\(\int e^{\frac {\arctan (x)}{3}} x^m \, dx\) [151]

Optimal result

Integrand size = 12, antiderivative size = 38 \[ \int e^{\frac {\arctan (x)}{3}} x^m \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,-\frac {i}{6},\frac {i}{6},2+m,i x,-i x\right )}{1+m} \]

[Out]

x^(1+m)*AppellF1(1+m,1/6*I,-1/6*I,2+m,-I*x,I*x)/(1+m)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5170, 138} \[ \int e^{\frac {\arctan (x)}{3}} x^m \, dx=\frac {x^{m+1} \operatorname {AppellF1}\left (m+1,-\frac {i}{6},\frac {i}{6},m+2,i x,-i x\right )}{m+1} \]

[In]

Int[E^(ArcTan[x]/3)*x^m,x]

[Out]

(x^(1 + m)*AppellF1[1 + m, -1/6*I, I/6, 2 + m, I*x, (-I)*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 (1-i x)^{\frac {i}{6}} (1+i x)^{-\frac {i}{6}} x^m \, dx \\ & = \frac {x^{1+m} \operatorname {AppellF1}\left (1+m,-\frac {i}{6},\frac {i}{6},2+m,i x,-i x\right )}{1+m} \\ \end{align*}

Mathematica [F]

\[ \int e^{\frac {\arctan (x)}{3}} x^m \, dx=\int e^{\frac {\arctan (x)}{3}} x^m \, dx \]

[In]

Integrate[E^(ArcTan[x]/3)*x^m,x]

[Out]

Integrate[E^(ArcTan[x]/3)*x^m, x]

Maple [F]

\[\int {\mathrm e}^{\frac {\arctan \left (x \right )}{3}} x^{m}d x\]

[In]

int(exp(1/3*arctan(x))*x^m,x)

[Out]

int(exp(1/3*arctan(x))*x^m,x)

Fricas [F]

\[ \int e^{\frac {\arctan (x)}{3}} x^m \, dx=\int { x^{m} e^{\left (\frac {1}{3} \, \arctan \left (x\right )\right )} \,d x } \]

[In]

integrate(exp(1/3*arctan(x))*x^m,x, algorithm="fricas")

[Out]

integral(x^m*e^(1/3*arctan(x)), x)

Sympy [F]

\[ \int e^{\frac {\arctan (x)}{3}} x^m \, dx=\int x^{m} e^{\frac {\operatorname {atan}{\left (x \right )}}{3}}\, dx \]

[In]

integrate(exp(1/3*atan(x))*x**m,x)

[Out]

Integral(x**m*exp(atan(x)/3), x)

Maxima [F]

\[ \int e^{\frac {\arctan (x)}{3}} x^m \, dx=\int { x^{m} e^{\left (\frac {1}{3} \, \arctan \left (x\right )\right )} \,d x } \]

[In]

integrate(exp(1/3*arctan(x))*x^m,x, algorithm="maxima")

[Out]

integrate(x^m*e^(1/3*arctan(x)), x)

Giac [F]

\[ \int e^{\frac {\arctan (x)}{3}} x^m \, dx=\int { x^{m} e^{\left (\frac {1}{3} \, \arctan \left (x\right )\right )} \,d x } \]

[In]

integrate(exp(1/3*arctan(x))*x^m,x, algorithm="giac")

[Out]

integrate(x^m*e^(1/3*arctan(x)), x)

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {\arctan (x)}{3}} x^m \, dx=\int x^m\,{\mathrm {e}}^{\frac {\mathrm {atan}\left (x\right )}{3}} \,d x \]

[In]

int(x^m*exp(atan(x)/3),x)

[Out]

int(x^m*exp(atan(x)/3), x)