3.99 \(\int \frac {e^{\tan ^{-1}(x)}}{(1+x^2)^{3/2}} \, dx\)

Optimal. Leaf size=20 \[ \frac {(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt {x^2+1}} \]

[Out]

1/2*exp(arctan(x))*(1+x)/(x^2+1)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 20, 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 = {5069} \[ \frac {(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt {x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTan[x]/(1 + x^2)^(3/2),x]

[Out]

(E^ArcTan[x]*(1 + x))/(2*Sqrt[1 + x^2])

Rule 5069

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((n + a*x)*E^(n*ArcTan[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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \[ \frac {(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt {x^2+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTan[x]/(1 + x^2)^(3/2),x]

[Out]

(E^ArcTan[x]*(1 + x))/(2*Sqrt[1 + x^2])

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fricas [A]  time = 0.44, size = 15, normalized size = 0.75 \[ \frac {{\left (x + 1\right )} e^{\arctan \relax (x)}}{2 \, \sqrt {x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(x))/(x^2+1)^(3/2),x, algorithm="fricas")

[Out]

1/2*(x + 1)*e^arctan(x)/sqrt(x^2 + 1)

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giac [A]  time = 0.01, size = 22, normalized size = 1.10 \[ \frac {1}{2} \, {\left (\frac {x}{\sqrt {x^{2} + 1}} + \frac {1}{\sqrt {x^{2} + 1}}\right )} e^{\arctan \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(x))/(x^2+1)^(3/2),x, algorithm="giac")

[Out]

1/2*(x/sqrt(x^2 + 1) + 1/sqrt(x^2 + 1))*e^arctan(x)

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maple [A]  time = 0.00, size = 16, normalized size = 0.80 \[ \frac {\left (x +1\right ) {\mathrm e}^{\arctan \relax (x )}}{2 \sqrt {x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arctan(x))/(x^2+1)^(3/2),x)

[Out]

1/2*exp(arctan(x))*(x+1)/(x^2+1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\arctan \relax (x)}}{{\left (x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(x))/(x^2+1)^(3/2),x, algorithm="maxima")

[Out]

integrate(e^arctan(x)/(x^2 + 1)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\mathrm {e}}^{\mathrm {atan}\relax (x)}}{{\left (x^2+1\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(atan(x))/(x^2 + 1)^(3/2),x)

[Out]

int(exp(atan(x))/(x^2 + 1)^(3/2), x)

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sympy [A]  time = 17.37, size = 31, normalized size = 1.55 \[ \frac {x e^{\operatorname {atan}{\relax (x )}}}{2 \sqrt {x^{2} + 1}} + \frac {e^{\operatorname {atan}{\relax (x )}}}{2 \sqrt {x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(atan(x))/(x**2+1)**(3/2),x)

[Out]

x*exp(atan(x))/(2*sqrt(x**2 + 1)) + exp(atan(x))/(2*sqrt(x**2 + 1))

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