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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 31

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

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Mathematica [A]  time = 0.01, size = 6, normalized size = 1.00 \[ \log (\log (\sin (x))+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + Log[Sin[x]]]

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fricas [A]  time = 0.44, size = 6, normalized size = 1.00 \[ \log \left (\log \left (E \sin \relax (x)\right )\right ) \]

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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giac [B]  time = 0.17, size = 32, normalized size = 5.33 \[ \frac {1}{2} \, \log \left (\frac {1}{4} \, {\left (\pi {\left (\mathrm {sgn}\relax (E) - 1\right )} + \pi {\left (\mathrm {sgn}\left (\sin \relax (x)\right ) - 1\right )}\right )}^{2} + {\left (\log \left ({\left | E \right |}\right ) + \log \left ({\left | \sin \relax (x) \right |}\right )\right )}^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(1/4*(pi*(sgn(E) - 1) + pi*(sgn(sin(x)) - 1))^2 + (log(abs(E)) + log(abs(sin(x))))^2)

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maple [A]  time = 0.40, size = 7, normalized size = 1.17 \[ \ln \left (\ln \left (E \sin \relax (x )\right )\right ) \]

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 = 0.44, size = 6, normalized size = 1.00 \[ \log \left (\log \left (E \sin \relax (x)\right )\right ) \]

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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mupad [B]  time = 0.50, size = 6, normalized size = 1.00 \[ \ln \left (\ln \left (\sin \relax (x)\right )+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/log(exp(1)*sin(x)),x)

[Out]

log(log(sin(x)) + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot {\relax (x )}}{\log {\left (\sin {\relax (x )} \right )} + 1}\, dx \]

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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