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

Optimal. Leaf size=62 \[ \frac {4 a^3}{a x+i}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)+\frac {8 a^2}{x}-\frac {2 i a}{x^2}-\frac {1}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, 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, 88} \[ \frac {4 a^3}{a x+i}+\frac {8 a^2}{x}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)-\frac {2 i a}{x^2}-\frac {1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*x^3) - ((2*I)*a)/x^2 + (8*a^2)/x + (4*a^3)/(I + a*x) - (12*I)*a^3*Log[x] + (12*I)*a^3*Log[I + a*x]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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]

Rubi steps

\begin {align*} \int \frac {e^{4 i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1+i a x)^2}{x^4 (1-i a x)^2} \, dx\\ &=\int \left (\frac {1}{x^4}+\frac {4 i a}{x^3}-\frac {8 a^2}{x^2}-\frac {12 i a^3}{x}-\frac {4 a^4}{(i+a x)^2}+\frac {12 i a^4}{i+a x}\right ) \, dx\\ &=-\frac {1}{3 x^3}-\frac {2 i a}{x^2}+\frac {8 a^2}{x}+\frac {4 a^3}{i+a x}-12 i a^3 \log (x)+12 i a^3 \log (i+a x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 62, normalized size = 1.00 \[ \frac {4 a^3}{a x+i}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)+\frac {8 a^2}{x}-\frac {2 i a}{x^2}-\frac {1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*1/x^3 - ((2*I)*a)/x^2 + (8*a^2)/x + (4*a^3)/(I + a*x) - (12*I)*a^3*Log[x] + (12*I)*a^3*Log[I + a*x]

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fricas [A]  time = 0.47, size = 86, normalized size = 1.39 \[ \frac {36 \, a^{3} x^{3} + 18 i \, a^{2} x^{2} + 5 \, a x - 36 \, {\left (i \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \relax (x) - 36 \, {\left (-i \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\frac {a x + i}{a}\right ) - i}{3 \, a x^{4} + 3 i \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(36*a^3*x^3 + 18*I*a^2*x^2 + 5*a*x - 36*(I*a^4*x^4 - a^3*x^3)*log(x) - 36*(-I*a^4*x^4 + a^3*x^3)*log((a*x + I)
/a) - I)/(3*a*x^4 + 3*I*x^3)

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giac [A]  time = 0.14, size = 59, normalized size = 0.95 \[ 12 \, a^{3} i \log \left (a x + i\right ) - 12 \, a^{3} i \log \left ({\left | x \right |}\right ) + \frac {36 \, a^{3} x^{3} + 18 \, a^{2} i x^{2} + 5 \, a x - i}{3 \, {\left (a x + i\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12*a^3*i*log(a*x + i) - 12*a^3*i*log(abs(x)) + 1/3*(36*a^3*x^3 + 18*a^2*i*x^2 + 5*a*x - i)/((a*x + i)*x^3)

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maple [A]  time = 0.06, size = 68, normalized size = 1.10 \[ -\frac {1}{3 x^{3}}-12 i a^{3} \ln \relax (x )-\frac {2 i a}{x^{2}}+\frac {8 a^{2}}{x}+\frac {4 a^{3}}{a x +i}+6 i a^{3} \ln \left (a^{2} x^{2}+1\right )+12 a^{3} \arctan \left (a x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/x^3-12*I*a^3*ln(x)-2*I*a/x^2+8*a^2/x+4*a^3/(I+a*x)+6*I*a^3*ln(a^2*x^2+1)+12*a^3*arctan(a*x)

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maxima [A]  time = 0.42, size = 77, normalized size = 1.24 \[ 12 \, a^{3} \arctan \left (a x\right ) + 6 i \, a^{3} \log \left (a^{2} x^{2} + 1\right ) - 12 i \, a^{3} \log \relax (x) + \frac {72 \, a^{4} x^{4} - 36 i \, a^{3} x^{3} + 46 \, a^{2} x^{2} - 12 i \, a x - 2}{6 \, {\left (a^{2} x^{5} + x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12*a^3*arctan(a*x) + 6*I*a^3*log(a^2*x^2 + 1) - 12*I*a^3*log(x) + 1/6*(72*a^4*x^4 - 36*I*a^3*x^3 + 46*a^2*x^2
- 12*I*a*x - 2)/(a^2*x^5 + x^3)

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mupad [B]  time = 0.13, size = 55, normalized size = 0.89 \[ 24\,a^3\,\mathrm {atan}\left (2\,a\,x+1{}\mathrm {i}\right )+\frac {\frac {5\,x}{3}+12\,a^2\,x^3+a\,x^2\,6{}\mathrm {i}-\frac {1{}\mathrm {i}}{3\,a}}{x^4+\frac {x^3\,1{}\mathrm {i}}{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

24*a^3*atan(2*a*x + 1i) + ((5*x)/3 + a*x^2*6i - 1i/(3*a) + 12*a^2*x^3)/(x^4 + (x^3*1i)/a)

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sympy [A]  time = 0.38, size = 71, normalized size = 1.15 \[ 12 a^{3} \left (- i \log {\left (24 a^{4} x \right )} + i \log {\left (24 a^{4} x + 24 i a^{3} \right )}\right ) + \frac {- 36 a^{3} x^{3} - 18 i a^{2} x^{2} - 5 a x + i}{- 3 a x^{4} - 3 i x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12*a**3*(-I*log(24*a**4*x) + I*log(24*a**4*x + 24*I*a**3)) + (-36*a**3*x**3 - 18*I*a**2*x**2 - 5*a*x + I)/(-3*
a*x**4 - 3*I*x**3)

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