\(\int \frac {e^{-2 i \arctan (a+b x)}}{x^2} \, dx\) [204]

Optimal result

Integrand size = 16, antiderivative size = 62 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^2} \, dx=-\frac {i+a}{(i-a) x}+\frac {2 i b \log (x)}{(i-a)^2}-\frac {2 i b \log (i-a-b x)}{(i-a)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5203, 78} \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^2} \, dx=\frac {2 i b \log (x)}{(-a+i)^2}-\frac {2 i b \log (-a-b x+i)}{(-a+i)^2}-\frac {a+i}{(-a+i) x} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^2} \, dx=\frac {1+a^2+2 i b x \log (x)-2 i b x \log (i-a-b x)}{(-i+a)^2 x} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {-a^{2}-1}{x \left (i-a \right )^{2}}-\frac {2 b \left (i a +1\right ) \ln \left (x \right )}{\left (i-a \right )^{3}}+\frac {2 b \left (i a +1\right ) \ln \left (-b x -a +i\right )}{\left (i-a \right )^{3}}\)	\(69\)
risch	\(\frac {i}{\left (a -i\right ) x}+\frac {a}{\left (a -i\right ) x}+\frac {b \ln \left (4 a^{4} b^{2} x^{2}+8 a^{5} b x +4 a^{6}+8 a^{2} b^{2} x^{2}+16 a^{3} b x +12 a^{4}+4 b^{2} x^{2}+8 a b x +12 a^{2}+4\right )}{i a^{2}+2 a -i}-\frac {2 i b \arctan \left (\frac {\left (2 a^{2} b +2 b \right ) x +2 a^{3}+2 a}{-2 a^{2}-2}\right )}{i a^{2}+2 a -i}-\frac {2 b \ln \left (\left (-2 a^{2} b -2 b \right ) x \right )}{i a^{2}+2 a -i}\)	\(188\)
parallelrisch	\(\frac {-2 \ln \left (x \right ) x \,b^{2}-2 i x \,a^{3} b^{2}-2 i x a \,b^{2}-2 i \ln \left (x \right ) x^{2} b^{3}+2 i \ln \left (b x +a -i\right ) x^{2} b^{3}-6 \ln \left (b x +a -i\right ) x \,a^{2} b^{2}+4 \ln \left (x \right ) x^{2} a \,b^{3}-4 \ln \left (b x +a -i\right ) x^{2} a \,b^{3}+a^{5} b -2 a^{3} b -3 a b -6 i \ln \left (x \right ) x a \,b^{2}+6 i \ln \left (b x +a -i\right ) x a \,b^{2}-2 i \ln \left (b x +a -i\right ) x^{2} a^{2} b^{3}+2 i \ln \left (x \right ) x \,a^{3} b^{2}-2 i \ln \left (b x +a -i\right ) x \,a^{3} b^{2}+2 i \ln \left (x \right ) x^{2} a^{2} b^{3}-b^{2} x +a^{4} b^{2} x +2 \ln \left (b x +a -i\right ) x \,b^{2}+i b +6 \ln \left (x \right ) x \,a^{2} b^{2}-3 i a^{4} b -2 i a^{2} b}{b \left (-a^{4}+4 i a^{3}+6 a^{2}-4 i a -1\right ) \left (-b x -a +i\right ) x}\)	\(306\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.65 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^2} \, dx=\frac {2 i \, b x \log \left (x\right ) - 2 i \, b x \log \left (\frac {b x + a - i}{b}\right ) + a^{2} + 1}{{\left (a^{2} - 2 i \, a - 1\right )} x} \]

[In]

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

[Out]

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

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. 158 vs. \(2 (41) = 82\).

Time = 0.31 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.55 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^2} \, dx=\frac {2 i b \log {\left (- \frac {2 a^{3} b}{\left (a - i\right )^{2}} + \frac {6 i a^{2} b}{\left (a - i\right )^{2}} + 2 a b + \frac {6 a b}{\left (a - i\right )^{2}} + 4 b^{2} x - 2 i b - \frac {2 i b}{\left (a - i\right )^{2}} \right )}}{\left (a - i\right )^{2}} - \frac {2 i b \log {\left (\frac {2 a^{3} b}{\left (a - i\right )^{2}} - \frac {6 i a^{2} b}{\left (a - i\right )^{2}} + 2 a b - \frac {6 a b}{\left (a - i\right )^{2}} + 4 b^{2} x - 2 i b + \frac {2 i b}{\left (a - i\right )^{2}} \right )}}{\left (a - i\right )^{2}} - \frac {- a - i}{x \left (a - i\right )} \]

[In]

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

[Out]

2*I*b*log(-2*a**3*b/(a - I)**2 + 6*I*a**2*b/(a - I)**2 + 2*a*b + 6*a*b/(a - I)**2 + 4*b**2*x - 2*I*b - 2*I*b/(
a - I)**2)/(a - I)**2 - 2*I*b*log(2*a**3*b/(a - I)**2 - 6*I*a**2*b/(a - I)**2 + 2*a*b - 6*a*b/(a - I)**2 + 4*b
**2*x - 2*I*b + 2*I*b/(a - I)**2)/(a - I)**2 - (-a - I)/(x*(a - 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. 110 vs. \(2 (41) = 82\).

Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.77 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^2} \, dx=-\frac {2 \, {\left (a - i\right )} b \log \left (i \, b x + i \, a + 1\right )}{-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1} + \frac {2 \, {\left (a - i\right )} b \log \left (x\right )}{-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1} + \frac {a^{3} + {\left (a^{2} + 1\right )} b x - i \, a^{2} + a - i}{{\left (a^{2} - 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} x} \]

[In]

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

[Out]

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

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. 95 vs. \(2 (41) = 82\).

Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.53 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^2} \, dx=\frac {2 \, b^{2} \log \left (-\frac {i \, a}{i \, b x + i \, a + 1} - \frac {1}{i \, b x + i \, a + 1} + 1\right )}{-i \, a^{2} b - 2 \, a b + i \, b} - \frac {a b + i \, b}{{\left (a - i\right )}^{2} {\left (\frac {i \, a}{i \, b x + i \, a + 1} + \frac {1}{i \, b x + i \, a + 1} - 1\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.61 \[ \int \frac {e^{-2 i \arctan (a+b x)}}{x^2} \, dx=\frac {-1+a\,1{}\mathrm {i}}{x\,\left (1+a\,1{}\mathrm {i}\right )}-\frac {4\,b\,\mathrm {atan}\left (\frac {a^2\,1{}\mathrm {i}+2\,a-\mathrm {i}}{{\left (a-\mathrm {i}\right )}^2}+\frac {x\,\left (2\,a^4\,b^2+4\,a^2\,b^2+2\,b^2\right )}{{\left (a-\mathrm {i}\right )}^2\,\left (-1{}\mathrm {i}\,b\,a^3+b\,a^2-1{}\mathrm {i}\,b\,a+b\right )}\right )}{{\left (a-\mathrm {i}\right )}^2} \]

[In]

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

[Out]

(a*1i - 1)/(x*(a*1i + 1)) - (4*b*atan((2*a + a^2*1i - 1i)/(a - 1i)^2 + (x*(2*b^2 + 4*a^2*b^2 + 2*a^4*b^2))/((a
- 1i)^2*(b - a*b*1i + a^2*b - a^3*b*1i))))/(a - 1i)^2