3.3.47 \(\int e^{\text {ArcTan}(a x)} \, dx\) [247]

Optimal. Leaf size=60 \[ \frac {\left (\frac {1}{5}+\frac {2 i}{5}\right ) 2^{1-\frac {i}{2}} (1-i a x)^{1+\frac {i}{2}} \, _2F_1\left (\frac {i}{2},1+\frac {i}{2};2+\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a} \]

[Out]

(1/5+2/5*I)*2^(1-1/2*I)*(1-I*a*x)^(1+1/2*I)*hypergeom([1/2*I, 1+1/2*I],[2+1/2*I],1/2-1/2*I*a*x)/a

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Rubi [A]
time = 0.01, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5169, 71} \begin {gather*} \frac {\left (\frac {1}{5}+\frac {2 i}{5}\right ) 2^{1-\frac {i}{2}} (1-i a x)^{1+\frac {i}{2}} \, _2F_1\left (\frac {i}{2},1+\frac {i}{2};2+\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcTan[a*x],x]

[Out]

((1/5 + (2*I)/5)*2^(1 - I/2)*(1 - I*a*x)^(1 + I/2)*Hypergeometric2F1[I/2, 1 + I/2, 2 + I/2, (1 - I*a*x)/2])/a

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 5169

Int[E^(ArcTan[(a_.)*(x_)]*(n_.)), x_Symbol] :> Int[(1 - I*a*x)^(I*(n/2))/(1 + I*a*x)^(I*(n/2)), x] /; FreeQ[{a
, n}, x] &&  !IntegerQ[(I*n - 1)/2]

Rubi steps

\begin {align*} \int e^{\tan ^{-1}(a x)} \, dx &=\int (1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} \, dx\\ &=\frac {\left (\frac {1}{5}+\frac {2 i}{5}\right ) 2^{1-\frac {i}{2}} (1-i a x)^{1+\frac {i}{2}} \, _2F_1\left (\frac {i}{2},1+\frac {i}{2};2+\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 45, normalized size = 0.75 \begin {gather*} \frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \text {ArcTan}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \text {ArcTan}(a x)}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTan[a*x],x]

[Out]

((4/5 - (8*I)/5)*E^((1 + 2*I)*ArcTan[a*x])*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, -E^((2*I)*ArcTan[a*x])])/a

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\arctan \left (a x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arctan(a*x)),x)

[Out]

int(exp(arctan(a*x)),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(a*x)),x, algorithm="maxima")

[Out]

integrate(e^(arctan(a*x)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(a*x)),x, algorithm="fricas")

[Out]

integral(e^(arctan(a*x)), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{\operatorname {atan}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(atan(a*x)),x)

[Out]

Integral(exp(atan(a*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(a*x)),x, algorithm="giac")

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(atan(a*x)),x)

[Out]

int(exp(atan(a*x)), x)

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