3.190 \(\int \cos (x) \log (\sin (x)) \, dx\)

Optimal. Leaf size=11 \[ \sin (x) \log (\sin (x))-\sin (x) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2637, 2554} \[ \sin (x) \log (\sin (x))-\sin (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Log[Sin[x]],x]

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

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 \cos (x) \log (\sin (x)) \, dx &=\log (\sin (x)) \sin (x)-\int \cos (x) \, dx\\ &=-\sin (x)+\log (\sin (x)) \sin (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ \sin (x) \log (\sin (x))-\sin (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Log[Sin[x]],x]

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

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fricas [A]  time = 0.48, size = 11, normalized size = 1.00 \[ \log \left (\sin \relax (x)\right ) \sin \relax (x) - \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*log(sin(x)),x, algorithm="fricas")

[Out]

log(sin(x))*sin(x) - sin(x)

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giac [A]  time = 0.16, size = 11, normalized size = 1.00 \[ \log \left (\sin \relax (x)\right ) \sin \relax (x) - \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*log(sin(x)),x, algorithm="giac")

[Out]

log(sin(x))*sin(x) - sin(x)

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maple [A]  time = 0.19, size = 12, normalized size = 1.09 \[ \ln \left (\sin \relax (x )\right ) \sin \relax (x )-\sin \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*ln(sin(x)),x)

[Out]

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

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maxima [A]  time = 0.44, size = 11, normalized size = 1.00 \[ \log \left (\sin \relax (x)\right ) \sin \relax (x) - \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*log(sin(x)),x, algorithm="maxima")

[Out]

log(sin(x))*sin(x) - sin(x)

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mupad [B]  time = 0.41, size = 8, normalized size = 0.73 \[ \sin \relax (x)\,\left (\ln \left (\sin \relax (x)\right )-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(sin(x))*cos(x),x)

[Out]

sin(x)*(log(sin(x)) - 1)

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sympy [A]  time = 0.87, size = 10, normalized size = 0.91 \[ \log {\left (\sin {\relax (x )} \right )} \sin {\relax (x )} - \sin {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*ln(sin(x)),x)

[Out]

log(sin(x))*sin(x) - sin(x)

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