Integrand size = 23, antiderivative size = 255 \[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\frac {i a b d x}{2 c^2}+\frac {b^2 d x}{3 c^2}+\frac {i b^2 d x^2}{12 c}-\frac {b^2 d \arctan (c x)}{3 c^3}+\frac {i b^2 d x \arctan (c x)}{2 c^2}-\frac {b d x^2 (a+b \arctan (c x))}{3 c}-\frac {1}{6} i b d x^3 (a+b \arctan (c x))-\frac {7 i d (a+b \arctan (c x))^2}{12 c^3}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{4} i c d x^4 (a+b \arctan (c x))^2-\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}-\frac {i b^2 d \log \left (1+c^2 x^2\right )}{3 c^3}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \]
1/2*I*a*b*d*x/c^2+1/3*b^2*d*x/c^2+1/12*I*b^2*d*x^2/c-1/3*b^2*d*arctan(c*x) /c^3+1/2*I*b^2*d*x*arctan(c*x)/c^2-1/3*b*d*x^2*(a+b*arctan(c*x))/c-1/6*I*b *d*x^3*(a+b*arctan(c*x))-7/12*I*d*(a+b*arctan(c*x))^2/c^3+1/3*d*x^3*(a+b*a rctan(c*x))^2+1/4*I*c*d*x^4*(a+b*arctan(c*x))^2-2/3*b*d*(a+b*arctan(c*x))* ln(2/(1+I*c*x))/c^3-1/3*I*b^2*d*ln(c^2*x^2+1)/c^3-1/3*I*b^2*d*polylog(2,1- 2/(1+I*c*x))/c^3
Time = 0.49 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.95 \[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\frac {i d \left (b^2+6 a b c x-4 i b^2 c x+4 i a b c^2 x^2+b^2 c^2 x^2-4 i a^2 c^3 x^3-2 a b c^3 x^3+3 a^2 c^4 x^4+b^2 \left (1-4 i c^3 x^3+3 c^4 x^4\right ) \arctan (c x)^2+2 b \arctan (c x) \left (b \left (2 i+3 c x+2 i c^2 x^2-c^3 x^3\right )+a \left (-3-4 i c^3 x^3+3 c^4 x^4\right )+4 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-4 i a b \log \left (1+c^2 x^2\right )-4 b^2 \log \left (1+c^2 x^2\right )+4 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{12 c^3} \]
((I/12)*d*(b^2 + 6*a*b*c*x - (4*I)*b^2*c*x + (4*I)*a*b*c^2*x^2 + b^2*c^2*x ^2 - (4*I)*a^2*c^3*x^3 - 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + b^2*(1 - (4*I)*c^ 3*x^3 + 3*c^4*x^4)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(2*I + 3*c*x + (2*I) *c^2*x^2 - c^3*x^3) + a*(-3 - (4*I)*c^3*x^3 + 3*c^4*x^4) + (4*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (4*I)*a*b*Log[1 + c^2*x^2] - 4*b^2*Log[1 + c^2* x^2] + 4*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c^3
Time = 0.67 (sec) , antiderivative size = 255, 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.
| \(\displaystyle \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (d x^2 (a+b \arctan (c x))^2+i c d x^3 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {7 i d (a+b \arctan (c x))^2}{12 c^3}-\frac {2 b d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^3}+\frac {1}{4} i c d x^4 (a+b \arctan (c x))^2+\frac {1}{3} d x^3 (a+b \arctan (c x))^2-\frac {1}{6} i b d x^3 (a+b \arctan (c x))-\frac {b d x^2 (a+b \arctan (c x))}{3 c}+\frac {i a b d x}{2 c^2}-\frac {b^2 d \arctan (c x)}{3 c^3}+\frac {i b^2 d x \arctan (c x)}{2 c^2}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {b^2 d x}{3 c^2}-\frac {i b^2 d \log \left (c^2 x^2+1\right )}{3 c^3}+\frac {i b^2 d x^2}{12 c}\) |
((I/2)*a*b*d*x)/c^2 + (b^2*d*x)/(3*c^2) + ((I/12)*b^2*d*x^2)/c - (b^2*d*Ar cTan[c*x])/(3*c^3) + ((I/2)*b^2*d*x*ArcTan[c*x])/c^2 - (b*d*x^2*(a + b*Arc Tan[c*x]))/(3*c) - (I/6)*b*d*x^3*(a + b*ArcTan[c*x]) - (((7*I)/12)*d*(a + b*ArcTan[c*x])^2)/c^3 + (d*x^3*(a + b*ArcTan[c*x])^2)/3 + (I/4)*c*d*x^4*(a + b*ArcTan[c*x])^2 - (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^ 3) - ((I/3)*b^2*d*Log[1 + c^2*x^2])/c^3 - ((I/3)*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3
3.1.69.3.1 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])
Time = 1.42 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.38
| method | result | size |
| parts | \(a^{2} d \left (\frac {1}{4} i c \,x^{4}+\frac {1}{3} x^{3}\right )+\frac {d \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{6}-\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \arctan \left (c x \right ) c x}{2}-\frac {c^{2} x^{2} \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c x}{3}+\frac {i c^{2} x^{2}}{12}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}\right )}{c^{3}}+\frac {2 a b d \left (\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{4}-\frac {i c^{3} x^{3}}{12}-\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{4}\right )}{c^{3}}\) | \(352\) |
| derivativedivides | \(\frac {a^{2} d \left (\frac {1}{4} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+d \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{6}-\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \arctan \left (c x \right ) c x}{2}-\frac {c^{2} x^{2} \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c x}{3}+\frac {i c^{2} x^{2}}{12}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}\right )+2 a b d \left (\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{4}-\frac {i c^{3} x^{3}}{12}-\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{4}\right )}{c^{3}}\) | \(355\) |
| default | \(\frac {a^{2} d \left (\frac {1}{4} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+d \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{6}-\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \arctan \left (c x \right ) c x}{2}-\frac {c^{2} x^{2} \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c x}{3}+\frac {i c^{2} x^{2}}{12}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}\right )+2 a b d \left (\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{4}-\frac {i c^{3} x^{3}}{12}-\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{4}\right )}{c^{3}}\) | \(355\) |
| risch | \(\frac {b^{2} d x}{3 c^{2}}-\frac {115 b^{2} d \arctan \left (c x \right )}{288 c^{3}}-\frac {a b d}{c^{3}}+\frac {a^{2} d \,x^{3}}{3}+\frac {b d a \ln \left (c^{2} x^{2}+1\right )}{3 c^{3}}-\frac {d a b \,x^{2}}{3 c}+\frac {i d a b \ln \left (-i c x +1\right ) x^{3}}{3}-\frac {i d c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}-\frac {i b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}+\frac {i b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{3 c^{3}}-\frac {i b d a \arctan \left (c x \right )}{2 c^{3}}-\frac {i d \,b^{2} \ln \left (-i c x +1\right ) x^{2}}{6 c}-\frac {d c a b \ln \left (-i c x +1\right ) x^{4}}{4}+\frac {i a b d x}{2 c^{2}}-\frac {i d \,b^{2} \left (3 c^{4} x^{4}-4 i c^{3} x^{3}+1\right ) \ln \left (i c x +1\right )^{2}}{48 c^{3}}+\left (\frac {i d \,b^{2} \left (3 c \,x^{4}-4 i x^{3}\right ) \ln \left (-i c x +1\right )}{24}+\frac {b d \left (6 a \,c^{4} x^{4}-8 i a \,c^{3} x^{3}-2 b \,c^{3} x^{3}+4 i b \,c^{2} x^{2}-7 i b \ln \left (-i c x +1\right )+6 x b c \right )}{24 c^{3}}\right ) \ln \left (i c x +1\right )-\frac {211 i b^{2} d \ln \left (c^{2} x^{2}+1\right )}{576 c^{3}}-\frac {i d a b \,x^{3}}{6}+\frac {19 i b^{2} d \ln \left (-i c x +1\right )}{288 c^{3}}-\frac {i b^{2} d \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}-\frac {d \,b^{2} \ln \left (-i c x +1\right ) x}{4 c^{2}}+\frac {7 i d \,b^{2} \ln \left (-i c x +1\right )^{2}}{48 c^{3}}+\frac {i a^{2} c d \,x^{4}}{4}+\frac {i b^{2} d \,x^{2}}{12 c}+\frac {5 i b^{2} d}{12 c^{3}}-\frac {7 i d \,a^{2}}{12 c^{3}}-\frac {d \,b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}+\frac {d \,b^{2} \ln \left (-i c x +1\right ) x^{3}}{12}\) | \(545\) |
a^2*d*(1/4*I*c*x^4+1/3*x^3)+d*b^2/c^3*(1/4*I*arctan(c*x)^2*c^4*x^4+1/3*c^3 *x^3*arctan(c*x)^2-1/6*I*arctan(c*x)*c^3*x^3-1/4*I*arctan(c*x)^2+1/2*I*arc tan(c*x)*c*x-1/3*c^2*x^2*arctan(c*x)+1/3*arctan(c*x)*ln(c^2*x^2+1)+1/3*c*x +1/12*I*c^2*x^2-1/3*I*ln(c^2*x^2+1)-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)-1/6*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))+2*a*b*d/c^3*(1/4*I*arctan(c*x)*c^4*x^4+1/3*c^3* x^3*arctan(c*x)+1/4*I*c*x-1/12*I*c^3*x^3-1/6*c^2*x^2+1/6*ln(c^2*x^2+1)-1/4 *I*arctan(c*x))
\[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
1/48*(-3*I*b^2*c*d*x^4 - 4*b^2*d*x^3)*log(-(c*x + I)/(c*x - I))^2 + integr al(1/12*(12*I*a^2*c^3*d*x^5 + 12*a^2*c^2*d*x^4 + 12*I*a^2*c*d*x^3 + 12*a^2 *d*x^2 - (12*a*b*c^3*d*x^5 + 3*(-4*I*a*b - b^2)*c^2*d*x^4 + 4*(3*a*b + I*b ^2)*c*d*x^3 - 12*I*a*b*d*x^2)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)
Timed out. \[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]
\[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
1/4*I*a^2*c*d*x^4 + 1/3*a^2*d*x^3 + 1/6*I*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a*b*c*d + 1/3*(2*x^3*arctan(c*x) - c*(x^ 2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*d - 1/48*(-3*I*b^2*c*d*x^4 - 4*b^2*d*x^ 3)*arctan(c*x)^2 - 1/48*(3*b^2*c*d*x^4 - 4*I*b^2*d*x^3)*arctan(c*x)*log(c^ 2*x^2 + 1) + 1/192*(-3*I*b^2*c*d*x^4 - 4*b^2*d*x^3)*log(c^2*x^2 + 1)^2 + I *integrate(-1/48*(14*b^2*c^2*d*x^4*arctan(c*x) - 36*(b^2*c^3*d*x^5 + b^2*c *d*x^3)*arctan(c*x)^2 - 3*(b^2*c^3*d*x^5 + b^2*c*d*x^3)*log(c^2*x^2 + 1)^2 - (3*b^2*c^3*d*x^5 - 4*b^2*c*d*x^3 - 12*(b^2*c^2*d*x^4 + b^2*d*x^2)*arcta n(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x) + integrate(1/48*(36*(b^2*c^2* d*x^4 + b^2*d*x^2)*arctan(c*x)^2 + 3*(b^2*c^2*d*x^4 + b^2*d*x^2)*log(c^2*x ^2 + 1)^2 + 2*(3*b^2*c^3*d*x^5 - 4*b^2*c*d*x^3)*arctan(c*x) + (7*b^2*c^2*d *x^4 + 12*(b^2*c^3*d*x^5 + b^2*c*d*x^3)*arctan(c*x))*log(c^2*x^2 + 1))/(c^ 2*x^2 + 1), x)
\[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\,1{}\mathrm {i}\right ) \,d x \]