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

Optimal result

Integrand size = 14, antiderivative size = 9 \[ \int \frac {e^{\cot ^{-1}(x)}}{a+a x^2} \, dx=-\frac {e^{\cot ^{-1}(x)}}{a} \]

[Out]

-exp(arccot(x))/a

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5221} \[ \int \frac {e^{\cot ^{-1}(x)}}{a+a x^2} \, dx=-\frac {e^{\cot ^{-1}(x)}}{a} \]

[In]

Int[E^ArcCot[x]/(a + a*x^2),x]

[Out]

-(E^ArcCot[x]/a)

Rule 5221

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*} \text {integral}& = -\frac {e^{\cot ^{-1}(x)}}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\cot ^{-1}(x)}}{a+a x^2} \, dx=-\frac {e^{\cot ^{-1}(x)}}{a} \]

[In]

Integrate[E^ArcCot[x]/(a + a*x^2),x]

[Out]

-(E^ArcCot[x]/a)

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00

method	result	size
gosper	\(-\frac {{\mathrm e}^{\operatorname {arccot}\left (x \right )}}{a}\)	\(9\)
parallelrisch	\(-\frac {{\mathrm e}^{\operatorname {arccot}\left (x \right )}}{a}\)	\(9\)
risch	\(-\frac {\left (-i x +1\right )^{-\frac {i}{2}} \left (i x +1\right )^{\frac {i}{2}} {\mathrm e}^{\frac {\pi }{2}}}{a}\)	\(28\)

[In]

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

[Out]

-exp(arccot(x))/a

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\cot ^{-1}(x)}}{a+a x^2} \, dx=-\frac {e^{\operatorname {arccot}\left (x\right )}}{a} \]

[In]

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

[Out]

-e^arccot(x)/a

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\cot ^{-1}(x)}}{a+a x^2} \, dx=- \frac {e^{\operatorname {acot}{\left (x \right )}}}{a} \]

[In]

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

[Out]

-exp(acot(x))/a

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\cot ^{-1}(x)}}{a+a x^2} \, dx=-\frac {e^{\left (\arctan \left (1, x\right )\right )}}{a} \]

[In]

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

[Out]

-e^(arctan2(1, x))/a

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\cot ^{-1}(x)}}{a+a x^2} \, dx=-\frac {e^{\arctan \left (\frac {1}{x}\right )}}{a} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\cot ^{-1}(x)}}{a+a x^2} \, dx=-\frac {{\mathrm {e}}^{\mathrm {acot}\left (x\right )}}{a} \]

[In]

int(exp(acot(x))/(a + a*x^2),x)

[Out]

-exp(acot(x))/a