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

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

[Out]

exp(-cot(x))

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Rubi [A]  time = 0.01, antiderivative size = 6, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.07, size = 6, normalized size = 1.00 \[ e^{-\cot (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-Cot[x])

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fricas [A]  time = 0.64, size = 9, normalized size = 1.50 \[ e^{\left (-\frac {\cos \relax (x)}{\sin \relax (x)}\right )} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \relax (x)^{2} e^{\left (-\cot \relax (x)\right )}\,{d x} \]

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)

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maple [A]  time = 0.05, size = 6, normalized size = 1.00 \[ {\mathrm e}^{-\cot \relax (x )} \]

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 = 0.32, size = 5, normalized size = 0.83 \[ e^{\left (-\cot \relax (x)\right )} \]

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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mupad [B]  time = 2.94, size = 7, normalized size = 1.17 \[ {\mathrm {e}}^{-\frac {1}{\mathrm {tan}\relax (x)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-cot(x))/sin(x)^2,x)

[Out]

exp(-1/tan(x))

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sympy [A]  time = 18.87, size = 5, normalized size = 0.83 \[ e^{- \cot {\relax (x )}} \]

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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