\(\int \frac {\cot (x)}{\log (e \sin (x))} \, dx\) [180]

Optimal result

Integrand size = 10, antiderivative size = 6 \[ \int \frac {\cot (x)}{\log (e \sin (x))} \, dx=\log (\log (e \sin (x))) \]

[Out]

ln(ln(exp(1)*sin(x)))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4423, 31} \[ \int \frac {\cot (x)}{\log (e \sin (x))} \, dx=\log (\log (\sin (x))+1) \]

[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 4423

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*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x+x \log (x)} \, dx,x,\sin (x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\log (\sin (x))\right ) \\ & = \log (1+\log (\sin (x))) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\log (e \sin (x))} \, dx=\log (1+\log (\sin (x))) \]

[In]

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

[Out]

Log[1 + Log[Sin[x]]]

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.33

method	result	size
derivativedivides	\(\ln \left (\ln \left ({\mathrm e} \sin \left (x \right )\right )\right )\)	\(8\)
default	\(\ln \left (\ln \left ({\mathrm e} \sin \left (x \right )\right )\right )\)	\(8\)
risch	\(\ln \left (-\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\sin \left (x \right )\right )^{3}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{3}}{2}+\frac {i \pi }{2}+\ln \left (2\right )-\ln \left ({\mathrm e}^{2 i x}-1\right )+\ln \left ({\mathrm e}^{i x}\right )-1\right )\)	\(152\)

[In]

int(cot(x)/ln(exp(1)*sin(x)),x,method=_RETURNVERBOSE)

[Out]

ln(ln(exp(1)*sin(x)))

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17 \[ \int \frac {\cot (x)}{\log (e \sin (x))} \, dx=\log \left (\log \left (e \sin \left (x\right )\right )\right ) \]

[In]

integrate(cot(x)/log(exp(1)*sin(x)),x, algorithm="fricas")

[Out]

log(log(e*sin(x)))

Sympy [F]

\[ \int \frac {\cot (x)}{\log (e \sin (x))} \, dx=\int \frac {\cot {\left (x \right )}}{\log {\left (\sin {\left (x \right )} \right )} + 1}\, dx \]

[In]

integrate(cot(x)/ln(exp(1)*sin(x)),x)

[Out]

Integral(cot(x)/(log(sin(x)) + 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17 \[ \int \frac {\cot (x)}{\log (e \sin (x))} \, dx=\log \left (\log \left (e \sin \left (x\right )\right )\right ) \]

[In]

integrate(cot(x)/log(exp(1)*sin(x)),x, algorithm="maxima")

[Out]

log(log(e*sin(x)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (7) = 14\).

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 4.00 \[ \int \frac {\cot (x)}{\log (e \sin (x))} \, dx=\frac {1}{2} \, \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\left (\sin \left (x\right )\right ) - 1\right )}^{2} + {\left (\log \left ({\left | \sin \left (x\right ) \right |}\right ) + 1\right )}^{2}\right ) \]

[In]

integrate(cot(x)/log(exp(1)*sin(x)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.72 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\log (e \sin (x))} \, dx=\ln \left (\ln \left (\sin \left (x\right )\right )+1\right ) \]

[In]

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

[Out]

log(log(sin(x)) + 1)