3.633 \(\int \frac{\cot (x)}{\log (\sin (x))} \, dx\)

Optimal. Leaf size=4 \[ \log (\log (\sin (x))) \]

[Out]

Log[Log[Sin[x]]]

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Rubi [A]  time = 0.0200642, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4338, 2302, 29} \[ \log (\log (\sin (x))) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[Sin[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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0105987, size = 4, normalized size = 1. \[ \log (\log (\sin (x))) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[Sin[x]]]

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Maple [A]  time = 0.008, size = 5, normalized size = 1.3 \begin{align*} \ln \left ( \ln \left ( \sin \left ( x \right ) \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(ln(sin(x)))

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Maxima [A]  time = 0.937304, size = 5, normalized size = 1.25 \begin{align*} \log \left (\log \left (\sin \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(sin(x)))

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Fricas [A]  time = 2.3799, size = 24, normalized size = 6. \begin{align*} \log \left (\log \left (\sin \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(sin(x)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )}}{\log{\left (\sin{\left (x \right )} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)/log(sin(x)), x)

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Giac [A]  time = 1.06849, size = 7, normalized size = 1.75 \begin{align*} \log \left ({\left | \log \left (\sin \left (x\right )\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(log(sin(x))))