Integrand size = 16, antiderivative size = 36 \[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,\frac {1}{4},-\frac {1}{4},2+m,i a x,-i a x\right )}{1+m} \] Output:
x^(1+m)*AppellF1(1+m,-1/4,1/4,2+m,-I*a*x,I*a*x)/(1+m)
\[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx \] Input:
Integrate[E^((I/2)*ArcTan[a*x])*x^m,x]
Output:
Integrate[E^((I/2)*ArcTan[a*x])*x^m, x]
Time = 0.31 (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 = {5585, 150}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^m e^{\frac {1}{2} i \arctan (a x)} \, dx\) |
\(\Big \downarrow \) 5585 |
\(\displaystyle \int \frac {\sqrt [4]{1+i a x} x^m}{\sqrt [4]{1-i a x}}dx\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {x^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{4},-\frac {1}{4},m+2,i a x,-i a x\right )}{m+1}\) |
Input:
Int[E^((I/2)*ArcTan[a*x])*x^m,x]
Output:
(x^(1 + m)*AppellF1[1 + m, 1/4, -1/4, 2 + m, I*a*x, (-I)*a*x])/(1 + m)
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> 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] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
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] && !Intege rQ[(I*n - 1)/2]
\[\int \sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}\, x^{m}d x\]
Input:
int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x)
Output:
int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x)
\[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\int { x^{m} \sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}} \,d x } \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x, algorithm="fricas")
Output:
integral(x^m*sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)), x)
\[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\int x^{m} \sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}\, dx \] Input:
integrate(((1+I*a*x)/(a**2*x**2+1)**(1/2))**(1/2)*x**m,x)
Output:
Integral(x**m*sqrt(I*(a*x - I)/sqrt(a**2*x**2 + 1)), x)
\[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\int { x^{m} \sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}} \,d x } \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x, algorithm="maxima")
Output:
integrate(x^m*sqrt((I*a*x + 1)/sqrt(a^2*x^2 + 1)), x)
Exception generated. \[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo r the root of a polynomial with parameters. This might be wrong.The choice was done
Timed out. \[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\int x^m\,\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}} \,d x \] Input:
int(x^m*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/2),x)
Output:
int(x^m*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/2), x)
\[ \int e^{\frac {1}{2} i \arctan (a x)} x^m \, dx=\int \frac {x^{m} \sqrt {a i x +1}}{\left (a^{2} x^{2}+1\right )^{\frac {1}{4}}}d x \] Input:
int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x)
Output:
int((x**m*sqrt(a*i*x + 1)*(a**2*x**2 + 1)**(3/4))/(a**2*x**2 + 1),x)