3.157 \(\int \log (e^{\cos (x)}) \, dx\)

Optimal. Leaf size=15 \[ \sin (x)-x \cos (x)+x \log \left (e^{\cos (x)}\right ) \]

[Out]

-x*cos(x)+x*ln(exp(cos(x)))+sin(x)

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2548, 3296, 2637} \[ \sin (x)-x \cos (x)+x \log \left (e^{\cos (x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[E^Cos[x]],x]

[Out]

-(x*Cos[x]) + x*Log[E^Cos[x]] + Sin[x]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \log \left (e^{\cos (x)}\right ) \, dx &=x \log \left (e^{\cos (x)}\right )+\int x \sin (x) \, dx\\ &=-x \cos (x)+x \log \left (e^{\cos (x)}\right )+\int \cos (x) \, dx\\ &=-x \cos (x)+x \log \left (e^{\cos (x)}\right )+\sin (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \sin (x)+x \left (\log \left (e^{\cos (x)}\right )-\cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[E^Cos[x]],x]

[Out]

x*(-Cos[x] + Log[E^Cos[x]]) + Sin[x]

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fricas [A]  time = 0.40, size = 2, normalized size = 0.13 \[ \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(exp(cos(x))),x, algorithm="fricas")

[Out]

sin(x)

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giac [A]  time = 0.00, size = 2, normalized size = 0.13 \[ \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(exp(cos(x))),x, algorithm="giac")

[Out]

sin(x)

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maple [A]  time = 0.02, size = 15, normalized size = 1.00 \[ -x \cos \relax (x )+x \ln \left ({\mathrm e}^{\cos \relax (x )}\right )+\sin \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(exp(cos(x))),x)

[Out]

-x*cos(x)+x*ln(exp(cos(x)))+sin(x)

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maxima [A]  time = 0.63, size = 2, normalized size = 0.13 \[ \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(exp(cos(x))),x, algorithm="maxima")

[Out]

sin(x)

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mupad [B]  time = 0.10, size = 2, normalized size = 0.13 \[ \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(exp(cos(x))),x)

[Out]

sin(x)

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sympy [A]  time = 0.20, size = 2, normalized size = 0.13 \[ \sin {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(exp(cos(x))),x)

[Out]

sin(x)

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