Integrand size = 22, antiderivative size = 122 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {b^2}{2 c d^2 (i-c x)}-\frac {b^2 \arctan (c x)}{2 c d^2}+\frac {i b (a+b \arctan (c x))}{c d^2 (i-c x)}-\frac {i (a+b \arctan (c x))^2}{2 c d^2}+\frac {i (a+b \arctan (c x))^2}{c d^2 (1+i c x)} \] Output:
1/2*b^2/c/d^2/(I-c*x)-1/2*b^2*arctan(c*x)/c/d^2+I*b*(a+b*arctan(c*x))/c/d^ 2/(I-c*x)-1/2*I*(a+b*arctan(c*x))^2/c/d^2+I*(a+b*arctan(c*x))^2/c/d^2/(1+I *c*x)
Time = 0.59 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.59 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=-\frac {-2 a^2+2 i a b+b^2+b (2 i a+b) (i+c x) \arctan (c x)+b^2 (-1+i c x) \arctan (c x)^2}{2 c d^2 (-i+c x)} \] Input:
Integrate[(a + b*ArcTan[c*x])^2/(d + I*c*d*x)^2,x]
Output:
-1/2*(-2*a^2 + (2*I)*a*b + b^2 + b*((2*I)*a + b)*(I + c*x)*ArcTan[c*x] + b ^2*(-1 + I*c*x)*ArcTan[c*x]^2)/(c*d^2*(-I + c*x))
Time = 0.35 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5389, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {i (a+b \arctan (c x))^2}{c d^2 (1+i c x)}-\frac {2 i b \int \left (\frac {a+b \arctan (c x)}{2 d \left (c^2 x^2+1\right )}-\frac {a+b \arctan (c x)}{2 d (i-c x)^2}\right )dx}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i (a+b \arctan (c x))^2}{c d^2 (1+i c x)}-\frac {2 i b \left (\frac {(a+b \arctan (c x))^2}{4 b c d}-\frac {a+b \arctan (c x)}{2 c d (-c x+i)}-\frac {i b \arctan (c x)}{4 c d}+\frac {i b}{4 c d (-c x+i)}\right )}{d}\) |
Input:
Int[(a + b*ArcTan[c*x])^2/(d + I*c*d*x)^2,x]
Output:
(I*(a + b*ArcTan[c*x])^2)/(c*d^2*(1 + I*c*x)) - ((2*I)*b*(((I/4)*b)/(c*d*( I - c*x)) - ((I/4)*b*ArcTan[c*x])/(c*d) - (a + b*ArcTan[c*x])/(2*c*d*(I - c*x)) + (a + b*ArcTan[c*x])^2/(4*b*c*d)))/d
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTan[c*x])^p/(e*(q + 1))), x] - S imp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (110 ) = 220\).
Time = 0.82 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.94
| method | result | size |
| derivativedivides | \(\frac {\frac {i a^{2}}{d^{2} \left (i c x +1\right )}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{i c x +1}-2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{4}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{4}+\frac {\arctan \left (c x \right )}{2 c x -2 i}+\frac {\ln \left (c x -i\right )^{2}}{16}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{8}-\frac {i \arctan \left (c x \right )}{4}-\frac {i}{4 \left (c x -i\right )}+\frac {\ln \left (c x +i\right )^{2}}{16}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{8}\right )\right )}{d^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} \left (i c x +1\right )}-\frac {i a b \arctan \left (c x \right )}{d^{2}}-\frac {i a b}{d^{2} \left (c x -i\right )}}{c}\) | \(237\) |
| default | \(\frac {\frac {i a^{2}}{d^{2} \left (i c x +1\right )}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{i c x +1}-2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{4}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{4}+\frac {\arctan \left (c x \right )}{2 c x -2 i}+\frac {\ln \left (c x -i\right )^{2}}{16}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{8}-\frac {i \arctan \left (c x \right )}{4}-\frac {i}{4 \left (c x -i\right )}+\frac {\ln \left (c x +i\right )^{2}}{16}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{8}\right )\right )}{d^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} \left (i c x +1\right )}-\frac {i a b \arctan \left (c x \right )}{d^{2}}-\frac {i a b}{d^{2} \left (c x -i\right )}}{c}\) | \(237\) |
| parts | \(\frac {i a^{2}}{d^{2} \left (i c x +1\right ) c}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{i c x +1}-2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{4}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{4}+\frac {\arctan \left (c x \right )}{2 c x -2 i}+\frac {\ln \left (c x -i\right )^{2}}{16}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{8}-\frac {i \arctan \left (c x \right )}{4}-\frac {i}{4 \left (c x -i\right )}+\frac {\ln \left (c x +i\right )^{2}}{16}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{8}\right )\right )}{d^{2} c}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} c \left (i c x +1\right )}-\frac {i a b \arctan \left (c x \right )}{d^{2} c}-\frac {i a b}{d^{2} c \left (c x -i\right )}\) | \(248\) |
| orering | \(-\frac {i \left (3 c^{4} x^{4}+6 i c^{3} x^{3}-2 c^{2} x^{2}+6 i c x -5\right ) \left (a +b \arctan \left (c x \right )\right )^{2}}{4 c \left (i c d x +d \right )^{2}}-\frac {3 i \left (c^{5} x^{5}+i c^{4} x^{4}+2 c^{3} x^{3}+2 i c^{2} x^{2}+c x +i\right ) \left (\frac {2 \left (a +b \arctan \left (c x \right )\right ) b c}{\left (i c d x +d \right )^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 i \left (a +b \arctan \left (c x \right )\right )^{2} c d}{\left (i c d x +d \right )^{3}}\right )}{4 c^{2}}-\frac {i \left (c^{2} x^{2}+1\right )^{3} \left (\frac {2 b^{2} c^{2}}{\left (c^{2} x^{2}+1\right )^{2} \left (i c d x +d \right )^{2}}-\frac {8 i \left (a +b \arctan \left (c x \right )\right ) b \,c^{2} d}{\left (i c d x +d \right )^{3} \left (c^{2} x^{2}+1\right )}-\frac {4 \left (a +b \arctan \left (c x \right )\right ) b \,c^{3} x}{\left (i c d x +d \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}-\frac {6 \left (a +b \arctan \left (c x \right )\right )^{2} c^{2} d^{2}}{\left (i c d x +d \right )^{4}}\right )}{8 c^{3}}\) | \(312\) |
| risch | \(\frac {i b^{2} \left (c x +i\right ) \ln \left (i c x +1\right )^{2}}{8 d^{2} \left (c x -i\right ) c}-\frac {i b \left (b c x \ln \left (-i c x +1\right )+i b \ln \left (-i c x +1\right )-2 i b +4 a \right ) \ln \left (i c x +1\right )}{4 d^{2} \left (c x -i\right ) c}+\frac {8 i a b \ln \left (-i c x +1\right )+4 b^{2} \ln \left (-i c x +1\right )+i b^{2} c x \ln \left (-i c x +1\right )^{2}-b^{2} \ln \left (-i c x +1\right )^{2}-2 i \ln \left (\left (i c b -2 a c \right ) x -2 i a -b \right ) b^{2} c x +2 i \ln \left (\left (-i c b +2 a c \right ) x -2 i a -b \right ) b^{2} c x +4 \ln \left (\left (i c b -2 a c \right ) x -2 i a -b \right ) a b c x -4 \ln \left (\left (-i c b +2 a c \right ) x -2 i a -b \right ) a b c x -4 i \ln \left (\left (i c b -2 a c \right ) x -2 i a -b \right ) a b +4 i \ln \left (\left (-i c b +2 a c \right ) x -2 i a -b \right ) a b -2 \ln \left (\left (i c b -2 a c \right ) x -2 i a -b \right ) b^{2}+2 \ln \left (\left (-i c b +2 a c \right ) x -2 i a -b \right ) b^{2}-8 i a b +8 a^{2}-4 b^{2}}{8 d^{2} \left (c x -i\right ) c}\) | \(403\) |
Input:
int((a+b*arctan(c*x))^2/(d+I*c*d*x)^2,x,method=_RETURNVERBOSE)
Output:
1/c*(I*a^2/d^2/(1+I*c*x)+b^2/d^2*(I/(1+I*c*x)*arctan(c*x)^2-2*I*(1/4*I*arc tan(c*x)*ln(c*x+I)-1/4*I*arctan(c*x)*ln(c*x-I)+1/2*arctan(c*x)/(c*x-I)+1/1 6*ln(c*x-I)^2-1/8*ln(c*x-I)*ln(-1/2*I*(c*x+I))-1/4*I*arctan(c*x)-1/4*I/(c* x-I)+1/16*ln(c*x+I)^2-1/8*(ln(c*x+I)-ln(-1/2*I*(c*x+I)))*ln(-1/2*I*(-c*x+I ))))+2*I*a*b/d^2/(1+I*c*x)*arctan(c*x)-I*a*b/d^2*arctan(c*x)-I*a*b/d^2/(c* x-I))
Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {{\left (i \, b^{2} c x - b^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + 8 \, a^{2} - 8 i \, a b - 4 \, b^{2} + 2 \, {\left ({\left (2 \, a b - i \, b^{2}\right )} c x + 2 i \, a b + b^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{8 \, {\left (c^{2} d^{2} x - i \, c d^{2}\right )}} \] Input:
integrate((a+b*arctan(c*x))^2/(d+I*c*d*x)^2,x, algorithm="fricas")
Output:
1/8*((I*b^2*c*x - b^2)*log(-(c*x + I)/(c*x - I))^2 + 8*a^2 - 8*I*a*b - 4*b ^2 + 2*((2*a*b - I*b^2)*c*x + 2*I*a*b + b^2)*log(-(c*x + I)/(c*x - I)))/(c ^2*d^2*x - I*c*d^2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (94) = 188\).
Time = 5.20 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.48 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=- \frac {b \left (2 a - i b\right ) \log {\left (- \frac {i b \left (2 a - i b\right )}{c} + x \left (2 a b - i b^{2}\right ) \right )}}{4 c d^{2}} + \frac {b \left (2 a - i b\right ) \log {\left (\frac {i b \left (2 a - i b\right )}{c} + x \left (2 a b - i b^{2}\right ) \right )}}{4 c d^{2}} + \frac {\left (- 2 i a b - b^{2}\right ) \log {\left (i c x + 1 \right )}}{2 c^{2} d^{2} x - 2 i c d^{2}} + \frac {\left (i b^{2} c x - b^{2}\right ) \log {\left (- i c x + 1 \right )}^{2}}{8 c^{2} d^{2} x - 8 i c d^{2}} + \frac {\left (i b^{2} c x - b^{2}\right ) \log {\left (i c x + 1 \right )}^{2}}{8 c^{2} d^{2} x - 8 i c d^{2}} + \frac {\left (4 i a b - i b^{2} c x \log {\left (i c x + 1 \right )} + b^{2} \log {\left (i c x + 1 \right )} + 2 b^{2}\right ) \log {\left (- i c x + 1 \right )}}{4 c^{2} d^{2} x - 4 i c d^{2}} - \frac {- 2 a^{2} + 2 i a b + b^{2}}{2 c^{2} d^{2} x - 2 i c d^{2}} \] Input:
integrate((a+b*atan(c*x))**2/(d+I*c*d*x)**2,x)
Output:
-b*(2*a - I*b)*log(-I*b*(2*a - I*b)/c + x*(2*a*b - I*b**2))/(4*c*d**2) + b *(2*a - I*b)*log(I*b*(2*a - I*b)/c + x*(2*a*b - I*b**2))/(4*c*d**2) + (-2* I*a*b - b**2)*log(I*c*x + 1)/(2*c**2*d**2*x - 2*I*c*d**2) + (I*b**2*c*x - b**2)*log(-I*c*x + 1)**2/(8*c**2*d**2*x - 8*I*c*d**2) + (I*b**2*c*x - b**2 )*log(I*c*x + 1)**2/(8*c**2*d**2*x - 8*I*c*d**2) + (4*I*a*b - I*b**2*c*x*l og(I*c*x + 1) + b**2*log(I*c*x + 1) + 2*b**2)*log(-I*c*x + 1)/(4*c**2*d**2 *x - 4*I*c*d**2) - (-2*a**2 + 2*I*a*b + b**2)/(2*c**2*d**2*x - 2*I*c*d**2)
Exception generated. \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arctan(c*x))^2/(d+I*c*d*x)^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.15 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=-\frac {{\left (-2 i \, b^{2} \arctan \left (-\frac {i \, {\left (i \, c d x + d\right )} {\left (\frac {d}{i \, c d x + d} - 1\right )}}{d}\right )^{2} + 4 i \, a b \arctan \left (-\frac {i \, {\left (i \, c d x + d\right )} {\left (\frac {d}{i \, c d x + d} - 1\right )}}{d}\right ) + 2 \, b^{2} \arctan \left (-\frac {i \, {\left (i \, c d x + d\right )} {\left (\frac {d}{i \, c d x + d} - 1\right )}}{d}\right ) - 2 i \, a^{2} - 2 \, a b + i \, b^{2}\right )} e^{\left (2 i \, \arctan \left (-\frac {i \, {\left (i \, c d x + d\right )} {\left (\frac {d}{i \, c d x + d} - 1\right )}}{d}\right )\right )}}{4 \, c d^{2}} \] Input:
integrate((a+b*arctan(c*x))^2/(d+I*c*d*x)^2,x, algorithm="giac")
Output:
-1/4*(-2*I*b^2*arctan(-I*(I*c*d*x + d)*(d/(I*c*d*x + d) - 1)/d)^2 + 4*I*a* b*arctan(-I*(I*c*d*x + d)*(d/(I*c*d*x + d) - 1)/d) + 2*b^2*arctan(-I*(I*c* d*x + d)*(d/(I*c*d*x + d) - 1)/d) - 2*I*a^2 - 2*a*b + I*b^2)*e^(2*I*arctan (-I*(I*c*d*x + d)*(d/(I*c*d*x + d) - 1)/d))/(c*d^2)
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \] Input:
int((a + b*atan(c*x))^2/(d + c*d*x*1i)^2,x)
Output:
int((a + b*atan(c*x))^2/(d + c*d*x*1i)^2, x)
\[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {-2 \left (\int \frac {\mathit {atan} \left (c x \right )}{c^{2} x^{2}-2 c i x -1}d x \right ) a b -\left (\int \frac {\mathit {atan} \left (c x \right )^{2}}{c^{2} x^{2}-2 c i x -1}d x \right ) b^{2}-\left (\int \frac {1}{c^{2} x^{2}-2 c i x -1}d x \right ) a^{2}}{d^{2}} \] Input:
int((a+b*atan(c*x))^2/(d+I*c*d*x)^2,x)
Output:
( - 2*int(atan(c*x)/(c**2*x**2 - 2*c*i*x - 1),x)*a*b - int(atan(c*x)**2/(c **2*x**2 - 2*c*i*x - 1),x)*b**2 - int(1/(c**2*x**2 - 2*c*i*x - 1),x)*a**2) /d**2