Integrand size = 21, antiderivative size = 122 \[ \int \frac {x (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=-\frac {b}{2 c^2 d^2 (i-c x)}+\frac {b \arctan (c x)}{2 c^2 d^2}-\frac {i (a+b \arctan (c x))}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2} \] Output:
-1/2*b/c^2/d^2/(I-c*x)+1/2*b*arctan(c*x)/c^2/d^2-I*(a+b*arctan(c*x))/c^2/d ^2/(I-c*x)+(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^2/d^2+1/2*I*b*polylog(2,1-2 /(1+I*c*x))/c^2/d^2
Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.05 \[ \int \frac {x (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=-\frac {i (a+b \arctan (c x))}{c^2 d^2 (i-c x)}-\frac {b \left (\frac {1}{c (i-c x)}-\frac {\arctan (c x)}{c}\right )}{2 c d^2}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )}{c^2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i+c x}{i-c x}\right )}{2 c^2 d^2} \] Input:
Integrate[(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2,x]
Output:
((-I)*(a + b*ArcTan[c*x]))/(c^2*d^2*(I - c*x)) - (b*(1/(c*(I - c*x)) - Arc Tan[c*x]/c))/(2*c*d^2) + ((a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)])/(c^2*d ^2) + ((I/2)*b*PolyLog[2, -((I + c*x)/(I - c*x))])/(c^2*d^2)
Time = 0.36 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5411, 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.
| \(\displaystyle \int \frac {x (a+b \arctan (c x))}{(d+i c d x)^2} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-\frac {a+b \arctan (c x)}{c d^2 (c x-i)}-\frac {i (a+b \arctan (c x))}{c d^2 (c x-i)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i (a+b \arctan (c x))}{c^2 d^2 (-c x+i)}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^2 d^2}+\frac {b \arctan (c x)}{2 c^2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c^2 d^2}-\frac {b}{2 c^2 d^2 (-c x+i)}\) |
Input:
Int[(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2,x]
Output:
-1/2*b/(c^2*d^2*(I - c*x)) + (b*ArcTan[c*x])/(2*c^2*d^2) - (I*(a + b*ArcTa n[c*x]))/(c^2*d^2*(I - c*x)) + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c ^2*d^2) + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2)
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f* x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] & & IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (110 ) = 220\).
Time = 0.43 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.07
| method | result | size |
| derivativedivides | \(\frac {-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a \arctan \left (c x \right )}{d^{2}}+\frac {i a}{d^{2} \left (c x -i\right )}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}+\frac {i b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}+\frac {b}{2 d^{2} \left (c x -i\right )}-\frac {i b \ln \left (c^{2} x^{2}+1\right )}{8 d^{2}}+\frac {b \arctan \left (c x \right )}{4 d^{2}}+\frac {i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 d^{2}}-\frac {b \arctan \left (\frac {c x}{2}\right )}{8 d^{2}}+\frac {b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 d^{2}}+\frac {b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 d^{2}}-\frac {i b \ln \left (c x -i\right )^{2}}{4 d^{2}}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d^{2}}+\frac {i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{2}}}{c^{2}}\) | \(252\) |
| default | \(\frac {-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a \arctan \left (c x \right )}{d^{2}}+\frac {i a}{d^{2} \left (c x -i\right )}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}+\frac {i b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}+\frac {b}{2 d^{2} \left (c x -i\right )}-\frac {i b \ln \left (c^{2} x^{2}+1\right )}{8 d^{2}}+\frac {b \arctan \left (c x \right )}{4 d^{2}}+\frac {i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 d^{2}}-\frac {b \arctan \left (\frac {c x}{2}\right )}{8 d^{2}}+\frac {b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 d^{2}}+\frac {b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 d^{2}}-\frac {i b \ln \left (c x -i\right )^{2}}{4 d^{2}}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d^{2}}+\frac {i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{2}}}{c^{2}}\) | \(252\) |
| risch | \(\frac {i b \ln \left (i c x +1\right )^{2}}{4 c^{2} d^{2}}+\frac {b \ln \left (i c x +1\right )}{2 c^{2} d^{2} \left (c x -i\right )}-\frac {i b \ln \left (c^{2} x^{2}+1\right )}{8 d^{2} c^{2}}+\frac {b \arctan \left (c x \right )}{4 c^{2} d^{2}}+\frac {b \ln \left (-i c x +1\right ) x}{4 d^{2} c \left (-i c x -1\right )}+\frac {i b \ln \left (-i c x +1\right )}{4 d^{2} c^{2} \left (-i c x -1\right )}-\frac {i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d^{2} c^{2}}+\frac {i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{2} c^{2}}+\frac {i b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{2} c^{2}}+\frac {a}{d^{2} c^{2} \left (-i c x -1\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2} c^{2}}-\frac {i a \arctan \left (c x \right )}{d^{2} c^{2}}-\frac {b}{2 c^{2} d^{2} \left (-c x +i\right )}\) | \(272\) |
| parts | \(-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2} c^{2}}-\frac {i a \arctan \left (c x \right )}{d^{2} c^{2}}-\frac {i a}{d^{2} c^{2} \left (-c x +i\right )}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{2} d^{2}}+\frac {i b \arctan \left (c x \right )}{c^{2} d^{2} \left (c x -i\right )}+\frac {i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 c^{2} d^{2}}-\frac {b \arctan \left (\frac {c x}{2}\right )}{8 c^{2} d^{2}}+\frac {b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 c^{2} d^{2}}+\frac {b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 c^{2} d^{2}}+\frac {b}{2 c^{2} d^{2} \left (c x -i\right )}-\frac {i b \ln \left (c^{2} x^{2}+1\right )}{8 d^{2} c^{2}}+\frac {b \arctan \left (c x \right )}{4 c^{2} d^{2}}+\frac {i b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{2} d^{2}}+\frac {i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{2} d^{2}}-\frac {i b \ln \left (c x -i\right )^{2}}{4 c^{2} d^{2}}\) | \(294\) |
Input:
int(x*(a+b*arctan(c*x))/(d+I*c*d*x)^2,x,method=_RETURNVERBOSE)
Output:
1/c^2*(-1/2*a/d^2*ln(c^2*x^2+1)-I*a/d^2*arctan(c*x)+I*a/d^2/(c*x-I)-b/d^2* arctan(c*x)*ln(c*x-I)+I*b/d^2*arctan(c*x)/(c*x-I)+1/2*b/d^2/(c*x-I)-1/8*I* b/d^2*ln(c^2*x^2+1)+1/4*b*arctan(c*x)/d^2+1/16*I*b/d^2*ln(c^4*x^4+10*c^2*x ^2+9)-1/8*b/d^2*arctan(1/2*c*x)+1/8*b/d^2*arctan(1/6*c^3*x^3+7/6*c*x)+1/4* b/d^2*arctan(1/2*c*x-1/2*I)-1/4*I*b/d^2*ln(c*x-I)^2+1/2*I*b/d^2*ln(-1/2*I* (c*x+I))*ln(c*x-I)+1/2*I*b/d^2*dilog(-1/2*I*(c*x+I)))
\[ \int \frac {x (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x}{{\left (i \, c d x + d\right )}^{2}} \,d x } \] Input:
integrate(x*(a+b*arctan(c*x))/(d+I*c*d*x)^2,x, algorithm="fricas")
Output:
integral(1/2*(-I*b*x*log(-(c*x + I)/(c*x - I)) - 2*a*x)/(c^2*d^2*x^2 - 2*I *c*d^2*x - d^2), x)
\[ \int \frac {x (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\frac {\left (- i b c x \log {\left (i c x + 1 \right )} - b \log {\left (i c x + 1 \right )} - b\right ) \log {\left (- i c x + 1 \right )}}{2 c^{3} d^{2} x - 2 i c^{2} d^{2}} - \frac {\int \frac {i b}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \frac {i b \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \frac {2 a c^{2} x^{2}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \left (- \frac {b c x}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \frac {2 i a c x}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \left (- \frac {b c x \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \left (- \frac {2 i b c^{2} x^{2} \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx}{2 c d^{2}} \] Input:
integrate(x*(a+b*atan(c*x))/(d+I*c*d*x)**2,x)
Output:
(-I*b*c*x*log(I*c*x + 1) - b*log(I*c*x + 1) - b)*log(-I*c*x + 1)/(2*c**3*d **2*x - 2*I*c**2*d**2) - (Integral(I*b/(c**3*x**3 - I*c**2*x**2 + c*x - I) , x) + Integral(I*b*log(I*c*x + 1)/(c**3*x**3 - I*c**2*x**2 + c*x - I), x) + Integral(2*a*c**2*x**2/(c**3*x**3 - I*c**2*x**2 + c*x - I), x) + Integr al(-b*c*x/(c**3*x**3 - I*c**2*x**2 + c*x - I), x) + Integral(2*I*a*c*x/(c* *3*x**3 - I*c**2*x**2 + c*x - I), x) + Integral(-b*c*x*log(I*c*x + 1)/(c** 3*x**3 - I*c**2*x**2 + c*x - I), x) + Integral(-2*I*b*c**2*x**2*log(I*c*x + 1)/(c**3*x**3 - I*c**2*x**2 + c*x - I), x))/(2*c*d**2)
\[ \int \frac {x (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x}{{\left (i \, c d x + d\right )}^{2}} \,d x } \] Input:
integrate(x*(a+b*arctan(c*x))/(d+I*c*d*x)^2,x, algorithm="maxima")
Output:
a*(I/(c^3*d^2*x - I*c^2*d^2) - log(c*x - I)/(c^2*d^2)) - 1/8*(4*(I*c*x + 1 )*arctan(c*x)^2 + 4*c*x*arctan2(1, c*x) - (-I*c*x - 1)*log(c^2*x^2 + 1)^2 - (c^3*d^2*x - I*c^2*d^2)*((c*(x/(c^5*d^2*x^2 + c^3*d^2) + arctan(c*x)/(c^ 4*d^2)) - 2*arctan(c*x)/(c^5*d^2*x^2 + c^3*d^2))*c + 8*integrate(1/4*log(c ^2*x^2 + 1)/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x)) - (I*c^3*d^2*x + c^ 2*d^2)*(c*(c^2/(c^7*d^2*x^2 + c^5*d^2) + log(c^2*x^2 + 1)/(c^5*d^2*x^2 + c ^3*d^2)) + 16*integrate(1/4*arctan(c*x)/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d ^2), x)) + (c^4*d^2*x - I*c^3*d^2)*(c*(x/(c^5*d^2*x^2 + c^3*d^2) + arctan( c*x)/(c^4*d^2)) - 8*c*integrate(1/4*x^2*log(c^2*x^2 + 1)/(c^5*d^2*x^4 + 2* c^3*d^2*x^2 + c*d^2), x) - 2*arctan(c*x)/(c^5*d^2*x^2 + c^3*d^2)) + (-I*c^ 4*d^2*x - c^3*d^2)*(16*c*integrate(1/4*x^2*arctan(c*x)/(c^5*d^2*x^4 + 2*c^ 3*d^2*x^2 + c*d^2), x) - c^2/(c^7*d^2*x^2 + c^5*d^2) - log(c^2*x^2 + 1)/(c ^5*d^2*x^2 + c^3*d^2)) + 16*(c^5*d^2*x - I*c^4*d^2)*integrate(1/4*(2*c*x^3 *arctan(c*x) + x^2*log(c^2*x^2 + 1))/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2) , x) + 16*(-I*c^5*d^2*x - c^4*d^2)*integrate(1/4*(c*x^3*log(c^2*x^2 + 1) - 2*x^2*arctan(c*x))/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x) - 4*I*arctan (c*x) - 4*I*arctan2(1, c*x) + 2*log(c^2*x^2 + 1))*b/(c^3*d^2*x - I*c^2*d^2 )
\[ \int \frac {x (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x}{{\left (i \, c d x + d\right )}^{2}} \,d x } \] Input:
integrate(x*(a+b*arctan(c*x))/(d+I*c*d*x)^2,x, algorithm="giac")
Output:
integrate((b*arctan(c*x) + a)*x/(I*c*d*x + d)^2, x)
Timed out. \[ \int \frac {x (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \] Input:
int((x*(a + b*atan(c*x)))/(d + c*d*x*1i)^2,x)
Output:
int((x*(a + b*atan(c*x)))/(d + c*d*x*1i)^2, x)
\[ \int \frac {x (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\frac {-\left (\int \frac {\mathit {atan} \left (c x \right ) x}{c^{2} x^{2}-2 c i x -1}d x \right ) b -\left (\int \frac {x}{c^{2} x^{2}-2 c i x -1}d x \right ) a}{d^{2}} \] Input:
int(x*(a+b*atan(c*x))/(d+I*c*d*x)^2,x)
Output:
( - (int((atan(c*x)*x)/(c**2*x**2 - 2*c*i*x - 1),x)*b + int(x/(c**2*x**2 - 2*c*i*x - 1),x)*a))/d**2