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

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

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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5169, 71} \begin {gather*} \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 {gather*}

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 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^{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*}

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Mathematica [A]
time = 0.02, size = 37, normalized size = 0.80 \begin {gather*} \frac {(1-i) e^{(2+2 i) \text {ArcTan}(a x)} \, _2F_1\left (1-i,2;2-i;-e^{2 i \text {ArcTan}(a x)}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.00, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{2 \arctan \left (a x \right )}\, dx\]

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 = 0.00, 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(2*arctan(a*x)),x, algorithm="maxima")

[Out]

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

[Out]

integral(e^(2*arctan(a*x)), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{2 \operatorname {atan}{\left (a x \right )}}\, dx \end {gather*}

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 = 0.00, 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(2*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}}^{2\,\mathrm {atan}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(exp(2*atan(a*x)), x)

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