\(\int \frac {e^{-2 i \arctan (a+b x)}}{x^4} \, dx\) [206]

Optimal result

Integrand size = 16, antiderivative size = 104 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^4} \, dx=\frac {-i-a}{3 (i-a) x^3}-\frac {i b}{(i-a)^2 x^2}+\frac {2 b^2}{(1+i a)^3 x}+\frac {2 i b^3 \log (x)}{(i-a)^4}-\frac {2 i b^3 \log (i-a-b x)}{(i-a)^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5203, 78} \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^4} \, dx=\frac {2 i b^3 \log (x)}{(-a+i)^4}-\frac {2 i b^3 \log (-a-b x+i)}{(-a+i)^4}+\frac {2 b^2}{(1+i a)^3 x}-\frac {i b}{(-a+i)^2 x^2}-\frac {a+i}{3 (-a+i) x^3} \]

[In]

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

[Out]

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

Rule 78

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]))))

Rule 5203

Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[(d + e*x)^m*((1 -
I*a*c - I*b*c*x)^(I*(n/2))/(1 + I*a*c + I*b*c*x)^(I*(n/2))), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^4} \, dx=\frac {(-i+a) \left (-i+a-i a^2+a^3-3 b x-3 i a b x+6 i b^2 x^2\right )+6 i b^3 x^3 \log (x)-6 i b^3 x^3 \log (i-a-b x)}{3 (-i+a)^4 x^3} \]

[In]

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

[Out]

((-I + a)*(-I + a - I*a^2 + a^3 - 3*b*x - (3*I)*a*b*x + (6*I)*b^2*x^2) + (6*I)*b^3*x^3*Log[x] - (6*I)*b^3*x^3*
Log[I - a - b*x])/(3*(-I + a)^4*x^3)

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.44

method	result	size
default	\(-\frac {2 b^{3} \left (i a +1\right ) \ln \left (x \right )}{\left (i-a \right )^{5}}-\frac {2 b^{2} \left (i a^{2}+2 a -i\right )}{\left (i-a \right )^{5} x}-\frac {a^{5}-3 i a^{4}-2 a^{3}-2 i a^{2}-3 a +i}{3 \left (i-a \right )^{5} x^{3}}+\frac {b \left (i a^{3}+3 a^{2}-3 i a -1\right )}{\left (i-a \right )^{5} x^{2}}+\frac {2 b^{3} \left (i a +1\right ) \ln \left (-b x -a +i\right )}{\left (i-a \right )^{5}}\)	\(150\)
risch	\(\frac {\frac {2 i b^{2} x^{2}}{\left (a^{2}-2 i a -1\right ) \left (a -i\right )}-\frac {i b x}{a^{2}-2 i a -1}+\frac {i+a}{3 a -3 i}}{x^{3}}-\frac {2 b^{3} \ln \left (\left (-2 a^{6} b -6 a^{4} b -6 a^{2} b -2 b \right ) x \right )}{i a^{4}+4 a^{3}-6 i a^{2}-4 a +i}+\frac {b^{3} \ln \left (4 a^{12} b^{2} x^{2}+8 a^{13} b x +4 a^{14}+24 a^{10} b^{2} x^{2}+48 a^{11} b x +28 a^{12}+60 a^{8} b^{2} x^{2}+120 a^{9} b x +84 a^{10}+80 a^{6} b^{2} x^{2}+160 a^{7} b x +140 a^{8}+60 a^{4} b^{2} x^{2}+120 a^{5} b x +140 a^{6}+24 a^{2} b^{2} x^{2}+48 a^{3} b x +84 a^{4}+4 b^{2} x^{2}+8 a b x +28 a^{2}+4\right )}{i a^{4}+4 a^{3}-6 i a^{2}-4 a +i}-\frac {2 i b^{3} \arctan \left (\frac {\left (2 a^{6} b +6 a^{4} b +6 a^{2} b +2 b \right ) x +2 a^{7}+6 a^{5}+6 a^{3}+2 a}{-2 a^{6}-6 a^{4}-6 a^{2}-2}\right )}{i a^{4}+4 a^{3}-6 i a^{2}-4 a +i}\)	\(399\)
parallelrisch	\(-\frac {168 a^{3} b^{3} x^{2}-9 i a^{10} b +75 i a^{8} b -42 i a^{6} b +6 \ln \left (b x +a -i\right ) x^{3} b^{4}+3 i x^{2} b^{3}+42 a^{5} b -24 a \,b^{3} x^{2}+81 a^{2} b^{2} x -35 a^{9} b +90 a^{7} b -90 i a^{4} b +35 i a^{2} b -75 a^{3} b +42 x^{3} a^{6} b^{4}+24 x^{2} a^{7} b^{3}+x \,a^{10} b^{2}-54 x \,a^{8} b^{2}-6 \ln \left (x \right ) x^{3} b^{4}+9 a b +294 a^{6} b^{2} x -2 b^{2} x -336 a^{4} b^{2} x -126 i \ln \left (x \right ) x^{3} a^{5} b^{4}-120 \ln \left (x \right ) x^{4} a^{3} b^{5}-210 \ln \left (x \right ) x^{3} a^{4} b^{4}+36 \ln \left (x \right ) x^{4} a^{5} b^{5}+42 \ln \left (x \right ) x^{3} a^{6} b^{4}+36 \ln \left (x \right ) x^{4} a \,b^{5}+126 \ln \left (x \right ) x^{3} a^{2} b^{4}+120 \ln \left (b x +a -i\right ) x^{4} a^{3} b^{5}+210 \ln \left (b x +a -i\right ) x^{3} a^{4} b^{4}-36 \ln \left (b x +a -i\right ) x^{4} a^{5} b^{5}-42 \ln \left (b x +a -i\right ) x^{3} a^{6} b^{4}-126 i x^{3} a^{5} b^{4}-36 \ln \left (b x +a -i\right ) x^{4} a \,b^{5}-126 \ln \left (b x +a -i\right ) x^{3} a^{2} b^{4}-42 i x^{3} a \,b^{4}-84 i x^{2} a^{2} b^{3}-84 i x^{2} a^{6} b^{3}+210 i x^{3} a^{3} b^{4}+210 i x^{2} a^{4} b^{3}+6 i x^{3} a^{7} b^{4}+3 i x^{2} a^{8} b^{3}-11 i x \,a^{9} b^{2}-6 i \ln \left (x \right ) x^{4} b^{5}+6 i \ln \left (b x +a -i\right ) x^{4} b^{5}+156 i x \,a^{7} b^{2}-378 i x \,a^{5} b^{2}+204 i x \,a^{3} b^{2}+126 i \ln \left (b x +a -i\right ) x^{3} a^{5} b^{4}-i b +6 i \ln \left (x \right ) x^{4} a^{6} b^{5}+a^{11} b +6 i \ln \left (x \right ) x^{3} a^{7} b^{4}-6 i \ln \left (b x +a -i\right ) x^{3} a^{7} b^{4}-90 i \ln \left (x \right ) x^{4} a^{4} b^{5}+90 i \ln \left (b x +a -i\right ) x^{4} a^{4} b^{5}+90 i \ln \left (x \right ) x^{4} a^{2} b^{5}-90 i \ln \left (b x +a -i\right ) x^{4} a^{2} b^{5}-6 i \ln \left (b x +a -i\right ) x^{4} a^{6} b^{5}-6 x^{3} b^{4}-210 i \ln \left (b x +a -i\right ) x^{3} a^{3} b^{4}-42 i \ln \left (x \right ) x^{3} a \,b^{4}+42 i \ln \left (b x +a -i\right ) x^{3} a \,b^{4}+210 i \ln \left (x \right ) x^{3} a^{3} b^{4}-168 a^{5} b^{3} x^{2}-210 a^{4} b^{4} x^{3}+126 a^{2} b^{4} x^{3}-19 i x a \,b^{2}}{3 \left (-a^{7}+7 i a^{6}+21 a^{5}-35 i a^{4}-35 a^{3}+21 i a^{2}+7 a -i\right ) b \left (-a^{3}+3 i a^{2}+3 a -i\right ) \left (-b x -a +i\right ) x^{3}}\)	\(914\)

[In]

int(1/(1+I*(b*x+a))^2*(1+(b*x+a)^2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-2*b^3*(1+I*a)/(I-a)^5*ln(x)-2*b^2*(I*a^2-I+2*a)/(I-a)^5/x-1/3/(I-a)^5*(-3*I*a^4+a^5-2*I*a^2-2*a^3+I-3*a)/x^3+
b*(I*a^3-3*I*a+3*a^2-1)/(I-a)^5/x^2+2*b^3*(1+I*a)/(I-a)^5*ln(I-a-b*x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^4} \, dx=\frac {6 i \, b^{3} x^{3} \log \left (x\right ) - 6 i \, b^{3} x^{3} \log \left (\frac {b x + a - i}{b}\right ) - 6 \, {\left (-i \, a - 1\right )} b^{2} x^{2} + a^{4} - 2 i \, a^{3} - 3 \, {\left (i \, a^{2} + 2 \, a - i\right )} b x - 2 i \, a - 1}{3 \, {\left (a^{4} - 4 i \, a^{3} - 6 \, a^{2} + 4 i \, a + 1\right )} x^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (75) = 150\).

Time = 0.57 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.75 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^4} \, dx=\frac {2 i b^{3} \log {\left (- \frac {2 a^{5} b^{3}}{\left (a - i\right )^{4}} + \frac {10 i a^{4} b^{3}}{\left (a - i\right )^{4}} + \frac {20 a^{3} b^{3}}{\left (a - i\right )^{4}} - \frac {20 i a^{2} b^{3}}{\left (a - i\right )^{4}} + 2 a b^{3} - \frac {10 a b^{3}}{\left (a - i\right )^{4}} + 4 b^{4} x - 2 i b^{3} + \frac {2 i b^{3}}{\left (a - i\right )^{4}} \right )}}{\left (a - i\right )^{4}} - \frac {2 i b^{3} \log {\left (\frac {2 a^{5} b^{3}}{\left (a - i\right )^{4}} - \frac {10 i a^{4} b^{3}}{\left (a - i\right )^{4}} - \frac {20 a^{3} b^{3}}{\left (a - i\right )^{4}} + \frac {20 i a^{2} b^{3}}{\left (a - i\right )^{4}} + 2 a b^{3} + \frac {10 a b^{3}}{\left (a - i\right )^{4}} + 4 b^{4} x - 2 i b^{3} - \frac {2 i b^{3}}{\left (a - i\right )^{4}} \right )}}{\left (a - i\right )^{4}} - \frac {- a^{3} + i a^{2} - a - 6 i b^{2} x^{2} + x \left (3 i a b + 3 b\right ) + i}{x^{3} \cdot \left (3 a^{3} - 9 i a^{2} - 9 a + 3 i\right )} \]

[In]

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

[Out]

2*I*b**3*log(-2*a**5*b**3/(a - I)**4 + 10*I*a**4*b**3/(a - I)**4 + 20*a**3*b**3/(a - I)**4 - 20*I*a**2*b**3/(a
- I)**4 + 2*a*b**3 - 10*a*b**3/(a - I)**4 + 4*b**4*x - 2*I*b**3 + 2*I*b**3/(a - I)**4)/(a - I)**4 - 2*I*b**3*
log(2*a**5*b**3/(a - I)**4 - 10*I*a**4*b**3/(a - I)**4 - 20*a**3*b**3/(a - I)**4 + 20*I*a**2*b**3/(a - I)**4 +
2*a*b**3 + 10*a*b**3/(a - I)**4 + 4*b**4*x - 2*I*b**3 - 2*I*b**3/(a - I)**4)/(a - I)**4 - (-a**3 + I*a**2 - a
- 6*I*b**2*x**2 + x*(3*I*a*b + 3*b) + I)/(x**3*(3*a**3 - 9*I*a**2 - 9*a + 3*I))

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (72) = 144\).

Time = 0.19 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.10 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^4} \, dx=\frac {2 \, {\left (a - i\right )} b^{3} \log \left (i \, b x + i \, a + 1\right )}{i \, a^{5} + 5 \, a^{4} - 10 i \, a^{3} - 10 \, a^{2} + 5 i \, a + 1} - \frac {2 \, {\left (a - i\right )} b^{3} \log \left (x\right )}{i \, a^{5} + 5 \, a^{4} - 10 i \, a^{3} - 10 \, a^{2} + 5 i \, a + 1} + \frac {6 \, {\left (a - i\right )} b^{3} x^{3} - i \, a^{5} + 3 \, {\left (a^{2} - 2 i \, a - 1\right )} b^{2} x^{2} - 3 \, a^{4} + 2 i \, a^{3} - {\left (i \, a^{4} + 5 \, a^{3} - 9 i \, a^{2} - 7 \, a + 2 i\right )} b x - 2 \, a^{2} + 3 i \, a + 1}{3 \, {\left ({\left (-i \, a^{4} - 4 \, a^{3} + 6 i \, a^{2} + 4 \, a - i\right )} b x^{4} + {\left (-i \, a^{5} - 5 \, a^{4} + 10 i \, a^{3} + 10 \, a^{2} - 5 i \, a - 1\right )} x^{3}\right )}} \]

[In]

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

[Out]

2*(a - I)*b^3*log(I*b*x + I*a + 1)/(I*a^5 + 5*a^4 - 10*I*a^3 - 10*a^2 + 5*I*a + 1) - 2*(a - I)*b^3*log(x)/(I*a
^5 + 5*a^4 - 10*I*a^3 - 10*a^2 + 5*I*a + 1) + 1/3*(6*(a - I)*b^3*x^3 - I*a^5 + 3*(a^2 - 2*I*a - 1)*b^2*x^2 - 3
*a^4 + 2*I*a^3 - (I*a^4 + 5*a^3 - 9*I*a^2 - 7*a + 2*I)*b*x - 2*a^2 + 3*I*a + 1)/((-I*a^4 - 4*a^3 + 6*I*a^2 + 4
*a - I)*b*x^4 + (-I*a^5 - 5*a^4 + 10*I*a^3 + 10*a^2 - 5*I*a - 1)*x^3)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (72) = 144\).

Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.76 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^4} \, dx=\frac {2 \, b^{4} \log \left (-\frac {i \, a}{i \, b x + i \, a + 1} - \frac {1}{i \, b x + i \, a + 1} + 1\right )}{-i \, a^{4} b - 4 \, a^{3} b + 6 i \, a^{2} b + 4 \, a b - i \, b} + \frac {\frac {-i \, a b^{3} + 10 \, b^{3}}{i \, a + 1} + \frac {3 i \, {\left (a b^{4} + 8 i \, b^{4}\right )}}{{\left (i \, b x + i \, a + 1\right )} b} + \frac {3 \, {\left (a^{2} b^{5} + 4 i \, a b^{5} + 5 \, b^{5}\right )}}{{\left (i \, b x + i \, a + 1\right )}^{2} b^{2}}}{3 \, {\left (a - i\right )}^{3} {\left (\frac {i \, a}{i \, b x + i \, a + 1} + \frac {1}{i \, b x + i \, a + 1} - 1\right )}^{3}} \]

[In]

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

[Out]

2*b^4*log(-I*a/(I*b*x + I*a + 1) - 1/(I*b*x + I*a + 1) + 1)/(-I*a^4*b - 4*a^3*b + 6*I*a^2*b + 4*a*b - I*b) + 1
/3*((-I*a*b^3 + 10*b^3)/(I*a + 1) + 3*I*(a*b^4 + 8*I*b^4)/((I*b*x + I*a + 1)*b) + 3*(a^2*b^5 + 4*I*a*b^5 + 5*b
^5)/((I*b*x + I*a + 1)^2*b^2))/((a - I)^3*(I*a/(I*b*x + I*a + 1) + 1/(I*b*x + I*a + 1) - 1)^3)

Mupad [B] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.91 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^4} \, dx=\frac {\frac {a+1{}\mathrm {i}}{3\,\left (a-\mathrm {i}\right )}+\frac {b^2\,x^2\,2{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^3}-\frac {b\,x\,1{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^2}}{x^3}-\frac {4\,b^3\,\mathrm {atan}\left (\frac {\left (a^4-a^3\,4{}\mathrm {i}-6\,a^2+a\,4{}\mathrm {i}+1\right )\,1{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^4}+\frac {x\,\left (2\,a^{12}\,b^2+12\,a^{10}\,b^2+30\,a^8\,b^2+40\,a^6\,b^2+30\,a^4\,b^2+12\,a^2\,b^2+2\,b^2\right )}{{\left (a-\mathrm {i}\right )}^4\,\left (-1{}\mathrm {i}\,b\,a^9+3\,b\,a^8+8\,b\,a^6+6{}\mathrm {i}\,b\,a^5+6\,b\,a^4+8{}\mathrm {i}\,b\,a^3+3{}\mathrm {i}\,b\,a-b\right )}\right )}{{\left (a-\mathrm {i}\right )}^4} \]

[In]

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

[Out]

((a + 1i)/(3*(a - 1i)) + (b^2*x^2*2i)/(a - 1i)^3 - (b*x*1i)/(a - 1i)^2)/x^3 - (4*b^3*atan(((a*4i - 6*a^2 - a^3
*4i + a^4 + 1)*1i)/(a - 1i)^4 + (x*(2*b^2 + 12*a^2*b^2 + 30*a^4*b^2 + 40*a^6*b^2 + 30*a^8*b^2 + 12*a^10*b^2 +
2*a^12*b^2))/((a - 1i)^4*(a*b*3i - b + a^3*b*8i + 6*a^4*b + a^5*b*6i + 8*a^6*b + 3*a^8*b - a^9*b*1i))))/(a - 1
i)^4