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

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

[Out]

-exp(arccot(x))/a

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Rubi [A]  time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, size = 9, normalized size = 1.00 \[ -\frac {e^{\cot ^{-1}(x)}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCot[x]/(a + a*x^2),x]

[Out]

-(E^ArcCot[x]/a)

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fricas [A]  time = 0.74, size = 8, normalized size = 0.89 \[ -\frac {e^{\operatorname {arccot}\relax (x)}}{a} \]

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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giac [A]  time = 0.13, size = 10, normalized size = 1.11 \[ -\frac {e^{\arctan \left (\frac {1}{x}\right )}}{a} \]

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

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maple [A]  time = 0.04, size = 9, normalized size = 1.00 \[ -\frac {{\mathrm e}^{\mathrm {arccot}\relax (x )}}{a} \]

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 = 0.42, size = 9, normalized size = 1.00 \[ -\frac {e^{\left (\arctan \left (1, x\right )\right )}}{a} \]

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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mupad [B]  time = 0.08, size = 8, normalized size = 0.89 \[ -\frac {{\mathrm {e}}^{\mathrm {acot}\relax (x)}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(acot(x))/(a + a*x^2),x)

[Out]

-exp(acot(x))/a

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sympy [A]  time = 0.75, size = 7, normalized size = 0.78 \[ - \frac {e^{\operatorname {acot}{\relax (x )}}}{a} \]

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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