3.723 \(\int e^{-\cot (x)} \csc ^2(x) \, dx\)

Optimal. Leaf size=6 \[ e^{-\cot (x)} \]

[Out]

E^(-Cot[x])

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Rubi [A]  time = 0.0147822, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4344, 2194} \[ e^{-\cot (x)} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^2/E^Cot[x],x]

[Out]

E^(-Cot[x])

Rule 4344

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -Dist[d
/(b*c), Subst[Int[SubstFor[1, Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d], x] /; FunctionOfQ[Cot[c*(
a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Csc] || EqQ[F, csc])

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int e^{-\cot (x)} \csc ^2(x) \, dx &=-\operatorname{Subst}\left (\int e^{-x} \, dx,x,\cot (x)\right )\\ &=e^{-\cot (x)}\\ \end{align*}

Mathematica [A]  time = 0.0762972, size = 6, normalized size = 1. \[ e^{-\cot (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^2/E^Cot[x],x]

[Out]

E^(-Cot[x])

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Maple [A]  time = 0.011, size = 6, normalized size = 1. \begin{align*} \left ({{\rm e}^{\cot \left ( x \right ) }} \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^2/exp(cot(x)),x)

[Out]

1/exp(cot(x))

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Maxima [A]  time = 0.978347, size = 7, normalized size = 1.17 \begin{align*} e^{\left (-\cot \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/exp(cot(x)),x, algorithm="maxima")

[Out]

e^(-cot(x))

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Fricas [A]  time = 2.23891, size = 27, normalized size = 4.5 \begin{align*} e^{\left (-\frac{\cos \left (x\right )}{\sin \left (x\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/exp(cot(x)),x, algorithm="fricas")

[Out]

e^(-cos(x)/sin(x))

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Sympy [A]  time = 115.646, size = 5, normalized size = 0.83 \begin{align*} e^{- \cot{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**2/exp(cot(x)),x)

[Out]

exp(-cot(x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (x\right )^{2} e^{\left (-\cot \left (x\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/exp(cot(x)),x, algorithm="giac")

[Out]

integrate(csc(x)^2*e^(-cot(x)), x)