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

Optimal. Leaf size=71 \[ \frac {i 2^{\frac {n}{2}+1} (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{a (2-n)} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5061, 69} \[ \frac {i 2^{\frac {n}{2}+1} (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{a (2-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]))

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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 53, normalized size = 0.75 \[ -\frac {4 i e^{i (n+2) \tan ^{-1}(a x)} \, _2F_1\left (2,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 i \tan ^{-1}(a x)}\right )}{a (n+2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\left (-\frac {a x + i}{a x - i}\right )^{\frac {1}{2} \, n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((-(a*x + I)/(a*x - I))^(1/2*n)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{i n \arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (i \, n \arctan \left (a x\right )\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(I*n*arctan(a*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*atan(a*x)*1i),x)

[Out]

int(exp(n*atan(a*x)*1i), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{i n \operatorname {atan}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(I*n*atan(a*x)), x)

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