3.5 \(\int \frac{e^{\cot ^{-1}(x)}}{(a+a x^2)^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{(1-x) e^{\cot ^{-1}(x)}}{2 a \sqrt{a x^2+a}} \]

[Out]

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

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Rubi [A]  time = 0.028173, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {5114} \[ -\frac{(1-x) e^{\cot ^{-1}(x)}}{2 a \sqrt{a x^2+a}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCot[x]/(a + a*x^2)^(3/2),x]

[Out]

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

Rule 5114

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

Rubi steps

\begin{align*} \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{3/2}} \, dx &=-\frac{e^{\cot ^{-1}(x)} (1-x)}{2 a \sqrt{a+a x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0673215, size = 25, normalized size = 0.93 \[ \frac{(x-1) e^{\cot ^{-1}(x)}}{2 a \sqrt{a \left (x^2+1\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCot[x]/(a + a*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.039, size = 23, normalized size = 0.9 \begin{align*}{\frac{ \left ({x}^{2}+1 \right ) \left ( x-1 \right ){{\rm e}^{{\rm arccot} \left (x\right )}}}{2} \left ( a{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arccot(x))/(a*x^2+a)^(3/2),x)

[Out]

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

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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 )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccot(x))/(a*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(e^arccot(x)/(a*x^2 + a)^(3/2), x)

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Fricas [A]  time = 2.44181, size = 77, normalized size = 2.85 \begin{align*} \frac{\sqrt{a x^{2} + a}{\left (x - 1\right )} e^{\operatorname{arccot}\left (x\right )}}{2 \,{\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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{acot}{\left (x \right )}}}{\left (a \left (x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(acot(x))/(a*x**2+a)**(3/2),x)

[Out]

Integral(exp(acot(x))/(a*(x**2 + 1))**(3/2), x)

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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 )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccot(x))/(a*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate(e^arccot(x)/(a*x^2 + a)^(3/2), x)