Integrand size = 16, antiderivative size = 76 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=-\frac {i-a}{2 (i+a) x^2}+\frac {2 i b}{(i+a)^2 x}-\frac {2 b^2 \log (x)}{(1-i a)^3}+\frac {2 b^2 \log (i+a+b x)}{(1-i a)^3} \] Output:
-1/2*(I-a)/(I+a)/x^2+2*I*b/(I+a)^2/x-2*b^2*ln(x)/(1-I*a)^3+2*b^2*ln(I+a+b* x)/(1-I*a)^3
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {(i+a) \left (1+a^2+4 i b x\right )+4 i b^2 x^2 \log (x)-4 i b^2 x^2 \log (i+a+b x)}{2 (i+a)^3 x^2} \] Input:
Integrate[E^((2*I)*ArcTan[a + b*x])/x^3,x]
Output:
((I + a)*(1 + a^2 + (4*I)*b*x) + (4*I)*b^2*x^2*Log[x] - (4*I)*b^2*x^2*Log[ I + a + b*x])/(2*(I + a)^3*x^2)
Time = 0.41 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5618, 86, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx\) |
\(\Big \downarrow \) 5618 |
\(\displaystyle \int \frac {i a+i b x+1}{x^3 (-i a-i b x+1)}dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (-\frac {2 i b^3}{(a+i)^3 (a+b x+i)}+\frac {2 i b^2}{(a+i)^3 x}-\frac {2 i b}{(a+i)^2 x^2}+\frac {-a+i}{(a+i) x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 b^2 \log (x)}{(1-i a)^3}+\frac {2 b^2 \log (a+b x+i)}{(1-i a)^3}+\frac {2 i b}{(a+i)^2 x}-\frac {-a+i}{2 (a+i) x^2}\) |
Input:
Int[E^((2*I)*ArcTan[a + b*x])/x^3,x]
Output:
-1/2*(I - a)/((I + a)*x^2) + ((2*I)*b)/((I + a)^2*x) - (2*b^2*Log[x])/(1 - I*a)^3 + (2*b^2*Log[I + a + b*x])/(1 - I*a)^3
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((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]))))
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (67 ) = 134\).
Time = 0.27 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.64
method | result | size |
parallelrisch | \(\frac {-2 i a^{5}+4 i a^{4} b x -1-2 i a -4 i a^{3}-4 i b x +12 \ln \left (x \right ) x^{2} a^{2} b^{2}-12 \ln \left (b x +a +i\right ) x^{2} a^{2} b^{2}-4 i \ln \left (b x +a +i\right ) x^{2} a^{3} b^{2}+a^{6}-4 \ln \left (x \right ) x^{2} b^{2}+4 \ln \left (b x +a +i\right ) x^{2} b^{2}+8 a^{3} b x +4 i \ln \left (x \right ) x^{2} a^{3} b^{2}+a^{4}+12 i \ln \left (b x +a +i\right ) x^{2} a \,b^{2}+8 a b x -12 i \ln \left (x \right ) x^{2} a \,b^{2}-a^{2}}{2 \left (a^{4}+2 a^{2}+1\right ) \left (a^{2}+1\right ) x^{2}}\) | \(201\) |
default | \(-\frac {2 b^{3} \left (\frac {\left (i a^{3} b +3 a^{2} b -3 i a b -b \right ) \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{2}}+\frac {\left (i a^{4}-6 i a^{2}+4 a^{3}+i-4 a -\frac {\left (i a^{3} b +3 a^{2} b -3 i a b -b \right ) a}{b}\right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}\right )}{\left (a^{2}+1\right )^{3}}-\frac {-a^{2}+2 i a +1}{2 \left (a^{2}+1\right ) x^{2}}+\frac {2 b \left (i a^{2}+2 a -i\right )}{\left (a^{2}+1\right )^{2} x}+\frac {2 b^{2} \left (i a^{3}+3 a^{2}-3 i a -1\right ) \ln \left (x \right )}{\left (a^{2}+1\right )^{3}}\) | \(211\) |
risch | \(\frac {\frac {2 i b x}{a^{2}+2 i a -1}+\frac {-i+a}{2 i+2 a}}{x^{2}}-\frac {2 b^{2} \ln \left (\left (2 a^{4} b +4 a^{2} b +2 b \right ) x \right )}{i a^{3}-3 a^{2}-3 i a +1}+\frac {b^{2} \ln \left (4 a^{8} b^{2} x^{2}+8 a^{9} b x +4 a^{10}+16 a^{6} b^{2} x^{2}+32 a^{7} b x +20 a^{8}+24 a^{4} b^{2} x^{2}+48 a^{5} b x +40 a^{6}+16 a^{2} b^{2} x^{2}+32 a^{3} b x +40 a^{4}+4 b^{2} x^{2}+8 a b x +20 a^{2}+4\right )}{i a^{3}-3 a^{2}-3 i a +1}-\frac {2 i b^{2} \arctan \left (\frac {\left (-2 a^{4} b -4 a^{2} b -2 b \right ) x -2 a^{5}-4 a^{3}-2 a}{-2 a^{4}-4 a^{2}-2}\right )}{i a^{3}-3 a^{2}-3 i a +1}\) | \(287\) |
Input:
int((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x^3,x,method=_RETURNVERBOSE)
Output:
1/2*(-2*I*a^5+4*I*x*a^4*b-1-2*I*a-4*I*a^3-4*I*b*x+12*ln(x)*x^2*a^2*b^2-12* ln(I+a+b*x)*x^2*a^2*b^2-4*I*ln(I+a+b*x)*x^2*a^3*b^2+a^6-4*ln(x)*x^2*b^2+4* ln(I+a+b*x)*x^2*b^2+8*a^3*b*x+4*I*ln(x)*x^2*a^3*b^2+a^4+12*I*ln(I+a+b*x)*x ^2*a*b^2+8*a*b*x-12*I*ln(x)*x^2*a*b^2-a^2)/(a^4+2*a^2+1)/(a^2+1)/x^2
Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {4 i \, b^{2} x^{2} \log \left (x\right ) - 4 i \, b^{2} x^{2} \log \left (\frac {b x + a + i}{b}\right ) + a^{3} - 4 \, {\left (-i \, a + 1\right )} b x + i \, a^{2} + a + i}{2 \, {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} x^{2}} \] Input:
integrate((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x^3,x, algorithm="fricas")
Output:
1/2*(4*I*b^2*x^2*log(x) - 4*I*b^2*x^2*log((b*x + a + I)/b) + a^3 - 4*(-I*a + 1)*b*x + I*a^2 + a + I)/((a^3 + 3*I*a^2 - 3*a - I)*x^2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (60) = 120\).
Time = 0.44 (sec) , antiderivative size = 228, normalized size of antiderivative = 3.00 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {2 i b^{2} \log {\left (- \frac {2 a^{4} b^{2}}{\left (a + i\right )^{3}} - \frac {8 i a^{3} b^{2}}{\left (a + i\right )^{3}} + \frac {12 a^{2} b^{2}}{\left (a + i\right )^{3}} + 2 a b^{2} + \frac {8 i a b^{2}}{\left (a + i\right )^{3}} + 4 b^{3} x + 2 i b^{2} - \frac {2 b^{2}}{\left (a + i\right )^{3}} \right )}}{\left (a + i\right )^{3}} - \frac {2 i b^{2} \log {\left (\frac {2 a^{4} b^{2}}{\left (a + i\right )^{3}} + \frac {8 i a^{3} b^{2}}{\left (a + i\right )^{3}} - \frac {12 a^{2} b^{2}}{\left (a + i\right )^{3}} + 2 a b^{2} - \frac {8 i a b^{2}}{\left (a + i\right )^{3}} + 4 b^{3} x + 2 i b^{2} + \frac {2 b^{2}}{\left (a + i\right )^{3}} \right )}}{\left (a + i\right )^{3}} - \frac {- a^{2} - 4 i b x - 1}{x^{2} \cdot \left (2 a^{2} + 4 i a - 2\right )} \] Input:
integrate((1+I*(b*x+a))**2/(1+(b*x+a)**2)/x**3,x)
Output:
2*I*b**2*log(-2*a**4*b**2/(a + I)**3 - 8*I*a**3*b**2/(a + I)**3 + 12*a**2* b**2/(a + I)**3 + 2*a*b**2 + 8*I*a*b**2/(a + I)**3 + 4*b**3*x + 2*I*b**2 - 2*b**2/(a + I)**3)/(a + I)**3 - 2*I*b**2*log(2*a**4*b**2/(a + I)**3 + 8*I *a**3*b**2/(a + I)**3 - 12*a**2*b**2/(a + I)**3 + 2*a*b**2 - 8*I*a*b**2/(a + I)**3 + 4*b**3*x + 2*I*b**2 + 2*b**2/(a + I)**3)/(a + I)**3 - (-a**2 - 4*I*b*x - 1)/(x**2*(2*a**2 + 4*I*a - 2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (58) = 116\).
Time = 0.11 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.47 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=-\frac {2 \, {\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} b^{2} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac {{\left (i \, a^{3} + 3 \, a^{2} - 3 i \, a - 1\right )} b^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac {2 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} b^{2} \log \left (x\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} + \frac {a^{4} - 2 i \, a^{3} - 4 \, {\left (-i \, a^{2} - 2 \, a + i\right )} b x - 2 i \, a - 1}{2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}} \] Input:
integrate((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x^3,x, algorithm="maxima")
Output:
-2*(a^3 - 3*I*a^2 - 3*a + I)*b^2*arctan((b^2*x + a*b)/b)/(a^6 + 3*a^4 + 3* a^2 + 1) - (I*a^3 + 3*a^2 - 3*I*a - 1)*b^2*log(b^2*x^2 + 2*a*b*x + a^2 + 1 )/(a^6 + 3*a^4 + 3*a^2 + 1) - 2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*b^2*log(x)/(a ^6 + 3*a^4 + 3*a^2 + 1) + 1/2*(a^4 - 2*I*a^3 - 4*(-I*a^2 - 2*a + I)*b*x - 2*I*a - 1)/((a^4 + 2*a^2 + 1)*x^2)
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.17 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {2 \, b^{3} \log \left (b x + a + i\right )}{i \, a^{3} b - 3 \, a^{2} b - 3 i \, a b + b} + \frac {2 \, b^{2} \log \left ({\left | x \right |}\right )}{-i \, a^{3} + 3 \, a^{2} + 3 i \, a - 1} + \frac {a^{3} + i \, a^{2} + 4 i \, {\left (a b + i \, b\right )} x + a + i}{2 \, {\left (a + i\right )}^{3} x^{2}} \] Input:
integrate((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x^3,x, algorithm="giac")
Output:
2*b^3*log(b*x + a + I)/(I*a^3*b - 3*a^2*b - 3*I*a*b + b) + 2*b^2*log(abs(x ))/(-I*a^3 + 3*a^2 + 3*I*a - 1) + 1/2*(a^3 + I*a^2 + 4*I*(a*b + I*b)*x + a + I)/((a + I)^3*x^2)
Time = 23.48 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.03 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {\frac {a-\mathrm {i}}{2\,\left (a+1{}\mathrm {i}\right )}+\frac {b\,x\,2{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^2}}{x^2}+\frac {b^2\,\mathrm {atanh}\left (\frac {-a^3-a^2\,3{}\mathrm {i}+3\,a+1{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^3}+\frac {x\,\left (2\,a^8\,b^2+8\,a^6\,b^2+12\,a^4\,b^2+8\,a^2\,b^2+2\,b^2\right )}{{\left (a+1{}\mathrm {i}\right )}^3\,\left (-b\,a^6+2{}\mathrm {i}\,b\,a^5-b\,a^4+4{}\mathrm {i}\,b\,a^3+b\,a^2+2{}\mathrm {i}\,b\,a+b\right )}\right )\,4{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^3} \] Input:
int((a*1i + b*x*1i + 1)^2/(x^3*((a + b*x)^2 + 1)),x)
Output:
((a - 1i)/(2*(a + 1i)) + (b*x*2i)/(a + 1i)^2)/x^2 + (b^2*atanh((3*a - a^2* 3i - a^3 + 1i)/(a + 1i)^3 + (x*(2*b^2 + 8*a^2*b^2 + 12*a^4*b^2 + 8*a^6*b^2 + 2*a^8*b^2))/((a + 1i)^3*(b + a*b*2i + a^2*b + a^3*b*4i - a^4*b + a^5*b* 2i - a^6*b)))*4i)/(a + 1i)^3
Time = 0.17 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.99 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {-4 \mathit {atan} \left (b x +a \right ) a^{3} b^{2} x^{2}+12 \mathit {atan} \left (b x +a \right ) a^{2} b^{2} i \,x^{2}+12 \mathit {atan} \left (b x +a \right ) a \,b^{2} x^{2}-4 \mathit {atan} \left (b x +a \right ) b^{2} i \,x^{2}-2 \,\mathrm {log}\left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{3} b^{2} i \,x^{2}-6 \,\mathrm {log}\left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{2} b^{2} x^{2}+6 \,\mathrm {log}\left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a \,b^{2} i \,x^{2}+2 \,\mathrm {log}\left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) b^{2} x^{2}+4 \,\mathrm {log}\left (x \right ) a^{3} b^{2} i \,x^{2}+12 \,\mathrm {log}\left (x \right ) a^{2} b^{2} x^{2}-12 \,\mathrm {log}\left (x \right ) a \,b^{2} i \,x^{2}-4 \,\mathrm {log}\left (x \right ) b^{2} x^{2}+a^{6}-2 a^{5} i +4 a^{4} b i x +a^{4}+8 a^{3} b x -4 a^{3} i -a^{2}+8 a b x -2 a i -4 b i x -1}{2 x^{2} \left (a^{6}+3 a^{4}+3 a^{2}+1\right )} \] Input:
int((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x^3,x)
Output:
( - 4*atan(a + b*x)*a**3*b**2*x**2 + 12*atan(a + b*x)*a**2*b**2*i*x**2 + 1 2*atan(a + b*x)*a*b**2*x**2 - 4*atan(a + b*x)*b**2*i*x**2 - 2*log(a**2 + 2 *a*b*x + b**2*x**2 + 1)*a**3*b**2*i*x**2 - 6*log(a**2 + 2*a*b*x + b**2*x** 2 + 1)*a**2*b**2*x**2 + 6*log(a**2 + 2*a*b*x + b**2*x**2 + 1)*a*b**2*i*x** 2 + 2*log(a**2 + 2*a*b*x + b**2*x**2 + 1)*b**2*x**2 + 4*log(x)*a**3*b**2*i *x**2 + 12*log(x)*a**2*b**2*x**2 - 12*log(x)*a*b**2*i*x**2 - 4*log(x)*b**2 *x**2 + a**6 - 2*a**5*i + 4*a**4*b*i*x + a**4 + 8*a**3*b*x - 4*a**3*i - a* *2 + 8*a*b*x - 2*a*i - 4*b*i*x - 1)/(2*x**2*(a**6 + 3*a**4 + 3*a**2 + 1))