Integrand size = 23, antiderivative size = 224 \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d}{3 x}-\frac {1}{3} b^2 c^3 d \arctan (c x)-\frac {b c d (a+b \arctan (c x))}{3 x^2}-\frac {i b c^2 d (a+b \arctan (c x))}{x}-\frac {1}{6} i c^3 d (a+b \arctan (c x))^2-\frac {d (a+b \arctan (c x))^2}{3 x^3}-\frac {i c d (a+b \arctan (c x))^2}{2 x^2}+i b^2 c^3 d \log (x)-\frac {1}{2} i b^2 c^3 d \log \left (1+c^2 x^2\right )-\frac {2}{3} b c^3 d (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )+\frac {1}{3} i b^2 c^3 d \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right ) \]
-1/3*b^2*c^2*d/x-1/3*b^2*c^3*d*arctan(c*x)-1/3*b*c*d*(a+b*arctan(c*x))/x^2 -I*b*c^2*d*(a+b*arctan(c*x))/x-1/6*I*c^3*d*(a+b*arctan(c*x))^2-1/3*d*(a+b* arctan(c*x))^2/x^3-1/2*I*c*d*(a+b*arctan(c*x))^2/x^2+I*b^2*c^3*d*ln(x)-1/2 *I*b^2*c^3*d*ln(c^2*x^2+1)-2/3*b*c^3*d*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x)) +1/3*I*b^2*c^3*d*polylog(2,-1+2/(1-I*c*x))
Time = 0.41 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.07 \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=\frac {d \left (-2 a^2-3 i a^2 c x-2 a b c x-6 i a b c^2 x^2-2 b^2 c^2 x^2-i b^2 \left (-2 i+3 c x+c^3 x^3\right ) \arctan (c x)^2-2 b \arctan (c x) \left (b c x \left (1+3 i c x+c^2 x^2\right )+a \left (2+3 i c x+3 i c^3 x^3\right )+2 b c^3 x^3 \log \left (1-e^{2 i \arctan (c x)}\right )\right )-4 a b c^3 x^3 \log (c x)+6 i b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+2 a b c^3 x^3 \log \left (1+c^2 x^2\right )+2 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{6 x^3} \]
(d*(-2*a^2 - (3*I)*a^2*c*x - 2*a*b*c*x - (6*I)*a*b*c^2*x^2 - 2*b^2*c^2*x^2 - I*b^2*(-2*I + 3*c*x + c^3*x^3)*ArcTan[c*x]^2 - 2*b*ArcTan[c*x]*(b*c*x*( 1 + (3*I)*c*x + c^2*x^2) + a*(2 + (3*I)*c*x + (3*I)*c^3*x^3) + 2*b*c^3*x^3 *Log[1 - E^((2*I)*ArcTan[c*x])]) - 4*a*b*c^3*x^3*Log[c*x] + (6*I)*b^2*c^3* x^3*Log[(c*x)/Sqrt[1 + c^2*x^2]] + 2*a*b*c^3*x^3*Log[1 + c^2*x^2] + (2*I)* b^2*c^3*x^3*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/(6*x^3)
Time = 0.61 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (\frac {d (a+b \arctan (c x))^2}{x^4}+\frac {i c d (a+b \arctan (c x))^2}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{6} i c^3 d (a+b \arctan (c x))^2-\frac {2}{3} b c^3 d \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))-\frac {i b c^2 d (a+b \arctan (c x))}{x}-\frac {d (a+b \arctan (c x))^2}{3 x^3}-\frac {i c d (a+b \arctan (c x))^2}{2 x^2}-\frac {b c d (a+b \arctan (c x))}{3 x^2}-\frac {1}{3} b^2 c^3 d \arctan (c x)+\frac {1}{3} i b^2 c^3 d \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )+i b^2 c^3 d \log (x)-\frac {b^2 c^2 d}{3 x}-\frac {1}{2} i b^2 c^3 d \log \left (c^2 x^2+1\right )\) |
-1/3*(b^2*c^2*d)/x - (b^2*c^3*d*ArcTan[c*x])/3 - (b*c*d*(a + b*ArcTan[c*x] ))/(3*x^2) - (I*b*c^2*d*(a + b*ArcTan[c*x]))/x - (I/6)*c^3*d*(a + b*ArcTan [c*x])^2 - (d*(a + b*ArcTan[c*x])^2)/(3*x^3) - ((I/2)*c*d*(a + b*ArcTan[c* x])^2)/x^2 + I*b^2*c^3*d*Log[x] - (I/2)*b^2*c^3*d*Log[1 + c^2*x^2] - (2*b* c^3*d*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/3 + (I/3)*b^2*c^3*d*Poly Log[2, -1 + 2/(1 - I*c*x)]
3.1.75.3.1 Defintions of rubi rules used
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. 407 vs. \(2 (198 ) = 396\).
Time = 3.14 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.82
| method | result | size |
| parts | \(a^{2} d \left (-\frac {i c}{2 x^{2}}-\frac {1}{3 x^{3}}\right )+d \,b^{2} c^{3} \left (-\frac {\arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{2}+i \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \arctan \left (c x \right ) \ln \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{3}-\frac {i \arctan \left (c x \right )}{c x}-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{3}-\frac {1}{3 c x}-\frac {i \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{3}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{3}\right )+2 a b d \,c^{3} \left (-\frac {\arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c x}-\frac {1}{6 c^{2} x^{2}}-\frac {\ln \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{2}\right )\) | \(408\) |
| derivativedivides | \(c^{3} \left (a^{2} d \left (-\frac {1}{3 c^{3} x^{3}}-\frac {i}{2 c^{2} x^{2}}\right )+d \,b^{2} \left (-\frac {\arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{2}+i \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \arctan \left (c x \right ) \ln \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{3}-\frac {i \arctan \left (c x \right )}{c x}-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{3}-\frac {1}{3 c x}-\frac {i \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{3}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{3}\right )+2 a b d \left (-\frac {\arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c x}-\frac {1}{6 c^{2} x^{2}}-\frac {\ln \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{2}\right )\right )\) | \(411\) |
| default | \(c^{3} \left (a^{2} d \left (-\frac {1}{3 c^{3} x^{3}}-\frac {i}{2 c^{2} x^{2}}\right )+d \,b^{2} \left (-\frac {\arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{2}+i \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \arctan \left (c x \right ) \ln \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{3}-\frac {i \arctan \left (c x \right )}{c x}-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{3}-\frac {1}{3 c x}-\frac {i \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{3}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{3}\right )+2 a b d \left (-\frac {\arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c x}-\frac {1}{6 c^{2} x^{2}}-\frac {\ln \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{2}\right )\right )\) | \(411\) |
a^2*d*(-1/2*I*c/x^2-1/3/x^3)+d*b^2*c^3*(-1/3*arctan(c*x)^2/c^3/x^3-1/2*I*l n(c^2*x^2+1)+I*ln(c*x)-1/3/c^2/x^2*arctan(c*x)-2/3*arctan(c*x)*ln(c*x)+1/3 *arctan(c*x)*ln(c^2*x^2+1)+1/3*I*ln(c*x)*ln(1-I*c*x)-I*arctan(c*x)/c/x-1/3 *I*ln(c*x)*ln(1+I*c*x)-1/3/c/x-1/2*I*arctan(c*x)^2/c^2/x^2-1/3*arctan(c*x) +1/6*I*(ln(c*x-I)*ln(c^2*x^2+1)-dilog(-1/2*I*(c*x+I))-ln(c*x-I)*ln(-1/2*I* (c*x+I))-1/2*ln(c*x-I)^2)-1/3*I*dilog(1+I*c*x)-1/6*I*(ln(c*x+I)*ln(c^2*x^2 +1)-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))-1/2*ln(c*x+I)^2)-1/2* I*arctan(c*x)^2+1/3*I*dilog(1-I*c*x))+2*a*b*d*c^3*(-1/3*arctan(c*x)/c^3/x^ 3-1/2*I*arctan(c*x)/c^2/x^2-1/2*I/c/x-1/6/c^2/x^2-1/3*ln(c*x)+1/6*ln(c^2*x ^2+1)-1/2*I*arctan(c*x))
\[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=\int { \frac {{\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
1/24*(24*x^3*integral(1/6*(6*I*a^2*c^3*d*x^3 + 6*a^2*c^2*d*x^2 + 6*I*a^2*c *d*x + 6*a^2*d - (6*a*b*c^3*d*x^3 + 3*(-2*I*a*b + b^2)*c^2*d*x^2 + 2*(3*a* b - I*b^2)*c*d*x - 6*I*a*b*d)*log(-(c*x + I)/(c*x - I)))/(c^2*x^6 + x^4), x) + (3*I*b^2*c*d*x + 2*b^2*d)*log(-(c*x + I)/(c*x - I))^2)/x^3
Timed out. \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=\int { \frac {{\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
-I*((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*a*b*c*d + 1/3*((c^2*log(c^2 *x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)*a*b*d - 1/2*I*a^2 *c*d/x^2 - 1/3*a^2*d/x^3 + 1/96*(96*I*x^3*integrate(1/48*(20*b^2*c^2*d*x^2 *arctan(c*x) + 36*(b^2*c^3*d*x^3 + b^2*c*d*x)*arctan(c*x)^2 + 3*(b^2*c^3*d *x^3 + b^2*c*d*x)*log(c^2*x^2 + 1)^2 - 2*(3*b^2*c^3*d*x^3 - 2*b^2*c*d*x + 6*(b^2*c^2*d*x^2 + b^2*d)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^6 + x^4), x) + 96*x^3*integrate(1/48*(36*(b^2*c^2*d*x^2 + b^2*d)*arctan(c*x)^2 + 3*( b^2*c^2*d*x^2 + b^2*d)*log(c^2*x^2 + 1)^2 - 4*(3*b^2*c^3*d*x^3 - 2*b^2*c*d *x)*arctan(c*x) - 2*(5*b^2*c^2*d*x^2 - 6*(b^2*c^3*d*x^3 + b^2*c*d*x)*arcta n(c*x))*log(c^2*x^2 + 1))/(c^2*x^6 + x^4), x) - 4*(3*I*b^2*c*d*x + 2*b^2*d )*arctan(c*x)^2 + 4*(3*b^2*c*d*x - 2*I*b^2*d)*arctan(c*x)*log(c^2*x^2 + 1) + (3*I*b^2*c*d*x + 2*b^2*d)*log(c^2*x^2 + 1)^2)/x^3
Timed out. \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\,1{}\mathrm {i}\right )}{x^4} \,d x \]