\(\int \cos (x) \log (\sin (x)) \, dx\) [190]

Optimal result

Integrand size = 6, antiderivative size = 11 \[ \int \cos (x) \log (\sin (x)) \, dx=-\sin (x)+\log (\sin (x)) \sin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2717, 2634} \[ \int \cos (x) \log (\sin (x)) \, dx=\sin (x) \log (\sin (x))-\sin (x) \]

[In]

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

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

Rule 2634

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 2717

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

Rubi steps \begin{align*} \text {integral}& = \log (\sin (x)) \sin (x)-\int \cos (x) \, dx \\ & = -\sin (x)+\log (\sin (x)) \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cos (x) \log (\sin (x)) \, dx=-\sin (x)+\log (\sin (x)) \sin (x) \]

[In]

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

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82

method	result	size
parallelrisch	\(\left (\ln \left (\sin \left (x \right )\right )-1\right ) \sin \left (x \right )\)	\(9\)
derivativedivides	\(-\sin \left (x \right )+\ln \left (\sin \left (x \right )\right ) \sin \left (x \right )\)	\(12\)
default	\(-\sin \left (x \right )+\ln \left (\sin \left (x \right )\right ) \sin \left (x \right )\)	\(12\)
norman	\(\frac {2 \tan \left (\frac {x}{2}\right ) \ln \left (\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan ^{2}\left (\frac {x}{2}\right )}\right )-2 \tan \left (\frac {x}{2}\right )}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(42\)
risch	\(-\frac {i {\mathrm e}^{-i x} \ln \left (2\right )}{2}+\frac {{\mathrm e}^{i x} \pi \operatorname {csgn}\left (\sin \left (x \right )\right )^{3}}{4}-\frac {{\mathrm e}^{-i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )}{4}+\frac {{\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )}{4}+\frac {i {\mathrm e}^{i x} \ln \left (2\right )}{2}+\frac {{\mathrm e}^{-i x} \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{3}}{4}-\frac {{\mathrm e}^{i x} \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{3}}{4}+\frac {{\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{4}-\frac {{\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{4}+\frac {{\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )}{4}-\frac {{\mathrm e}^{-i x} \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )}{4}-\frac {{\mathrm e}^{-i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{4}+\frac {{\mathrm e}^{-i x} \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{4}-\frac {{\mathrm e}^{-i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{4}+\frac {{\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{4}-\frac {{\mathrm e}^{-i x} \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{4}+\frac {{\mathrm e}^{i x} \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{4}-\frac {{\mathrm e}^{-i x} \pi \operatorname {csgn}\left (\sin \left (x \right )\right )^{3}}{4}+\frac {i {\mathrm e}^{i x}}{2}+\frac {{\mathrm e}^{-i x} \pi }{4}-\frac {i {\mathrm e}^{-i x}}{2}+\frac {i {\mathrm e}^{-i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{2}-\frac {i {\mathrm e}^{i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{2}-\frac {{\mathrm e}^{i x} \pi }{4}-\ln \left ({\mathrm e}^{i x}\right ) \sin \left (x \right )\)	\(416\)

[In]

int(cos(x)*ln(sin(x)),x,method=_RETURNVERBOSE)

[Out]

(ln(sin(x))-1)*sin(x)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cos (x) \log (\sin (x)) \, dx=\log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \]

[In]

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

[Out]

log(sin(x))*sin(x) - sin(x)

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \cos (x) \log (\sin (x)) \, dx=\log {\left (\sin {\left (x \right )} \right )} \sin {\left (x \right )} - \sin {\left (x \right )} \]

[In]

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

[Out]

log(sin(x))*sin(x) - sin(x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cos (x) \log (\sin (x)) \, dx=\log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \]

[In]

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

[Out]

log(sin(x))*sin(x) - sin(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cos (x) \log (\sin (x)) \, dx=\log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \]

[In]

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

[Out]

log(sin(x))*sin(x) - sin(x)

Mupad [B] (verification not implemented)

Time = 1.64 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \cos (x) \log (\sin (x)) \, dx=\sin \left (x\right )\,\left (\ln \left (\sin \left (x\right )\right )-1\right ) \]

[In]

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

[Out]

sin(x)*(log(sin(x)) - 1)