\(\int e^{2 \arctan (a x)} \, dx\) [262]

Optimal result

Integrand size = 8, antiderivative size = 46 \[ \int e^{2 \arctan (a x)} \, dx=\frac {(1+i) 2^{-1-i} (1-i a x)^{1+i} \operatorname {Hypergeometric2F1}\left (i,1+i,2+i,\frac {1}{2} (1-i a x)\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5169, 71} \[ \int e^{2 \arctan (a x)} \, dx=\frac {(1+i) 2^{-1-i} (1-i a x)^{1+i} \operatorname {Hypergeometric2F1}\left (i,1+i,2+i,\frac {1}{2} (1-i a x)\right )}{a} \]

[In]

Int[E^(2*ArcTan[a*x]),x]

[Out]

((1 + I)*(1 - I*a*x)^(1 + I)*Hypergeometric2F1[I, 1 + I, 2 + I, (1 - I*a*x)/2])/(2^(1 + I)*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)^i (1+i a x)^{-i} \, dx \\ & = \frac {(1+i) 2^{-1-i} (1-i a x)^{1+i} \operatorname {Hypergeometric2F1}\left (i,1+i,2+i,\frac {1}{2} (1-i a x)\right )}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int e^{2 \arctan (a x)} \, dx=\frac {(1-i) e^{(2+2 i) \arctan (a x)} \operatorname {Hypergeometric2F1}\left (1-i,2,2-i,-e^{2 i \arctan (a x)}\right )}{a} \]

[In]

Integrate[E^(2*ArcTan[a*x]),x]

[Out]

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

Maple [F]

\[\int {\mathrm e}^{2 \arctan \left (a x \right )}d x\]

[In]

int(exp(2*arctan(a*x)),x)

[Out]

int(exp(2*arctan(a*x)),x)

Fricas [F]

\[ \int e^{2 \arctan (a x)} \, dx=\int { e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \]

[In]

integrate(exp(2*arctan(a*x)),x, algorithm="fricas")

[Out]

integral(e^(2*arctan(a*x)), x)

Sympy [F]

\[ \int e^{2 \arctan (a x)} \, dx=\int e^{2 \operatorname {atan}{\left (a x \right )}}\, dx \]

[In]

integrate(exp(2*atan(a*x)),x)

[Out]

Integral(exp(2*atan(a*x)), x)

Maxima [F]

\[ \int e^{2 \arctan (a x)} \, dx=\int { e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \]

[In]

integrate(exp(2*arctan(a*x)),x, algorithm="maxima")

[Out]

integrate(e^(2*arctan(a*x)), x)

Giac [F]

\[ \int e^{2 \arctan (a x)} \, dx=\int { e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \]

[In]

integrate(exp(2*arctan(a*x)),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int e^{2 \arctan (a x)} \, dx=\int {\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )} \,d x \]

[In]

int(exp(2*atan(a*x)),x)

[Out]

int(exp(2*atan(a*x)), x)