3.139 \(\int e^{-6 i \tan ^{-1}(a x)} x^m \, dx\)

Optimal. Leaf size=115 \[ \frac{2 \left (2 m^2+4 m+3\right ) x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,-i a x)}{m+1}+\frac{4 i x^{m+1} \left (-a \left (m^2+3 m+3\right ) x+i (m+1)^2\right )}{(m+1) (1+i a x)^2}-\frac{(1-i a x)^2 x^{m+1}}{(m+1) (1+i a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0897069, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5062, 100, 145, 64} \[ \frac{2 \left (2 m^2+4 m+3\right ) x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,-i a x)}{m+1}+\frac{4 i x^{m+1} \left (-a \left (m^2+3 m+3\right ) x+i (m+1)^2\right )}{(m+1) (1+i a x)^2}-\frac{(1-i a x)^2 x^{m+1}}{(m+1) (1+i a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^m/E^((6*I)*ArcTan[a*x]),x]

[Out]

-((x^(1 + m)*(1 - I*a*x)^2)/((1 + m)*(1 + I*a*x)^2)) + ((4*I)*x^(1 + m)*(I*(1 + m)^2 - a*(3 + 3*m + m^2)*x))/(
(1 + m)*(1 + I*a*x)^2) + (2*(3 + 4*m + 2*m^2)*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-I)*a*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 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 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

Rule 64

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

Rubi steps

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

Mathematica [A]  time = 0.0344049, size = 94, normalized size = 0.82 \[ \frac{x^{m+1} \left (2 \left (2 m^2+4 m+3\right ) (a x-i)^2 \text{Hypergeometric2F1}(1,m+1,m+2,-i a x)-a^2 x^2+m^2 (4+4 i a x)+4 m (2+3 i a x)+10 i a x+5\right )}{(m+1) (a x-i)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/E^((6*I)*ArcTan[a*x]),x]

[Out]

(x^(1 + m)*(5 + (10*I)*a*x - a^2*x^2 + 4*m*(2 + (3*I)*a*x) + m^2*(4 + (4*I)*a*x) + 2*(3 + 4*m + 2*m^2)*(-I + a
*x)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (-I)*a*x]))/((1 + m)*(-I + a*x)^2)

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Maple [C]  time = 0.657, size = 1196, normalized size = 10.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(1+I*a*x)^6*(a^2*x^2+1)^3,x)

[Out]

1/120*I*(I*a)^(-m)/a*(x^m*(I*a)^m*(-720-1764*m-1624*m^2+7200*a^2*x^2-21*m^5-m^6+14400*a^2*m*x^2-3600*a^4*x^4-4
200*a^4*x^4*m+1112*a^2*x^2*m^4+4911*a^2*x^2*m^3+11722*a^2*x^2*m^2-735*m^3-175*m^4+6*a^2*x^2*m^6-120*a^6*x^6*m+
129*a^2*x^2*m^5-720*I*a^5*x^5+7200*I*a^3*x^3-3600*I*a*x-a^4*x^4*m^6-22*a^4*x^4*m^5-197*a^4*x^4*m^4-932*a^4*x^4
*m^3-2556*a^4*x^4*m^2+4*I*a^3*x^3*m^6+87*I*a^3*x^3*m^5+764*I*a^3*x^3*m^4-4*I*a*x*m^6-120*I*a^5*x^5*m+3483*I*a^
3*x^3*m^3-85*I*a*x*m^5+8802*I*a^3*x^3*m^2-720*I*a*x*m^4+12000*I*a^3*x^3*m-3095*I*a*x*m^3-7076*I*a*x*m^2-8100*I
*a*m*x)/(1+m)/m/(1+I*a*x)^5+x^m*(I*a)^m*(m^5+20*m^4+155*m^3+580*m^2+1044*m+720)*LerchPhi(-I*a*x,1,m))-1/40*I*(
I*a)^(-m)/a*(-x^m*(I*a)^m*(24+50*m+35*m^2-240*I*a^3*x^3+120*I*a*x-240*a^2*x^2-392*a^2*m*x^2+120*a^4*x^4+96*a^4
*x^4*m+149*I*a*x*m^2+226*I*a*m*x-6*a^2*x^2*m^4-63*a^2*x^2*m^3-239*a^2*x^2*m^2+10*m^3+m^4+a^4*x^4*m^4+11*a^4*x^
4*m^3+46*a^4*x^4*m^2-4*I*a^3*x^3*m^4-43*I*a^3*x^3*m^3-171*I*a^3*x^3*m^2+4*I*a*x*m^4-312*I*a^3*x^3*m+41*I*a*x*m
^3)/(1+I*a*x)^5+x^m*(I*a)^m*m*(m^4+10*m^3+35*m^2+50*m+24)*LerchPhi(-I*a*x,1,m))+1/40*I*(I*a)^(-m)/a*(-x^m*(I*a
)^m*(a^4*x^4*m^4+a^4*x^4*m^3+4*I*a*x*m^4-4*a^4*x^4*m^2-4*I*a*m*x-6*a^2*x^2*m^4-4*a^4*x^4*m+19*I*a^3*x^3*m^2-3*
a^2*x^2*m^3+20*I*a*x-3*I*a^3*x^3*m^3+31*a^2*x^2*m^2-21*I*a*x*m^2+m^4+18*a^2*m*x^2-4*I*a^3*x^3*m^4-40*a^2*x^2+I
*a*x*m^3-5*m^2+18*I*a^3*x^3*m+4)/(1+I*a*x)^5+x^m*(I*a)^m*(m^2-3*m+2)*m*(m^2+3*m+2)*LerchPhi(-I*a*x,1,m))-1/120
*I*(I*a)^(-m)/a*(-x^m*(I*a)^m*(a^4*x^4*m^4-9*a^4*x^4*m^3+4*I*a*x*m^4+26*a^4*x^4*m^2-154*I*a*m*x-6*a^2*x^2*m^4-
24*a^4*x^4*m-111*I*a^3*x^3*m^2+57*a^2*x^2*m^3+37*I*a^3*x^3*m^3+129*I*a*x*m^2-179*a^2*x^2*m^2-4*I*a^3*x^3*m^4+m
^4+188*a^2*m*x^2-39*I*a*x*m^3-10*m^3+108*I*a^3*x^3*m+35*m^2-50*m+24)/(1+I*a*x)^5+x^m*(I*a)^m*(m^4-10*m^3+35*m^
2-50*m+24)*m*LerchPhi(-I*a*x,1,m))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} x^{2} + 1\right )}^{3} x^{m}}{{\left (i \, a x + 1\right )}^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(1+I*a*x)^6*(a^2*x^2+1)^3,x, algorithm="maxima")

[Out]

integrate((a^2*x^2 + 1)^3*x^m/(I*a*x + 1)^6, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(1+I*a*x)^6*(a^2*x^2+1)^3,x, algorithm="fricas")

[Out]

integral(-(a^3*x^3 + 3*I*a^2*x^2 - 3*a*x - I)*x^m/(a^3*x^3 - 3*I*a^2*x^2 - 3*a*x + I), 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(x**m/(1+I*a*x)**6*(a**2*x**2+1)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} x^{2} + 1\right )}^{3} x^{m}}{{\left (i \, a x + 1\right )}^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(1+I*a*x)^6*(a^2*x^2+1)^3,x, algorithm="giac")

[Out]

integrate((a^2*x^2 + 1)^3*x^m/(I*a*x + 1)^6, x)