3.51 \(\int \frac{e^{-2 i \tan ^{-1}(a x)}}{x^4} \, dx\)

Optimal. Leaf size=49 \[ \frac{2 a^2}{x}+2 i a^3 \log (x)-2 i a^3 \log (-a x+i)+\frac{i a}{x^2}-\frac{1}{3 x^3} \]

[Out]

-1/(3*x^3) + (I*a)/x^2 + (2*a^2)/x + (2*I)*a^3*Log[x] - (2*I)*a^3*Log[I - a*x]

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Rubi [A]  time = 0.0317706, antiderivative size = 49, 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} \[ \frac{2 a^2}{x}+2 i a^3 \log (x)-2 i a^3 \log (-a x+i)+\frac{i a}{x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^((2*I)*ArcTan[a*x])*x^4),x]

[Out]

-1/(3*x^3) + (I*a)/x^2 + (2*a^2)/x + (2*I)*a^3*Log[x] - (2*I)*a^3*Log[I - a*x]

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

Mathematica [A]  time = 0.0148524, size = 49, normalized size = 1. \[ \frac{2 a^2}{x}+2 i a^3 \log (x)-2 i a^3 \log (-a x+i)+\frac{i a}{x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^((2*I)*ArcTan[a*x])*x^4),x]

[Out]

-1/(3*x^3) + (I*a)/x^2 + (2*a^2)/x + (2*I)*a^3*Log[x] - (2*I)*a^3*Log[I - a*x]

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Maple [A]  time = 0.048, size = 55, normalized size = 1.1 \begin{align*} 2\,{a}^{3}\arctan \left ( ax \right ) -i{a}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) -{\frac{1}{3\,{x}^{3}}}+{\frac{ia}{{x}^{2}}}+2\,i{a}^{3}\ln \left ( x \right ) +2\,{\frac{{a}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04681, size = 77, normalized size = 1.57 \begin{align*} -2 i \, a^{3} \log \left (i \, a x + 1\right ) + 2 i \, a^{3} \log \left (x\right ) + \frac{6 i \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 i \, a x - 1}{3 i \, a x^{4} + 3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*a^3*log(I*a*x + 1) + 2*I*a^3*log(x) + (6*I*a^3*x^3 + 3*a^2*x^2 + 2*I*a*x - 1)/(3*I*a*x^4 + 3*x^3)

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Fricas [A]  time = 1.64331, size = 117, normalized size = 2.39 \begin{align*} \frac{6 i \, a^{3} x^{3} \log \left (x\right ) - 6 i \, a^{3} x^{3} \log \left (\frac{a x - i}{a}\right ) + 6 \, a^{2} x^{2} + 3 i \, a x - 1}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(6*I*a^3*x^3*log(x) - 6*I*a^3*x^3*log((a*x - I)/a) + 6*a^2*x^2 + 3*I*a*x - 1)/x^3

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Sympy [A]  time = 0.520236, size = 39, normalized size = 0.8 \begin{align*} - 2 a^{3} \left (- i \log{\left (x \right )} + i \log{\left (x - \frac{i}{a} \right )}\right ) + \frac{6 a^{2} x^{2} + 3 i a x - 1}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**3*(-I*log(x) + I*log(x - I/a)) + (6*a**2*x**2 + 3*I*a*x - 1)/(3*x**3)

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Giac [A]  time = 1.10177, size = 104, normalized size = 2.12 \begin{align*} 2 \, a^{3} i \log \left (-i + \frac{i}{a i x + 1}\right ) + \frac{\frac{24 \, a^{3} i^{2}}{a i x + 1} + 10 \, a^{3} - \frac{15 \, a^{3} i^{2}}{{\left (a i x + 1\right )}^{2}}}{3 \,{\left (i - \frac{i}{a i x + 1}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a^3*i*log(-i + i/(a*i*x + 1)) + 1/3*(24*a^3*i^2/(a*i*x + 1) + 10*a^3 - 15*a^3*i^2/(a*i*x + 1)^2)/(i - i/(a*i
*x + 1))^3