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

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

[Out]

Log[Sin[x]]^2/2

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Rubi [A]  time = 0.0158706, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 3, integrand size = 6, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, 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 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])

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]

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 = 0.0040673, size = 9, normalized size = 1. $\frac{1}{2} \log ^2(\sin (x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[Sin[x]]^2/2

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Maple [A]  time = 0.011, size = 8, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( \sin \left ( x \right ) \right ) \right ) ^{2}}{2}} \end{align*}

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

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Maxima [A]  time = 0.995908, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \log \left (\sin \left (x\right )\right )^{2} \end{align*}

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

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Fricas [A]  time = 2.19348, size = 26, normalized size = 2.89 \begin{align*} \frac{1}{2} \, \log \left (\sin \left (x\right )\right )^{2} \end{align*}

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.16333, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \log \left (\sin \left (x\right )\right )^{2} \end{align*}

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