3.154 \(\int e^{i n \tan ^{-1}(a x)} x^3 \, dx\)

Optimal. Leaf size=171 \[ -\frac{2^{\frac{n}{2}-2} n \left (n^2+8\right ) (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right )}{3 a^4 (2-n)}-\frac{(1+i a x)^{\frac{n+2}{2}} \left (2 i a n x+n^2+6\right ) (1-i a x)^{1-\frac{n}{2}}}{24 a^4}+\frac{x^2 (1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{4 a^2} \]

[Out]

(x^2*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(4*a^2) - ((1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2)*(
6 + n^2 + (2*I)*a*n*x))/(24*a^4) - (2^(-2 + n/2)*n*(8 + n^2)*(1 - I*a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2,
-n/2, 2 - n/2, (1 - I*a*x)/2])/(3*a^4*(2 - n))

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Rubi [A]  time = 0.113763, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {5062, 100, 147, 69} \[ -\frac{2^{\frac{n}{2}-2} n \left (n^2+8\right ) (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right )}{3 a^4 (2-n)}-\frac{(1+i a x)^{\frac{n+2}{2}} \left (2 i a n x+n^2+6\right ) (1-i a x)^{1-\frac{n}{2}}}{24 a^4}+\frac{x^2 (1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(I*n*ArcTan[a*x])*x^3,x]

[Out]

(x^2*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(4*a^2) - ((1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2)*(
6 + n^2 + (2*I)*a*n*x))/(24*a^4) - (2^(-2 + n/2)*n*(8 + n^2)*(1 - I*a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2,
-n/2, 2 - n/2, (1 - I*a*x)/2])/(3*a^4*(2 - n))

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 100

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

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int e^{i n \tan ^{-1}(a x)} x^3 \, dx &=\int x^3 (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx\\ &=\frac{x^2 (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}}}{4 a^2}+\frac{\int x (1-i a x)^{-n/2} (1+i a x)^{n/2} (-2-i a n x) \, dx}{4 a^2}\\ &=\frac{x^2 (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}}}{4 a^2}-\frac{(1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}} \left (6+n^2+2 i a n x\right )}{24 a^4}+\frac{\left (i n \left (8+n^2\right )\right ) \int (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx}{24 a^3}\\ &=\frac{x^2 (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}}}{4 a^2}-\frac{(1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}} \left (6+n^2+2 i a n x\right )}{24 a^4}-\frac{2^{-2+\frac{n}{2}} n \left (8+n^2\right ) (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right )}{3 a^4 (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.180215, size = 210, normalized size = 1.23 \[ \frac{(a x+i) (1-i a x)^{-n/2} \left ((n-2) \left (a^2 x^2 (a x-i) (1+i a x)^{n/2}-i 2^{\frac{n}{2}+1} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right )\right )-i 2^{\frac{n}{2}+3} n \, _2F_1\left (-\frac{n}{2}-2,1-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right )+i 2^{\frac{n}{2}+3} (n-1) \, _2F_1\left (-\frac{n}{2}-1,1-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right )\right )}{4 a^4 (n-2)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(I*n*ArcTan[a*x])*x^3,x]

[Out]

((I + a*x)*((-I)*2^(3 + n/2)*n*Hypergeometric2F1[-2 - n/2, 1 - n/2, 2 - n/2, (1 - I*a*x)/2] + I*2^(3 + n/2)*(-
1 + n)*Hypergeometric2F1[-1 - n/2, 1 - n/2, 2 - n/2, (1 - I*a*x)/2] + (-2 + n)*(a^2*x^2*(1 + I*a*x)^(n/2)*(-I
+ a*x) - I*2^(1 + n/2)*Hypergeometric2F1[1 - n/2, -n/2, 2 - n/2, (1 - I*a*x)/2])))/(4*a^4*(-2 + n)*(1 - I*a*x)
^(n/2))

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{in\arctan \left ( ax \right ) }}{x}^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(I*n*arctan(a*x))*x^3,x)

[Out]

int(exp(I*n*arctan(a*x))*x^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(I*n*arctan(a*x))*x^3,x, algorithm="maxima")

[Out]

integrate(x^3*e^(I*n*arctan(a*x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{\left (-\frac{a x + i}{a x - i}\right )^{\frac{1}{2} \, n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(I*n*arctan(a*x))*x^3,x, algorithm="fricas")

[Out]

integral(x^3/(-(a*x + I)/(a*x - I))^(1/2*n), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(I*n*atan(a*x))*x**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(I*n*arctan(a*x))*x^3,x, algorithm="giac")

[Out]

integrate(x^3*e^(I*n*arctan(a*x)), x)