3.183 \(\int \cot (x) \log (\sin (x)) \, dx\)

Optimal. Leaf size=9 \[ \frac {1}{2} \log ^2(\sin (x)) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3475, 4338, 2301} \[ \frac {1}{2} \log ^2(\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Sin[x]]^2/2

Rule 2301

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 3475

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

Rule 4338

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*} \int \cot (x) \log (\sin (x)) \, dx &=\operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \log ^2(\sin (x))\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \[ \frac {1}{2} \log ^2(\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Sin[x]]^2/2

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \log \left (\sin \relax (x)\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.19, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \log \left (\sin \relax (x)\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.19, size = 8, normalized size = 0.89 \[ \frac {\ln \left (\sin \relax (x )\right )^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \log \left (\sin \relax (x)\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.38, size = 7, normalized size = 0.78 \[ \frac {{\ln \left (\sin \relax (x)\right )}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(x))^2/2

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________