3.150 \(\int e^{\frac{2}{3} \tan ^{-1}(x)} x^m \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0232761, antiderivative size = 38, normalized size of antiderivative = 1., 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 = {5062, 133} \[ \frac{x^{m+1} F_1\left (m+1;-\frac{i}{3},\frac{i}{3};m+2;i x,-i x\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5062

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]

Rule 133

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

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 = 0.19496, size = 0, normalized size = 0. \[ \int e^{\frac{2}{3} \tan ^{-1}(x)} x^m \, dx \]

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]

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Maple [F]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{\frac{2\,\arctan \left ( x \right ) }{3}}}}{x}^{m}\, dx \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} e^{\left (\frac{2}{3} \, \arctan \left (x\right )\right )}\,{d x} \end{align*}

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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} e^{\left (\frac{2}{3} \, \arctan \left (x\right )\right )}, x\right ) \end{align*}

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)

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

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} e^{\left (\frac{2}{3} \, \arctan \left (x\right )\right )}\,{d x} \end{align*}

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)