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

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

[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 = {2628, 3377, 2717} \begin {gather*} \sin (x)-x \cos (x)+x \log \left (e^{\cos (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[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*} \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 \begin {gather*} x \left (-\cos (x)+\log \left (e^{\cos (x)}\right )\right )+\sin (x) \end {gather*}

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 = 0.04, size = 15, normalized size = 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	\(-x \cos \left (x \right )+x \ln \left ({\mathrm e}^{\cos \left (x \right )}\right )+\sin \left (x \right )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 3.05, size = 2, normalized size = 0.13 \begin {gather*} \sin \left (x\right ) \end {gather*}

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 = 1.22, size = 2, normalized size = 0.13 \begin {gather*} \sin \left (x\right ) \end {gather*}

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 = 0.07, size = 15, normalized size = 1.00 \begin {gather*} x \log {\left (e^{\cos {\left (x \right )}} \right )} - x \cos {\left (x \right )} + \sin {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.46, size = 2, normalized size = 0.13 \begin {gather*} \sin \left (x\right ) \end {gather*}

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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Mupad [B]
time = 0.10, size = 2, normalized size = 0.13 \begin {gather*} \sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)

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