\(\int \log (e^{\cos (x)}) \, dx\) [157]

Optimal result

Integrand size = 5, antiderivative size = 15 \[ \int \log \left (e^{\cos (x)}\right ) \, dx=-x \cos (x)+x \log \left (e^{\cos (x)}\right )+\sin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2628, 3377, 2717} \[ \int \log \left (e^{\cos (x)}\right ) \, dx=\sin (x)-x \cos (x)+x \log \left (e^{\cos (x)}\right ) \]

[In]

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

[Out]

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

Rule 2628

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

Rule 2717

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

Rule 3377

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*} \text {integral}& = 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] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \log \left (e^{\cos (x)}\right ) \, dx=x \left (-\cos (x)+\log \left (e^{\cos (x)}\right )\right )+\sin (x) \]

[In]

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

[Out]

x*(-Cos[x] + Log[E^Cos[x]]) + Sin[x]

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00

method	result	size
default	\(-x \cos \left (x \right )+x \ln \left ({\mathrm e}^{\cos \left (x \right )}\right )+\sin \left (x \right )\)	\(15\)
risch	\(-x \cos \left (x \right )+x \ln \left ({\mathrm e}^{\cos \left (x \right )}\right )+\sin \left (x \right )\)	\(15\)
parallelrisch	\(-x \cos \left (x \right )+x \ln \left ({\mathrm e}^{\cos \left (x \right )}\right )+\sin \left (x \right )\)	\(15\)
parts	\(-x \cos \left (x \right )+x \ln \left ({\mathrm e}^{\cos \left (x \right )}\right )+\sin \left (x \right )\)	\(15\)

[In]

int(ln(exp(cos(x))),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.13 \[ \int \log \left (e^{\cos (x)}\right ) \, dx=\sin \left (x\right ) \]

[In]

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

[Out]

sin(x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \log \left (e^{\cos (x)}\right ) \, dx=x \log {\left (e^{\cos {\left (x \right )}} \right )} - x \cos {\left (x \right )} + \sin {\left (x \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.13 \[ \int \log \left (e^{\cos (x)}\right ) \, dx=\sin \left (x\right ) \]

[In]

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

[Out]

sin(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.13 \[ \int \log \left (e^{\cos (x)}\right ) \, dx=\sin \left (x\right ) \]

[In]

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

[Out]

sin(x)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.13 \[ \int \log \left (e^{\cos (x)}\right ) \, dx=\sin \left (x\right ) \]

[In]

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

[Out]

sin(x)