### 3.316 $$\int \frac{e^{\tan ^{-1}(x)}}{1+x^2} \, dx$$

Optimal. Leaf size=4 $e^{\tan ^{-1}(x)}$

[Out]

E^ArcTan[x]

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Rubi [A]  time = 0.0205325, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {5071} $e^{\tan ^{-1}(x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcTan[x]/(1 + x^2),x]

[Out]

E^ArcTan[x]

Rule 5071

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcTan[a*x])/(a*c*n), x] /; Fre
eQ[{a, c, d, n}, x] && EqQ[d, a^2*c]

Rubi steps

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

Mathematica [C]  time = 0.0042299, size = 27, normalized size = 6.75 $(1-i x)^{\frac{i}{2}} (1+i x)^{-\frac{i}{2}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTan[x]/(1 + x^2),x]

[Out]

(1 - I*x)^(I/2)/(1 + I*x)^(I/2)

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Maple [A]  time = 0.003, size = 4, normalized size = 1. \begin{align*}{{\rm e}^{\arctan \left ( x \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(arctan(x))

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Maxima [A]  time = 0.925476, size = 4, normalized size = 1. \begin{align*} e^{\arctan \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^arctan(x)

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Fricas [A]  time = 2.18507, size = 18, normalized size = 4.5 \begin{align*} e^{\arctan \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^arctan(x)

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Sympy [A]  time = 0.866199, size = 3, normalized size = 0.75 \begin{align*} e^{\operatorname{atan}{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(atan(x))

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Giac [A]  time = 1.04797, size = 4, normalized size = 1. \begin{align*} e^{\arctan \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^arctan(x)