3.262 \(\int e^{2 \tan ^{-1}(a x)} \, dx\)

Optimal. Leaf size=46 \[ \frac{(1+i) 2^{-1-i} (1-i a x)^{1+i} \, _2F_1\left (i,1+i;2+i;\frac{1}{2} (1-i a x)\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.0121184, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5061, 69} \[ \frac{(1+i) 2^{-1-i} (1-i a x)^{1+i} \, _2F_1\left (i,1+i;2+i;\frac{1}{2} (1-i a x)\right )}{a} \]

Antiderivative was successfully verified.

[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 5061

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]

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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]))

Rubi steps

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

Mathematica [A]  time = 0.0192597, size = 37, normalized size = 0.8 \[ \frac{(1-i) e^{(2+2 i) \tan ^{-1}(a x)} \, _2F_1\left (1-i,2;2-i;-e^{2 i \tan ^{-1}(a x)}\right )}{a} \]

Warning: Unable to verify antiderivative.

[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

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{2\,\arctan \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (2 \, \arctan \left (a x\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(2*arctan(a*x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (e^{\left (2 \, \arctan \left (a x\right )\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(e^(2*arctan(a*x)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{2 \operatorname{atan}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(2*atan(a*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (2 \, \arctan \left (a x\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(2*arctan(a*x)), x)