\(\int \frac {\cot (x)}{\log (e^{\sin (x)})} \, dx\) [181]

Optimal result

Integrand size = 10, antiderivative size = 37 \[ \int \frac {\cot (x)}{\log \left (e^{\sin (x)}\right )} \, dx=\frac {\log \left (\log \left (e^{\sin (x)}\right )\right )}{-\log \left (e^{\sin (x)}\right )+\sin (x)}-\frac {\log (\sin (x))}{-\log \left (e^{\sin (x)}\right )+\sin (x)} \]

[Out]

ln(ln(exp(sin(x))))/(-ln(exp(sin(x)))+sin(x))-ln(sin(x))/(-ln(exp(sin(x)))+sin(x))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4423, 2191, 2188, 29} \[ \int \frac {\cot (x)}{\log \left (e^{\sin (x)}\right )} \, dx=\frac {\log \left (\log \left (e^{\sin (x)}\right )\right )}{\sin (x)-\log \left (e^{\sin (x)}\right )}-\frac {\log (\sin (x))}{\sin (x)-\log \left (e^{\sin (x)}\right )} \]

[In]

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

[Out]

Log[Log[E^Sin[x]]]/(-Log[E^Sin[x]] + Sin[x]) - Log[Sin[x]]/(-Log[E^Sin[x]] + Sin[x])

Rule 29

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

Rule 2188

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 2191

Int[1/((u_)*(v_)), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Dist[b/(b*u - a*v), Int[1
/v, x], x] - Dist[a/(b*u - a*v), Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, 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 \log \left (e^x\right )} \, dx,x,\sin (x)\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,\sin (x)\right )}{-\log \left (e^{\sin (x)}\right )+\sin (x)}+\frac {\text {Subst}\left (\int \frac {1}{\log \left (e^x\right )} \, dx,x,\sin (x)\right )}{-\log \left (e^{\sin (x)}\right )+\sin (x)} \\ & = \frac {\log (\sin (x))}{\log \left (e^{\sin (x)}\right )-\sin (x)}+\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (e^{\sin (x)}\right )\right )}{-\log \left (e^{\sin (x)}\right )+\sin (x)} \\ & = -\frac {\log \left (\log \left (e^{\sin (x)}\right )\right )}{\log \left (e^{\sin (x)}\right )-\sin (x)}+\frac {\log (\sin (x))}{\log \left (e^{\sin (x)}\right )-\sin (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.68 \[ \int \frac {\cot (x)}{\log \left (e^{\sin (x)}\right )} \, dx=\frac {\log \left (\log \left (e^{\sin (x)}\right )\right )-\log (\sin (x))}{-\log \left (e^{\sin (x)}\right )+\sin (x)} \]

[In]

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

[Out]

(Log[Log[E^Sin[x]]] - Log[Sin[x]])/(-Log[E^Sin[x]] + Sin[x])

Maple [A] (verified)

Time = 2.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {\ln \left (\sin \left (x \right )\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}-\frac {\ln \left (\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}\)	\(35\)
risch	\(\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}-\frac {\ln \left ({\mathrm e}^{2 i x}+2 i \left (\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )\right ) {\mathrm e}^{i x}-1\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}\)	\(80\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.16 \[ \int \frac {\cot (x)}{\log \left (e^{\sin (x)}\right )} \, dx=-\frac {1}{\sin \left (x\right )} \]

[In]

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

[Out]

-1/sin(x)

Sympy [F]

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

[In]

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

[Out]

Integral(cot(x)/log(exp(sin(x))), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.16 \[ \int \frac {\cot (x)}{\log \left (e^{\sin (x)}\right )} \, dx=-\frac {1}{\sin \left (x\right )} \]

[In]

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

[Out]

-1/sin(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.16 \[ \int \frac {\cot (x)}{\log \left (e^{\sin (x)}\right )} \, dx=-\frac {1}{\sin \left (x\right )} \]

[In]

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

[Out]

-1/sin(x)

Mupad [B] (verification not implemented)

Time = 1.83 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.16 \[ \int \frac {\cot (x)}{\log \left (e^{\sin (x)}\right )} \, dx=-\frac {1}{\sin \left (x\right )} \]

[In]

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

[Out]

-1/sin(x)