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

Optimal. Leaf size=45 \[ -\frac{4 i}{a^2 (a x+i)}-\frac{8 \log (a x+i)}{a^2}-\frac{4 i x}{a}+\frac{x^2}{2} \]

[Out]

((-4*I)*x)/a + x^2/2 - (4*I)/(a^2*(I + a*x)) - (8*Log[I + a*x])/a^2

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Rubi [A]  time = 0.0262917, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5062, 77} \[ -\frac{4 i}{a^2 (a x+i)}-\frac{8 \log (a x+i)}{a^2}-\frac{4 i x}{a}+\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*I)*x)/a + x^2/2 - (4*I)/(a^2*(I + a*x)) - (8*Log[I + a*x])/a^2

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^{4 i \tan ^{-1}(a x)} x \, dx &=\int \frac{x (1+i a x)^2}{(1-i a x)^2} \, dx\\ &=\int \left (-\frac{4 i}{a}+x+\frac{4 i}{a (i+a x)^2}-\frac{8}{a (i+a x)}\right ) \, dx\\ &=-\frac{4 i x}{a}+\frac{x^2}{2}-\frac{4 i}{a^2 (i+a x)}-\frac{8 \log (i+a x)}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0240546, size = 45, normalized size = 1. \[ -\frac{4 i}{a^2 (a x+i)}-\frac{8 \log (a x+i)}{a^2}-\frac{4 i x}{a}+\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*I)*x)/a + x^2/2 - (4*I)/(a^2*(I + a*x)) - (8*Log[I + a*x])/a^2

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Maple [A]  time = 0.045, size = 53, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{2}}-{\frac{4\,ix}{a}}-{\frac{4\,i}{{a}^{2} \left ( ax+i \right ) }}-4\,{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}}}+{\frac{8\,i\arctan \left ( ax \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+I*a*x)^4/(a^2*x^2+1)^2*x,x)

[Out]

1/2*x^2-4*I*x/a-4*I/a^2/(a*x+I)-4/a^2*ln(a^2*x^2+1)+8*I/a^2*arctan(a*x)

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Maxima [A]  time = 1.50673, size = 81, normalized size = 1.8 \begin{align*} -\frac{4 \,{\left (i \, a x + 1\right )}}{a^{4} x^{2} + a^{2}} + \frac{a x^{2} - 8 i \, x}{2 \, a} + \frac{8 i \, \arctan \left (a x\right )}{a^{2}} - \frac{4 \, \log \left (a^{2} x^{2} + 1\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^4/(a^2*x^2+1)^2*x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.57681, size = 127, normalized size = 2.82 \begin{align*} \frac{a^{3} x^{3} - 7 i \, a^{2} x^{2} + 8 \, a x -{\left (16 \, a x + 16 i\right )} \log \left (\frac{a x + i}{a}\right ) - 8 i}{2 \,{\left (a^{3} x + i \, a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^4/(a^2*x^2+1)^2*x,x, algorithm="fricas")

[Out]

1/2*(a^3*x^3 - 7*I*a^2*x^2 + 8*a*x - (16*a*x + 16*I)*log((a*x + I)/a) - 8*I)/(a^3*x + I*a^2)

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Sympy [A]  time = 0.483821, size = 37, normalized size = 0.82 \begin{align*} - \frac{4 i a}{a^{4} x + i a^{3}} + \frac{x^{2}}{2} - \frac{4 i x}{a} - \frac{8 \log{\left (a x + i \right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)**4/(a**2*x**2+1)**2*x,x)

[Out]

-4*I*a/(a**4*x + I*a**3) + x**2/2 - 4*I*x/a - 8*log(a*x + I)/a**2

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Giac [A]  time = 1.11768, size = 61, normalized size = 1.36 \begin{align*} -\frac{8 \, \log \left (a x + i\right )}{a^{2}} - \frac{4 \, i}{{\left (a x + i\right )} a^{2}} + \frac{a^{4} x^{2} - 8 \, a^{3} i x}{2 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^4/(a^2*x^2+1)^2*x,x, algorithm="giac")

[Out]

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