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

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

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Rubi [A]  time = 0.02, antiderivative size = 4, normalized size of antiderivative = 1.00, 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 29

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

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 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 (\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*}

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Mathematica [A]  time = 0.01, size = 4, normalized size = 1.00 \[ \log (\log (\sin (x))) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[Sin[x]]]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot (x)}{\log (\sin (x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 1.19, size = 4, normalized size = 1.00 \[ \log \left (\log \left (\sin \relax (x)\right )\right ) \]

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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giac [A]  time = 0.95, size = 5, normalized size = 1.25 \[ \log \left ({\left | \log \left (\sin \relax (x)\right ) \right |}\right ) \]

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))))

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maple [A]  time = 0.18, size = 5, normalized size = 1.25

method	result	size
derivativedivides	\(\ln \left (\ln \left (\sin \relax (x )\right )\right )\)	\(5\)
default	\(\ln \left (\ln \left (\sin \relax (x )\right )\right )\)	\(5\)
risch	\(\ln \left (-\frac {i \pi \,\mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i \sin \relax (x )\right )}{2}-\frac {i \pi \mathrm {csgn}\left (i \sin \relax (x )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \mathrm {csgn}\left (\sin \relax (x )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right )}{2}-\frac {i \pi \mathrm {csgn}\left (\sin \relax (x )\right )^{3}}{2}-\frac {i \pi \mathrm {csgn}\left (\sin \relax (x )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i \sin \relax (x )\right )^{2}}{2}+\frac {i \pi \mathrm {csgn}\left (i \sin \relax (x )\right )^{3}}{2}+\frac {i \pi }{2}+\ln \relax (2)-\ln \left ({\mathrm e}^{2 i x}-1\right )+\ln \left ({\mathrm e}^{i x}\right )\right )\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/ln(sin(x)),x,method=_RETURNVERBOSE)

[Out]

ln(ln(sin(x)))

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maxima [A]  time = 0.42, size = 4, normalized size = 1.00 \[ \log \left (\log \left (\sin \relax (x)\right )\right ) \]

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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mupad [B]  time = 0.40, size = 4, normalized size = 1.00 \[ \ln \left (\ln \left (\sin \relax (x)\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(sin(x)))

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

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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