3.161 \(\int \frac{e^{i n \tan ^{-1}(a x)}}{x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0720683, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5062, 129, 151, 12, 131} \[ \frac{2 i a^3 \left (n^2+2\right ) (1+i a x)^{\frac{n-2}{2}} (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-i a x}{i a x+1}\right )}{3 (2-n)}-\frac{i a n (1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{6 x^2}-\frac{(1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[E^(I*n*ArcTan[a*x])/x^4,x]

[Out]

-((1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(3*x^3) - ((I/6)*a*n*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 +
n)/2))/x^2 + (((2*I)/3)*a^3*(2 + n^2)*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((-2 + n)/2)*Hypergeometric2F1[2, 1 -
n/2, 2 - n/2, (1 - I*a*x)/(1 + I*a*x)])/(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 129

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))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 151

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

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

Mathematica [A]  time = 0.0545292, size = 119, normalized size = 0.7 \[ -\frac{(a x+i) (1-i a x)^{-n/2} (1+i a x)^{\frac{n-2}{2}} \left (4 a^3 \left (n^2+2\right ) x^3 \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{a x+i}{i-a x}\right )-(n-2) (a x-i)^2 (a n x-2 i)\right )}{6 (n-2) x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(I*n*ArcTan[a*x])/x^4,x]

[Out]

-((1 + I*a*x)^((-2 + n)/2)*(I + a*x)*(-((-2 + n)*(-I + a*x)^2*(-2*I + a*n*x)) + 4*a^3*(2 + n^2)*x^3*Hypergeome
tric2F1[2, 1 - n/2, 2 - n/2, (I + a*x)/(I - a*x)]))/(6*(-2 + n)*x^3*(1 - I*a*x)^(n/2))

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Maple [F]  time = 0.19, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{in\arctan \left ( ax \right ) }}}{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(I*n*arctan(a*x))/x^4,x)

[Out]

int(exp(I*n*arctan(a*x))/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(I*n*arctan(a*x))/x^4,x, algorithm="maxima")

[Out]

integrate(e^(I*n*arctan(a*x))/x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x^{4} \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^4,x, algorithm="fricas")

[Out]

integral(1/(x^4*(-(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**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(I*n*arctan(a*x))/x^4,x, algorithm="giac")

[Out]

integrate(e^(I*n*arctan(a*x))/x^4, x)