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

Optimal. Leaf size=48 \[ \frac{2 a^2}{x}-2 i a^3 \log (x)+2 i a^3 \log (a x+i)-\frac{i a}{x^2}-\frac{1}{3 x^3} \]

[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 = 0.0304261, 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} \[ \frac{2 a^2}{x}-2 i a^3 \log (x)+2 i a^3 \log (a x+i)-\frac{i a}{x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[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 = 0.0124324, size = 48, 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[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 = 0.043, size = 55, normalized size = 1.2 \begin{align*} i{a}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) +2\,{a}^{3}\arctan \left ( ax \right ) -{\frac{1}{3\,{x}^{3}}}-2\,i{a}^{3}\ln \left ( x \right ) -{\frac{ia}{{x}^{2}}}+2\,{\frac{{a}^{2}}{x}} \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^4,x)

[Out]

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

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Maxima [A]  time = 1.50283, size = 69, normalized size = 1.44 \begin{align*} 2 \, a^{3} \arctan \left (a x\right ) + i \, a^{3} \log \left (a^{2} x^{2} + 1\right ) - 2 i \, a^{3} \log \left (x\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+I*a*x)^2/(a^2*x^2+1)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.71316, size = 119, normalized size = 2.48 \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+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 = 0.508047, size = 39, normalized size = 0.81 \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+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 = 1.0904, size = 57, normalized size = 1.19 \begin{align*} 2 \, a^{3} i \log \left (a x + i\right ) - 2 \, a^{3} i \log \left ({\left | x \right |}\right ) + \frac{6 \, a^{2} x^{2} - 3 \, a i x - 1}{3 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

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