\(\int e^{\arctan (a x)} \, dx\) [247]

Optimal result

Integrand size = 6, antiderivative size = 60 \[ \int e^{\arctan (a x)} \, dx=\frac {\left (\frac {1}{5}+\frac {2 i}{5}\right ) 2^{1-\frac {i}{2}} (1-i a x)^{1+\frac {i}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i}{2},1+\frac {i}{2},2+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a} \]

[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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5169, 71} \[ \int e^{\arctan (a x)} \, dx=\frac {\left (\frac {1}{5}+\frac {2 i}{5}\right ) 2^{1-\frac {i}{2}} (1-i a x)^{1+\frac {i}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i}{2},1+\frac {i}{2},2+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a} \]

[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*} \text {integral}& = \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}} \operatorname {Hypergeometric2F1}\left (\frac {i}{2},1+\frac {i}{2},2+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.75 \[ \int e^{\arctan (a x)} \, dx=\frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \arctan (a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},-e^{2 i \arctan (a x)}\right )}{a} \]

[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

Maple [F]

\[\int {\mathrm e}^{\arctan \left (a x \right )}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int e^{\arctan (a x)} \, dx=\int { e^{\left (\arctan \left (a x\right )\right )} \,d x } \]

[In]

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

[Out]

integral(e^(arctan(a*x)), x)

Sympy [F]

\[ \int e^{\arctan (a x)} \, dx=\int e^{\operatorname {atan}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral(exp(atan(a*x)), x)

Maxima [F]

\[ \int e^{\arctan (a x)} \, dx=\int { e^{\left (\arctan \left (a x\right )\right )} \,d x } \]

[In]

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

[Out]

integrate(e^(arctan(a*x)), x)

Giac [F]

\[ \int e^{\arctan (a x)} \, dx=\int { e^{\left (\arctan \left (a x\right )\right )} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int e^{\arctan (a x)} \, dx=\int {\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )} \,d x \]

[In]

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

[Out]

int(exp(atan(a*x)), x)