3.2.50 \(\int e^{\frac {2 \text {ArcTan}(x)}{3}} x^m \, dx\) [150]

Optimal. Leaf size=38 \[ \frac {x^{1+m} F_1\left (1+m;-\frac {i}{3},\frac {i}{3};2+m;i x,-i x\right )}{1+m} \]

[Out]

x^(1+m)*AppellF1(1+m,1/3*I,-1/3*I,2+m,-I*x,I*x)/(1+m)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5170, 138} \begin {gather*} \frac {x^{m+1} F_1\left (m+1;-\frac {i}{3},\frac {i}{3};m+2;i x,-i x\right )}{m+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((2*ArcTan[x])/3)*x^m,x]

[Out]

(x^(1 + m)*AppellF1[1 + m, -1/3*I, I/3, 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*} \int e^{\frac {2}{3} \tan ^{-1}(x)} x^m \, dx &=\int (1-i x)^{\frac {i}{3}} (1+i x)^{-\frac {i}{3}} x^m \, dx\\ &=\frac {x^{1+m} F_1\left (1+m;-\frac {i}{3},\frac {i}{3};2+m;i x,-i x\right )}{1+m}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]
time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int e^{\frac {2 \text {ArcTan}(x)}{3}} x^m \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[E^((2*ArcTan[x])/3)*x^m,x]

[Out]

Integrate[E^((2*ArcTan[x])/3)*x^m, x]

________________________________________________________________________________________

Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\frac {2 \arctan \left (x \right )}{3}} x^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2/3*arctan(x))*x^m,x)

[Out]

int(exp(2/3*arctan(x))*x^m,x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2/3*arctan(x))*x^m,x, algorithm="maxima")

[Out]

integrate(x^m*e^(2/3*arctan(x)), x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2/3*arctan(x))*x^m,x, algorithm="fricas")

[Out]

integral(x^m*e^(2/3*arctan(x)), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} e^{\frac {2 \operatorname {atan}{\left (x \right )}}{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2/3*atan(x))*x**m,x)

[Out]

Integral(x**m*exp(2*atan(x)/3), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2/3*arctan(x))*x^m,x, algorithm="giac")

[Out]

integrate(x^m*e^(2/3*arctan(x)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x^m\,{\mathrm {e}}^{\frac {2\,\mathrm {atan}\left (x\right )}{3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*exp((2*atan(x))/3),x)

[Out]

int(x^m*exp((2*atan(x))/3), x)

________________________________________________________________________________________