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

Optimal. Leaf size=35 \[ -\frac{(1-2 x) e^{\cot ^{-1}(x)}}{5 a^2 \left (x^2+1\right )}-\frac{2 e^{\cot ^{-1}(x)}}{5 a^2} \]

[Out]

(-2*E^ArcCot[x])/(5*a^2) - (E^ArcCot[x]*(1 - 2*x))/(5*a^2*(1 + x^2))

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Rubi [A]  time = 0.0431212, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5115, 5113} \[ -\frac{(1-2 x) e^{\cot ^{-1}(x)}}{5 a^2 \left (x^2+1\right )}-\frac{2 e^{\cot ^{-1}(x)}}{5 a^2} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCot[x]/(a + a*x^2)^2,x]

[Out]

(-2*E^ArcCot[x])/(5*a^2) - (E^ArcCot[x]*(1 - 2*x))/(5*a^2*(1 + x^2))

Rule 5115

Int[E^(ArcCot[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((n + 2*a*(p + 1)*x)*(c + d*x^
2)^(p + 1)*E^(n*ArcCot[a*x]))/(a*c*(n^2 + 4*(p + 1)^2)), x] + Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 + 4*(p + 1)^2
)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1]
&& NeQ[p, -3/2] && NeQ[n^2 + 4*(p + 1)^2, 0] &&  !(IntegerQ[p] && IntegerQ[(I*n)/2]) &&  !( !IntegerQ[p] && I
ntegerQ[(I*n - 1)/2])

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)}}{\left (a+a x^2\right )^2} \, dx &=-\frac{e^{\cot ^{-1}(x)} (1-2 x)}{5 a^2 \left (1+x^2\right )}+\frac{2 \int \frac{e^{\cot ^{-1}(x)}}{a+a x^2} \, dx}{5 a}\\ &=-\frac{2 e^{\cot ^{-1}(x)}}{5 a^2}-\frac{e^{\cot ^{-1}(x)} (1-2 x)}{5 a^2 \left (1+x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.106613, size = 28, normalized size = 0.8 \[ -\frac{\left (2 x^2-2 x+3\right ) e^{\cot ^{-1}(x)}}{5 a^2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCot[x]/(a + a*x^2)^2,x]

[Out]

-(E^ArcCot[x]*(3 - 2*x + 2*x^2))/(5*a^2*(1 + x^2))

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Maple [A]  time = 0.041, size = 26, normalized size = 0.7 \begin{align*} -{\frac{{{\rm e}^{{\rm arccot} \left (x\right )}} \left ( 2\,{x}^{2}-2\,x+3 \right ) }{ \left ( 5\,{x}^{2}+5 \right ){a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*exp(arccot(x))*(2*x^2-2*x+3)/(x^2+1)/a^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{arccot}\left (x\right )}}{{\left (a x^{2} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^arccot(x)/(a*x^2 + a)^2, x)

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Fricas [A]  time = 2.37982, size = 70, normalized size = 2. \begin{align*} -\frac{{\left (2 \, x^{2} - 2 \, x + 3\right )} e^{\operatorname{arccot}\left (x\right )}}{5 \,{\left (a^{2} x^{2} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(2*x^2 - 2*x + 3)*e^arccot(x)/(a^2*x^2 + a^2)

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Sympy [B]  time = 5.07343, size = 65, normalized size = 1.86 \begin{align*} - \frac{2 x^{2} e^{\operatorname{acot}{\left (x \right )}}}{5 a^{2} x^{2} + 5 a^{2}} + \frac{2 x e^{\operatorname{acot}{\left (x \right )}}}{5 a^{2} x^{2} + 5 a^{2}} - \frac{3 e^{\operatorname{acot}{\left (x \right )}}}{5 a^{2} x^{2} + 5 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x**2*exp(acot(x))/(5*a**2*x**2 + 5*a**2) + 2*x*exp(acot(x))/(5*a**2*x**2 + 5*a**2) - 3*exp(acot(x))/(5*a**2
*x**2 + 5*a**2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{arccot}\left (x\right )}}{{\left (a x^{2} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^arccot(x)/(a*x^2 + a)^2, x)