3.339 \(\int e^{n \tan ^{-1}(a x)} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2^{1-\frac{i n}{2}} (1-i a x)^{1+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (1-i a x)\right )}{a (-n+2 i)} \]

[Out]

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

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Rubi [A]  time = 0.0156714, antiderivative size = 81, 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} \[ -\frac{2^{1-\frac{i n}{2}} (1-i a x)^{1+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (1-i a x)\right )}{a (-n+2 i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0177068, size = 56, normalized size = 0.69 \[ \frac{4 e^{(n+2 i) \tan ^{-1}(a x)} \, _2F_1\left (2,1-\frac{i n}{2};2-\frac{i n}{2};-e^{2 i \tan ^{-1}(a x)}\right )}{a (n+2 i)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n\arctan \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(n*arctan(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(e^(n*arctan(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(n*atan(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(n*arctan(a*x)), x)