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]

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

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 0.53, size = 30, normalized size = 1.11 \[ \frac {\sqrt {a x^{2} + a} {\left (x - 1\right )} e^{\operatorname {arccot}\relax (x)}}{2 \, {\left (a^{2} x^{2} + a^{2}\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]

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)

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maple [A]  time = 0.04, size = 23, normalized size = 0.85 \[ \frac {\left (x^{2}+1\right ) \left (x -1\right ) {\mathrm e}^{\mathrm {arccot}\relax (x )}}{2 \left (a \,x^{2}+a \right )^{\frac {3}{2}}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]

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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mupad [B]  time = 0.68, size = 25, normalized size = 0.93 \[ \frac {{\mathrm {e}}^{\mathrm {acot}\relax (x)}\,\left (\frac {x}{2\,a}-\frac {1}{2\,a}\right )}{\sqrt {a\,x^2+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(acot(x))/(a + a*x^2)^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {acot}{\relax (x )}}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

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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