\(\int \cot (x) \log (\sin (x)) \, dx\) [183]

Optimal result

Integrand size = 6, antiderivative size = 9 \[ \int \cot (x) \log (\sin (x)) \, dx=\frac {1}{2} \log ^2(\sin (x)) \]

[Out]

1/2*ln(sin(x))^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3556, 4423, 2338} \[ \int \cot (x) \log (\sin (x)) \, dx=\frac {1}{2} \log ^2(\sin (x)) \]

[In]

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

[Out]

Log[Sin[x]]^2/2

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4423

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[1/(b
*c), Subst[Int[SubstFor[1/x, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cot] || EqQ[F, cot])

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sin (x)\right ) \\ & = \frac {1}{2} \log ^2(\sin (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \cot (x) \log (\sin (x)) \, dx=\frac {1}{2} \log ^2(\sin (x)) \]

[In]

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

[Out]

Log[Sin[x]]^2/2

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {\ln \left (\sin \left (x \right )\right )^{2}}{2}\)	\(8\)
default	\(\frac {\ln \left (\sin \left (x \right )\right )^{2}}{2}\)	\(8\)
risch	\(i x \ln \left (2\right )+\frac {x \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )}{2}+\frac {x \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{2}+\frac {x \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{2}+\frac {x \pi \operatorname {csgn}\left (\sin \left (x \right )\right )^{3}}{2}-\frac {x \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{2}+\frac {x \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )}{2}-\frac {x \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{3}}{2}+\frac {x \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{2}+i \left (i \ln \left ({\mathrm e}^{2 i x}-1\right )+x \right ) \ln \left ({\mathrm e}^{i x}\right )-\frac {i \pi \ln \left ({\mathrm e}^{2 i x}-1\right ) \operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{2}-\frac {i \pi \ln \left ({\mathrm e}^{2 i x}-1\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )^{3}}{2}+\frac {i \pi \ln \left ({\mathrm e}^{2 i x}-1\right ) \operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )}{2}+\frac {i \pi \ln \left ({\mathrm e}^{2 i x}-1\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{2}+\frac {i \pi \ln \left ({\mathrm e}^{2 i x}-1\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )}{2}+\frac {i \pi \ln \left ({\mathrm e}^{2 i x}-1\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{2}+\frac {i \pi \ln \left ({\mathrm e}^{2 i x}-1\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{3}}{2}-\ln \left (2\right ) \ln \left ({\mathrm e}^{2 i x}-1\right )-\frac {\pi x}{2}+\frac {x^{2}}{2}+\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )^{2}}{2}+\frac {i \pi \ln \left ({\mathrm e}^{2 i x}-1\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{2}-\frac {i \ln \left ({\mathrm e}^{2 i x}-1\right ) \pi }{2}\)	\(391\)

[In]

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

[Out]

1/2*ln(sin(x))^2

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cot (x) \log (\sin (x)) \, dx=\frac {1}{2} \, \log \left (\sin \left (x\right )\right )^{2} \]

[In]

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

[Out]

1/2*log(sin(x))^2

Sympy [F(-1)]

Timed out. \[ \int \cot (x) \log (\sin (x)) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cot (x) \log (\sin (x)) \, dx=\frac {1}{2} \, \log \left (\sin \left (x\right )\right )^{2} \]

[In]

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

[Out]

1/2*log(sin(x))^2

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cot (x) \log (\sin (x)) \, dx=\frac {1}{2} \, \log \left (\sin \left (x\right )\right )^{2} \]

[In]

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

[Out]

1/2*log(sin(x))^2

Mupad [B] (verification not implemented)

Time = 1.54 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cot (x) \log (\sin (x)) \, dx=\frac {{\ln \left (\sin \left (x\right )\right )}^2}{2} \]

[In]

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

[Out]

log(sin(x))^2/2