\(\int e^{-i \arctan (a x)} x^m \, dx\) [142]

Optimal result

Integrand size = 14, antiderivative size = 79 \[ \int e^{-i \arctan (a x)} x^m \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-a^2 x^2\right )}{1+m}-\frac {i a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{2+m} \]

[Out]

x^(1+m)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],-a^2*x^2)/(1+m)-I*a*x^(2+m)*hypergeom([1/2, 1+1/2*m],[2+1/2*m],
-a^2*x^2)/(2+m)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5168, 822, 371} \[ \int e^{-i \arctan (a x)} x^m \, dx=\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},-a^2 x^2\right )}{m+1}-\frac {i a x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-a^2 x^2\right )}{m+2} \]

[In]

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

[Out]

(x^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(a^2*x^2)])/(1 + m) - (I*a*x^(2 + m)*Hypergeometric2F
1[1/2, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + m)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 822

Int[((e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[f, Int[(e*x)^m*(a + c*
x^2)^p, x], x] + Dist[g/e, Int[(e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, p}, x] &&  !Ration
alQ[m] &&  !IGtQ[p, 0]

Rule 5168

Int[E^(ArcTan[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a*x)^((I*n + 1)/2)/((1 + I*a*x)^((I*n
- 1)/2)*Sqrt[1 + a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(I*n - 1)/2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m (1-i a x)}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\left ((i a) \int \frac {x^{1+m}}{\sqrt {1+a^2 x^2}} \, dx\right )+\int \frac {x^m}{\sqrt {1+a^2 x^2}} \, dx \\ & = \frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-a^2 x^2\right )}{1+m}-\frac {i a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{2+m} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.08 \[ \int e^{-i \arctan (a x)} x^m \, dx=-\frac {i x^{1+m} \sqrt {1+i a x} \sqrt {i+a x} \operatorname {AppellF1}\left (1+m,\frac {1}{2},-\frac {1}{2},2+m,-i a x,i a x\right )}{(1+m) \sqrt {1-i a x} \sqrt {-i+a x}} \]

[In]

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

[Out]

((-I)*x^(1 + m)*Sqrt[1 + I*a*x]*Sqrt[I + a*x]*AppellF1[1 + m, 1/2, -1/2, 2 + m, (-I)*a*x, I*a*x])/((1 + m)*Sqr
t[1 - I*a*x]*Sqrt[-I + a*x])

Maple [F]

\[\int \frac {x^{m} \sqrt {a^{2} x^{2}+1}}{i a x +1}d x\]

[In]

int(x^m/(1+I*a*x)*(a^2*x^2+1)^(1/2),x)

[Out]

int(x^m/(1+I*a*x)*(a^2*x^2+1)^(1/2),x)

Fricas [F]

\[ \int e^{-i \arctan (a x)} x^m \, dx=\int { \frac {\sqrt {a^{2} x^{2} + 1} x^{m}}{i \, a x + 1} \,d x } \]

[In]

integrate(x^m/(1+I*a*x)*(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int e^{-i \arctan (a x)} x^m \, dx=- i \int \frac {x^{m} \sqrt {a^{2} x^{2} + 1}}{a x - i}\, dx \]

[In]

integrate(x**m/(1+I*a*x)*(a**2*x**2+1)**(1/2),x)

[Out]

-I*Integral(x**m*sqrt(a**2*x**2 + 1)/(a*x - I), x)

Maxima [F]

\[ \int e^{-i \arctan (a x)} x^m \, dx=\int { \frac {\sqrt {a^{2} x^{2} + 1} x^{m}}{i \, a x + 1} \,d x } \]

[In]

integrate(x^m/(1+I*a*x)*(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a^2*x^2 + 1)*x^m/(I*a*x + 1), x)

Giac [F]

\[ \int e^{-i \arctan (a x)} x^m \, dx=\int { \frac {\sqrt {a^{2} x^{2} + 1} x^{m}}{i \, a x + 1} \,d x } \]

[In]

integrate(x^m/(1+I*a*x)*(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a^2*x^2 + 1)*x^m/(I*a*x + 1), x)

Mupad [F(-1)]

Timed out. \[ \int e^{-i \arctan (a x)} x^m \, dx=\int \frac {x^m\,\sqrt {a^2\,x^2+1}}{1+a\,x\,1{}\mathrm {i}} \,d x \]

[In]

int((x^m*(a^2*x^2 + 1)^(1/2))/(a*x*1i + 1),x)

[Out]

int((x^m*(a^2*x^2 + 1)^(1/2))/(a*x*1i + 1), x)