3.1.44 \(\int \cos (x) \log (\sin (x)) \, dx\) [44]

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 = {2717, 2634} \begin {gather*} \sin (x) \log (\sin (x))-\sin (x) \end {gather*}

Antiderivative was successfully verified.

[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*} \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 \begin {gather*} -\sin (x)+\log (\sin (x)) \sin (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

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Maple [A]
time = 0.13, size = 12, normalized size = 1.09

method	result	size
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}}{2}-\ln \left ({\mathrm e}^{i x}\right ) \sin \left (x \right )+\frac {i {\mathrm e}^{i x}}{2}-\frac {{\mathrm e}^{i x} \pi }{4}+\frac {{\mathrm e}^{-i x} \pi }{4}+\frac {{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )}{4}-\frac {{\mathrm e}^{-i x} \mathrm {csgn}\left (\sin \left (x \right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \pi }{4}-\frac {{\mathrm e}^{-i x} \mathrm {csgn}\left (\sin \left (x \right )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \pi }{4}+\frac {{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )^{2}}{4}+\frac {{\mathrm e}^{-i x} \mathrm {csgn}\left (i \sin \left (x \right )\right )^{2} \mathrm {csgn}\left (\sin \left (x \right )\right ) \pi }{4}-\frac {{\mathrm e}^{-i x} \mathrm {csgn}\left (\sin \left (x \right )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \pi }{4}-\frac {{\mathrm e}^{-i x} \mathrm {csgn}\left (i \sin \left (x \right )\right ) \mathrm {csgn}\left (\sin \left (x \right )\right ) \pi }{4}+\frac {{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )^{2}}{4}+\frac {{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (\sin \left (x \right )\right ) \mathrm {csgn}\left (i \sin \left (x \right )\right )}{4}-\frac {{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (\sin \left (x \right )\right ) \mathrm {csgn}\left (i \sin \left (x \right )\right )^{2}}{4}+\frac {{\mathrm e}^{i x} \pi \mathrm {csgn}\left (\sin \left (x \right )\right )^{3}}{4}-\frac {{\mathrm e}^{-i x} \mathrm {csgn}\left (\sin \left (x \right )\right )^{3} \pi }{4}+\frac {i {\mathrm e}^{i x} \ln \left (2\right )}{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 (2\right )}{2}+\frac {i {\mathrm e}^{-i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{2}+\frac {{\mathrm e}^{i x} \pi \mathrm {csgn}\left (i \sin \left (x \right )\right )^{2}}{4}-\frac {{\mathrm e}^{i x} \pi \mathrm {csgn}\left (i \sin \left (x \right )\right )^{3}}{4}-\frac {{\mathrm e}^{-i x} \mathrm {csgn}\left (i \sin \left (x \right )\right )^{2} \pi }{4}+\frac {{\mathrm e}^{-i x} \mathrm {csgn}\left (i \sin \left (x \right )\right )^{3} \pi }{4}\)	\(416\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.45, size = 11, normalized size = 1.00 \begin {gather*} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \end {gather*}

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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Fricas [A]
time = 0.86, size = 11, normalized size = 1.00 \begin {gather*} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \end {gather*}

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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Sympy [A]
time = 0.20, size = 10, normalized size = 0.91 \begin {gather*} \log {\left (\sin {\left (x \right )} \right )} \sin {\left (x \right )} - \sin {\left (x \right )} \end {gather*}

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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Giac [A]
time = 0.73, size = 11, normalized size = 1.00 \begin {gather*} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \end {gather*}

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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Mupad [B]
time = 0.20, size = 8, normalized size = 0.73 \begin {gather*} \sin \left (x\right )\,\left (\ln \left (\sin \left (x\right )\right )-1\right ) \end {gather*}

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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