\(\int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx\) [146]

Optimal result

Integrand size = 16, antiderivative size = 36 \[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,\frac {1}{4},-\frac {1}{4},2+m,i a x,-i a x\right )}{1+m} \]

[Out]

x^(1+m)*AppellF1(1+m,-1/4,1/4,2+m,-I*a*x,I*a*x)/(1+m)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5170, 138} \[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\frac {x^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{4},-\frac {1}{4},m+2,i a x,-i a x\right )}{m+1} \]

[In]

Int[E^((I/2)*ArcTan[a*x])*x^m,x]

[Out]

(x^(1 + m)*AppellF1[1 + m, 1/4, -1/4, 2 + m, I*a*x, (-I)*a*x])/(1 + m)

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 5170

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx \\ & = \frac {x^{1+m} \operatorname {AppellF1}\left (1+m,\frac {1}{4},-\frac {1}{4},2+m,i a x,-i a x\right )}{1+m} \\ \end{align*}

Mathematica [F]

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

[In]

Integrate[E^((I/2)*ArcTan[a*x])*x^m,x]

[Out]

Integrate[E^((I/2)*ArcTan[a*x])*x^m, x]

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:The choice was done assuming 0=[0,0]Warning, replacing 0 by -28, a substitution variable should perhaps be
purged.Warn

Mupad [F(-1)]

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

[In]

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

[Out]

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