log (6+3 x)) \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{2} (x+2) \sin (\log (3 (x+2)))+\frac{1}{2} (x+2) \cos (\log (3 (x+2))) \]

[Out]

((2 + x)*Cos[Log[3*(2 + x)]])/2 + ((2 + x)*Sin[Log[3*(2 + x)]])/2

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Rubi [A]  time = 0.0137795, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4476} \[ \frac{1}{2} (x+2) \sin (\log (3 (x+2)))+\frac{1}{2} (x+2) \cos (\log (3 (x+2))) \]

Antiderivative was successfully verified.

[In]

Int[Cos[Log[6 + 3*x]],x]

[Out]

((2 + x)*Cos[Log[3*(2 + x)]])/2 + ((2 + x)*Sin[Log[3*(2 + x)]])/2

Rule 4476

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[(x*Cos[d*(a + b*Log[c*x^n])])/(b^2*d^2
*n^2 + 1), x] + Simp[(b*d*n*x*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 + 1), x] /; FreeQ[{a, b, c, d, n}, x] &&
NeQ[b^2*d^2*n^2 + 1, 0]

Rubi steps

\begin{align*} \int \cos (\log (6+3 x)) \, dx &=\frac{1}{3} \operatorname{Subst}(\int \cos (\log (x)) \, dx,x,6+3 x)\\ &=\frac{1}{2} (2+x) \cos (\log (3 (2+x)))+\frac{1}{2} (2+x) \sin (\log (3 (2+x)))\\ \end{align*}

Mathematica [A]  time = 0.0119223, size = 22, normalized size = 0.76 \[ \frac{1}{2} (x+2) (\sin (\log (3 (x+2)))+\cos (\log (3 (x+2)))) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[Log[6 + 3*x]],x]

[Out]

((2 + x)*(Cos[Log[3*(2 + x)]] + Sin[Log[3*(2 + x)]]))/2

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Maple [C]  time = 0.037, size = 34, normalized size = 1.2 \begin{align*} \left ({\frac{1}{4}}-{\frac{i}{4}} \right ) \left ( 2+x \right ) \left ( 6+3\,x \right ) ^{i}+{\frac{ \left ({\frac{1}{4}}+{\frac{i}{4}} \right ) \left ( 2+x \right ) }{ \left ( 6+3\,x \right ) ^{i}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(ln(6+3*x)),x)

[Out]

(1/4-1/4*I)*(2+x)*(6+3*x)^I+(1/4+1/4*I)*(2+x)/((6+3*x)^I)

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Maxima [A]  time = 0.968133, size = 27, normalized size = 0.93 \begin{align*} \frac{1}{2} \,{\left (x + 2\right )}{\left (\cos \left (\log \left (3 \, x + 6\right )\right ) + \sin \left (\log \left (3 \, x + 6\right )\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(log(6+3*x)),x, algorithm="maxima")

[Out]

1/2*(x + 2)*(cos(log(3*x + 6)) + sin(log(3*x + 6)))

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Fricas [A]  time = 0.469818, size = 85, normalized size = 2.93 \begin{align*} \frac{1}{2} \,{\left (x + 2\right )} \cos \left (\log \left (3 \, x + 6\right )\right ) + \frac{1}{2} \,{\left (x + 2\right )} \sin \left (\log \left (3 \, x + 6\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(log(6+3*x)),x, algorithm="fricas")

[Out]

1/2*(x + 2)*cos(log(3*x + 6)) + 1/2*(x + 2)*sin(log(3*x + 6))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (\log{\left (3 x + 6 \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(ln(6+3*x)),x)

[Out]

Integral(cos(log(3*x + 6)), x)

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Giac [A]  time = 1.14413, size = 34, normalized size = 1.17 \begin{align*} \frac{1}{2} \,{\left (x + 2\right )} \cos \left (\log \left (3 \, x + 6\right )\right ) + \frac{1}{2} \,{\left (x + 2\right )} \sin \left (\log \left (3 \, x + 6\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(log(6+3*x)),x, algorithm="giac")

[Out]

1/2*(x + 2)*cos(log(3*x + 6)) + 1/2*(x + 2)*sin(log(3*x + 6))