3.4.16 \(\int \frac {e^{\tan ^{-1}(x)}}{1+x^2} \, dx\) [316]

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

[Out]

exp(arctan(x))

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Rubi [A]
time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, 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 = {5179} \begin {gather*} e^{\tan ^{-1}(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcTan[x]/(1 + x^2),x]

[Out]

E^ArcTan[x]

Rule 5179

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.00, size = 27, normalized size = 6.75 \begin {gather*} (1-i x)^{\frac {i}{2}} (1+i x)^{-\frac {i}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTan[x]/(1 + x^2),x]

[Out]

(1 - I*x)^(I/2)/(1 + I*x)^(I/2)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.71, size = 4, normalized size = 1.00 \begin {gather*} \text {ConditionalExpression}\left [E,\left \{\text {True}\right \}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

ConditionalExpression[E, {True}]

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Maple [A]
time = 0.06, size = 4, normalized size = 1.00

method	result	size
gosper	\({\mathrm e}^{\arctan \left (x \right )}\)	\(4\)
derivativedivides	\({\mathrm e}^{\arctan \left (x \right )}\)	\(4\)
default	\({\mathrm e}^{\arctan \left (x \right )}\)	\(4\)
risch	\(\left (i x +1\right )^{-\frac {i}{2}} \left (-i x +1\right )^{\frac {i}{2}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(arctan(x))

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Maxima [A]
time = 0.26, size = 3, normalized size = 0.75 \begin {gather*} e^{\arctan \left (x\right )} \end {gather*}

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 = 0.33, size = 3, normalized size = 0.75 \begin {gather*} e^{\arctan \left (x\right )} \end {gather*}

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 = 0.23, size = 3, normalized size = 0.75 \begin {gather*} e^{\operatorname {atan}{\left (x \right )}} \end {gather*}

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 = 0.00, size = 3, normalized size = 0.75 \begin {gather*} \mathrm {e}^{\arctan x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^arctan(x)

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Mupad [B]
time = 0.25, size = 3, normalized size = 0.75 \begin {gather*} {\mathrm {e}}^{\mathrm {atan}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(atan(x))/(x^2 + 1),x)

[Out]

exp(atan(x))

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