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

Optimal. Leaf size=6 \[ \log (\log (e \sin (x))) \]

[Out]

Log[Log[E*Sin[x]]]

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Rubi [A]  time = 0.0163892, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4338, 31} \[ \log (\log (\sin (x))+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0135556, size = 6, normalized size = 1. \[ \log (\log (\sin (x))+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + Log[Sin[x]]]

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Maple [A]  time = 0.013, size = 7, normalized size = 1.2 \begin{align*} \ln \left ( \ln \left ( E\sin \left ( x \right ) \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(ln(E*sin(x)))

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Maxima [A]  time = 1.00412, size = 8, normalized size = 1.33 \begin{align*} \log \left (\log \left (E \sin \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(E*sin(x)))

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Fricas [A]  time = 2.24403, size = 27, normalized size = 4.5 \begin{align*} \log \left (\log \left (E \sin \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(E*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 )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)/(log(sin(x)) + 1), x)

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Giac [A]  time = 1.25981, size = 8, normalized size = 1.33 \begin{align*} \log \left (\log \left (E \sin \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(E*sin(x)))