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*Log[E^Cos[x]] + Sin[x]

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Rubi [A]  time = 0.0098554, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, 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 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]

Rule 2637

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

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*}

Mathematica [A]  time = 0.009336, size = 15, normalized size = 1. \[ \sin (x)+x \left (\log \left (e^{\cos (x)}\right )-\cos (x)\right ) \]

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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Maple [A]  time = 0.012, size = 15, normalized size = 1. \begin{align*} -x\cos \left ( x \right ) +x\ln \left ({{\rm e}^{\cos \left ( x \right ) }} \right ) +\sin \left ( x \right ) \end{align*}

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 = 0.92759, size = 3, normalized size = 0.2 \begin{align*} \sin \left (x\right ) \end{align*}

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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Fricas [A]  time = 1.07367, size = 11, normalized size = 0.73 \begin{align*} \sin \left (x\right ) \end{align*}

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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Sympy [A]  time = 0.183113, size = 2, normalized size = 0.13 \begin{align*} \sin{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)

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Giac [A]  time = 1.12051, size = 3, normalized size = 0.2 \begin{align*} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)