3.11 \(\int e^{2 i \tan ^{-1}(a x)} x^3 \, dx\)

Optimal. Leaf size=48 \[ \frac{x^2}{a^2}-\frac{2 i x}{a^3}-\frac{2 \log (a x+i)}{a^4}+\frac{2 i x^3}{3 a}-\frac{x^4}{4} \]

[Out]

((-2*I)*x)/a^3 + x^2/a^2 + (((2*I)/3)*x^3)/a - x^4/4 - (2*Log[I + a*x])/a^4

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Rubi [A]  time = 0.0361756, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5062, 77} \[ \frac{x^2}{a^2}-\frac{2 i x}{a^3}-\frac{2 \log (a x+i)}{a^4}+\frac{2 i x^3}{3 a}-\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[E^((2*I)*ArcTan[a*x])*x^3,x]

[Out]

((-2*I)*x)/a^3 + x^2/a^2 + (((2*I)/3)*x^3)/a - x^4/4 - (2*Log[I + a*x])/a^4

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0196513, size = 48, normalized size = 1. \[ \frac{x^2}{a^2}-\frac{2 i x}{a^3}-\frac{2 \log (a x+i)}{a^4}+\frac{2 i x^3}{3 a}-\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^((2*I)*ArcTan[a*x])*x^3,x]

[Out]

((-2*I)*x)/a^3 + x^2/a^2 + (((2*I)/3)*x^3)/a - x^4/4 - (2*Log[I + a*x])/a^4

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Maple [A]  time = 0.038, size = 55, normalized size = 1.2 \begin{align*} -{\frac{{x}^{4}}{4}}+{\frac{{\frac{2\,i}{3}}{x}^{3}}{a}}+{\frac{{x}^{2}}{{a}^{2}}}-{\frac{2\,ix}{{a}^{3}}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{4}}}+{\frac{2\,i\arctan \left ( ax \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^4+2/3*I*x^3/a+x^2/a^2-2*I*x/a^3-1/a^4*ln(a^2*x^2+1)+2*I/a^4*arctan(a*x)

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Maxima [A]  time = 1.47701, size = 76, normalized size = 1.58 \begin{align*} -\frac{3 \, a^{3} x^{4} - 8 i \, a^{2} x^{3} - 12 \, a x^{2} + 24 i \, x}{12 \, a^{3}} + \frac{2 i \, \arctan \left (a x\right )}{a^{4}} - \frac{\log \left (a^{2} x^{2} + 1\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*a^3*x^4 - 8*I*a^2*x^3 - 12*a*x^2 + 24*I*x)/a^3 + 2*I*arctan(a*x)/a^4 - log(a^2*x^2 + 1)/a^4

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Fricas [A]  time = 1.49833, size = 112, normalized size = 2.33 \begin{align*} -\frac{3 \, a^{4} x^{4} - 8 i \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 24 i \, a x + 24 \, \log \left (\frac{a x + i}{a}\right )}{12 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*a^4*x^4 - 8*I*a^3*x^3 - 12*a^2*x^2 + 24*I*a*x + 24*log((a*x + I)/a))/a^4

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Sympy [A]  time = 0.348497, size = 41, normalized size = 0.85 \begin{align*} - \frac{x^{4}}{4} + \frac{2 i x^{3}}{3 a} + \frac{x^{2}}{a^{2}} - \frac{2 i x}{a^{3}} - \frac{2 \log{\left (a x + i \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**4/4 + 2*I*x**3/(3*a) + x**2/a**2 - 2*I*x/a**3 - 2*log(a*x + I)/a**4

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Giac [A]  time = 1.10319, size = 65, normalized size = 1.35 \begin{align*} -\frac{3 \, a^{4} x^{4} - 8 \, a^{3} i x^{3} - 12 \, a^{2} x^{2} + 24 \, a i x}{12 \, a^{4}} - \frac{2 \, \log \left (a x + i\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*a^4*x^4 - 8*a^3*i*x^3 - 12*a^2*x^2 + 24*a*i*x)/a^4 - 2*log(a*x + i)/a^4