3.2 \(\int \frac{e^{\cot ^{-1}(x)}}{a+a x^2} \, dx\)

Optimal. Leaf size=9 \[ -\frac{e^{\cot ^{-1}(x)}}{a} \]

[Out]

-(E^ArcCot[x]/a)

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Rubi [A]  time = 0.0204457, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {5113} \[ -\frac{e^{\cot ^{-1}(x)}}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCot[x]/(a + a*x^2),x]

[Out]

-(E^ArcCot[x]/a)

Rule 5113

Int[E^(ArcCot[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> -Simp[E^(n*ArcCot[a*x])/(a*c*n), x] /; Fr
eeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]

Rubi steps

\begin{align*} \int \frac{e^{\cot ^{-1}(x)}}{a+a x^2} \, dx &=-\frac{e^{\cot ^{-1}(x)}}{a}\\ \end{align*}

Mathematica [A]  time = 0.0319867, size = 9, normalized size = 1. \[ -\frac{e^{\cot ^{-1}(x)}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCot[x]/(a + a*x^2),x]

[Out]

-(E^ArcCot[x]/a)

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Maple [A]  time = 0.043, size = 9, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{{\rm arccot} \left (x\right )}}}{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arccot(x))/(a*x^2+a),x)

[Out]

-exp(arccot(x))/a

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Maxima [A]  time = 1.50966, size = 12, normalized size = 1.33 \begin{align*} -\frac{e^{\left (\arctan \left (1, x\right )\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccot(x))/(a*x^2+a),x, algorithm="maxima")

[Out]

-e^(arctan2(1, x))/a

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Fricas [A]  time = 2.53757, size = 22, normalized size = 2.44 \begin{align*} -\frac{e^{\operatorname{arccot}\left (x\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccot(x))/(a*x^2+a),x, algorithm="fricas")

[Out]

-e^arccot(x)/a

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Sympy [A]  time = 1.21094, size = 7, normalized size = 0.78 \begin{align*} - \frac{e^{\operatorname{acot}{\left (x \right )}}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(acot(x))/(a*x**2+a),x)

[Out]

-exp(acot(x))/a

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Giac [A]  time = 1.11449, size = 14, normalized size = 1.56 \begin{align*} -\frac{e^{\arctan \left (\frac{1}{x}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccot(x))/(a*x^2+a),x, algorithm="giac")

[Out]

-e^(arctan(1/x))/a