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

Optimal. Leaf size=104 \[ \frac {2 i b^3 \log (x)}{(-a+i)^4}-\frac {2 i b^3 \log (-a-b x+i)}{(-a+i)^4}+\frac {2 b^2}{(1+i a)^3 x}-\frac {i b}{(-a+i)^2 x^2}+\frac {-a-i}{3 (-a+i) x^3} \]

[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]  time = 0.06, antiderivative size = 102, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5095, 77} \[ \frac {2 b^2}{(1+i a)^3 x}+\frac {2 i b^3 \log (x)}{(-a+i)^4}-\frac {2 i b^3 \log (-a-b x+i)}{(-a+i)^4}-\frac {i b}{(-a+i)^2 x^2}-\frac {a+i}{3 (-a+i) x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(I + a)/(3*(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 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]))))

Rule 5095

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*} \int \frac {e^{-2 i \tan ^{-1}(a+b x)}}{x^4} \, dx &=\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]  time = 0.05, size = 91, normalized size = 0.88 \[ \frac {(a-i) \left (a^3-i a^2-3 i a b x+a+6 i b^2 x^2-3 b x-i\right )-6 i b^3 x^3 \log (-a-b x+i)+6 i b^3 x^3 \log (x)}{3 (a-i)^4 x^3} \]

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 94, normalized size = 0.90 \[ \frac {6 i \, b^{3} x^{3} \log \relax (x) - 6 i \, b^{3} x^{3} \log \left (\frac {b x + a - i}{b}\right ) - 6 \, {\left (-i \, a - 1\right )} b^{2} x^{2} + a^{4} - 2 i \, a^{3} + {\left (-3 i \, a^{2} - 6 \, a + 3 i\right )} b x - 2 i \, a - 1}{{\left (3 \, a^{4} - 12 i \, a^{3} - 18 \, a^{2} + 12 i \, a + 3\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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 - 6*
a + 3*I)*b*x - 2*I*a - 1)/((3*a^4 - 12*I*a^3 - 18*a^2 + 12*I*a + 3)*x^3)

________________________________________________________________________________________

giac [B]  time = 0.12, size = 202, normalized size = 1.94 \[ -\frac {2 \, b^{4} \log \left (-\frac {a i}{b i x + a i + 1} + \frac {i^{2}}{b i x + a i + 1} + 1\right )}{a^{4} b i + 4 \, a^{3} b - 6 \, a^{2} b i - 4 \, a b + b i} - \frac {\frac {3 \, {\left (a b^{4} i - 8 \, b^{4}\right )} i^{2}}{{\left (b i x + a i + 1\right )} b} + \frac {a b^{3} i - 10 \, b^{3}}{a i + 1} + \frac {3 \, {\left (a^{2} b^{5} + 4 \, a b^{5} i + 5 \, b^{5}\right )} i^{2}}{{\left (b i x + a i + 1\right )}^{2} b^{2}}}{3 \, {\left (a - i\right )}^{3} {\left (\frac {a i}{b i x + a i + 1} - \frac {i^{2}}{b i x + a i + 1} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.06, size = 349, normalized size = 3.36 \[ \frac {i a^{4}}{\left (i-a \right )^{5} x^{3}}-\frac {a^{5}}{3 \left (i-a \right )^{5} x^{3}}+\frac {2 i b^{3} \arctan \left (b x +a \right )}{\left (i-a \right )^{5}}+\frac {2 a^{3}}{3 \left (i-a \right )^{5} x^{3}}+\frac {i b^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a}{\left (i-a \right )^{5}}+\frac {a}{\left (i-a \right )^{5} x^{3}}-\frac {i}{3 \left (i-a \right )^{5} x^{3}}-\frac {2 b^{3} \ln \relax (x )}{\left (i-a \right )^{5}}+\frac {i b \,a^{3}}{\left (i-a \right )^{5} x^{2}}+\frac {2 i b^{2}}{\left (i-a \right )^{5} x}-\frac {4 b^{2} a}{\left (i-a \right )^{5} x}-\frac {2 i b^{2} a^{2}}{\left (i-a \right )^{5} x}-\frac {3 i b a}{\left (i-a \right )^{5} x^{2}}+\frac {3 b \,a^{2}}{\left (i-a \right )^{5} x^{2}}-\frac {b}{\left (i-a \right )^{5} x^{2}}+\frac {2 i a^{2}}{3 \left (i-a \right )^{5} x^{3}}+\frac {b^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{\left (i-a \right )^{5}}-\frac {2 b^{3} \arctan \left (b x +a \right ) a}{\left (i-a \right )^{5}}-\frac {2 i b^{3} \ln \relax (x ) a}{\left (i-a \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.34, size = 222, normalized size = 2.13 \[ \frac {{\left (2 \, a - 2 i\right )} b^{3} \log \left (i \, b x + i \, a + 1\right )}{i \, a^{5} + 5 \, a^{4} - 10 i \, a^{3} - 10 \, a^{2} + 5 i \, a + 1} - \frac {{\left (2 \, a - 2 i\right )} b^{3} \log \relax (x)}{i \, a^{5} + 5 \, a^{4} - 10 i \, a^{3} - 10 \, a^{2} + 5 i \, a + 1} - \frac {{\left (6 \, a - 6 i\right )} b^{3} x^{3} - i \, a^{5} + 3 \, {\left (a^{2} - 2 i \, a - 1\right )} b^{2} x^{2} - 3 \, a^{4} + 2 i \, a^{3} - {\left (i \, a^{4} + 5 \, a^{3} - 9 i \, a^{2} - 7 \, a + 2 i\right )} b x - 2 \, a^{2} + 3 i \, a + 1}{{\left (3 i \, a^{4} + 12 \, a^{3} - 18 i \, a^{2} - 12 \, a + 3 i\right )} b x^{4} + {\left (3 i \, a^{5} + 15 \, a^{4} - 30 i \, a^{3} - 30 \, a^{2} + 15 i \, a + 3\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a - 2*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 - 2*I)*b^3*log(x)/
(I*a^5 + 5*a^4 - 10*I*a^3 - 10*a^2 + 5*I*a + 1) - ((6*a - 6*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)/((3*I*a^4 + 12*a^3 - 18*I*a^
2 - 12*a + 3*I)*b*x^4 + (3*I*a^5 + 15*a^4 - 30*I*a^3 - 30*a^2 + 15*I*a + 3)*x^3)

________________________________________________________________________________________

mupad [B]  time = 0.77, size = 199, normalized size = 1.91 \[ \frac {\frac {a+1{}\mathrm {i}}{3\,\left (a-\mathrm {i}\right )}+\frac {b^2\,x^2\,2{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^3}-\frac {b\,x\,1{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^2}}{x^3}-\frac {4\,b^3\,\mathrm {atan}\left (\frac {\left (a^4-a^3\,4{}\mathrm {i}-6\,a^2+a\,4{}\mathrm {i}+1\right )\,1{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^4}+\frac {x\,\left (2\,a^{12}\,b^2+12\,a^{10}\,b^2+30\,a^8\,b^2+40\,a^6\,b^2+30\,a^4\,b^2+12\,a^2\,b^2+2\,b^2\right )}{{\left (a-\mathrm {i}\right )}^4\,\left (-1{}\mathrm {i}\,b\,a^9+3\,b\,a^8+8\,b\,a^6+6{}\mathrm {i}\,b\,a^5+6\,b\,a^4+8{}\mathrm {i}\,b\,a^3+3{}\mathrm {i}\,b\,a-b\right )}\right )}{{\left (a-\mathrm {i}\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [B]  time = 1.40, size = 286, normalized size = 2.75 \[ \frac {2 i b^{3} \log {\left (- \frac {2 a^{5} b^{3}}{\left (a - i\right )^{4}} + \frac {10 i a^{4} b^{3}}{\left (a - i\right )^{4}} + \frac {20 a^{3} b^{3}}{\left (a - i\right )^{4}} - \frac {20 i a^{2} b^{3}}{\left (a - i\right )^{4}} + 2 a b^{3} - \frac {10 a b^{3}}{\left (a - i\right )^{4}} + 4 b^{4} x - 2 i b^{3} + \frac {2 i b^{3}}{\left (a - i\right )^{4}} \right )}}{\left (a - i\right )^{4}} - \frac {2 i b^{3} \log {\left (\frac {2 a^{5} b^{3}}{\left (a - i\right )^{4}} - \frac {10 i a^{4} b^{3}}{\left (a - i\right )^{4}} - \frac {20 a^{3} b^{3}}{\left (a - i\right )^{4}} + \frac {20 i a^{2} b^{3}}{\left (a - i\right )^{4}} + 2 a b^{3} + \frac {10 a b^{3}}{\left (a - i\right )^{4}} + 4 b^{4} x - 2 i b^{3} - \frac {2 i b^{3}}{\left (a - i\right )^{4}} \right )}}{\left (a - i\right )^{4}} - \frac {- i a^{3} - a^{2} - i a + 6 b^{2} x^{2} + x \left (- 3 a b + 3 i b\right ) - 1}{x^{3} \left (3 i a^{3} + 9 a^{2} - 9 i a - 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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 - (-I*a**3 - a**2 - I
*a + 6*b**2*x**2 + x*(-3*a*b + 3*I*b) - 1)/(x**3*(3*I*a**3 + 9*a**2 - 9*I*a - 3))

________________________________________________________________________________________