### 3.181 $$\int \frac{\cot (x)}{\log (e^{\sin (x)})} \, dx$$

Optimal. Leaf size=37 $\frac{\log \left (\log \left (e^{\sin (x)}\right )\right )}{\sin (x)-\log \left (e^{\sin (x)}\right )}-\frac{\log (\sin (x))}{\sin (x)-\log \left (e^{\sin (x)}\right )}$

[Out]

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

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Rubi [A]  time = 0.0281907, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {4338, 2160, 2157, 29} $\frac{\log \left (\log \left (e^{\sin (x)}\right )\right )}{\sin (x)-\log \left (e^{\sin (x)}\right )}-\frac{\log (\sin (x))}{\sin (x)-\log \left (e^{\sin (x)}\right )}$

Antiderivative was successfully veriﬁed.

[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 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 2160

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 2157

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 29

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

Rubi steps

\begin{align*} \int \frac{\cot (x)}{\log \left (e^{\sin (x)}\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \log \left (e^x\right )} \, dx,x,\sin (x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sin (x)\right )}{-\log \left (e^{\sin (x)}\right )+\sin (x)}+\frac{\operatorname{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{\operatorname{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]  time = 0.0347289, size = 25, normalized size = 0.68 $\frac{\log \left (\log \left (e^{\sin (x)}\right )\right )-\log (\sin (x))}{\sin (x)-\log \left (e^{\sin (x)}\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.02, size = 35, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sin \left ( x \right ) \right ) }{\ln \left ({{\rm e}^{\sin \left ( x \right ) }} \right ) -\sin \left ( x \right ) }}-{\frac{\ln \left ( \ln \left ({{\rm e}^{\sin \left ( x \right ) }} \right ) \right ) }{\ln \left ({{\rm e}^{\sin \left ( x \right ) }} \right ) -\sin \left ( x \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00954, size = 8, normalized size = 0.22 \begin{align*} -\frac{1}{\sin \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/sin(x)

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Fricas [A]  time = 2.21378, size = 15, normalized size = 0.41 \begin{align*} -\frac{1}{\sin \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/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 (e^{\sin{\left (x \right )}} \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)/log(exp(sin(x))), x)

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Giac [A]  time = 1.29746, size = 8, normalized size = 0.22 \begin{align*} -\frac{1}{\sin \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/sin(x)