3.2.3 \(\int \cos (\log (6+3 x)) \, dx\) [103]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 = {4564} \begin {gather*} \frac {1}{2} (x+2) \sin (\log (3 (x+2)))+\frac {1}{2} (x+2) \cos (\log (3 (x+2))) \end {gather*}

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 4564

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} \text {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*}

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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.76 \begin {gather*} \frac {1}{2} (2+x) (\cos (\log (3 (2+x)))+\sin (\log (3 (2+x)))) \end {gather*}

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] Result contains complex when optimal does not.
time = 0.04, size = 34, normalized size = 1.17

method	result	size
risch	\(\left (\frac {1}{4}-\frac {i}{4}\right ) \left (2+x \right ) \left (6+3 x \right )^{i}+\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2+x \right ) \left (6+3 x \right )^{-i}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(ln(6+3*x)),x,method=_RETURNVERBOSE)

[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 = 0.27, size = 20, normalized size = 0.69 \begin {gather*} \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 {gather*}

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 = 1.96, size = 25, normalized size = 0.86 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cos {\left (\log {\left (3 x + 6 \right )} \right )}\, dx \end {gather*}

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 = 0.41, size = 25, normalized size = 0.86 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 2.17, size = 21, normalized size = 0.72 \begin {gather*} \frac {\sqrt {2}\,\sin \left (\frac {\pi }{4}+\ln \left (3\,x+6\right )\right )\,\left (3\,x+6\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(log(3*x + 6)),x)

[Out]

(2^(1/2)*sin(pi/4 + log(3*x + 6))*(3*x + 6))/6

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