Integrand size = 22, antiderivative size = 226 \[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=-\frac {7}{2} i a b d^3 x-b^2 d^3 x-\frac {1}{12} i b^2 c d^3 x^2+\frac {b^2 d^3 \arctan (c x)}{c}-\frac {7}{2} i b^2 d^3 x \arctan (c x)+b c d^3 x^2 (a+b \arctan (c x))+\frac {1}{6} i b c^2 d^3 x^3 (a+b \arctan (c x))-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}+\frac {4 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{c}+\frac {11 i b^2 d^3 \log \left (1+c^2 x^2\right )}{6 c}-\frac {2 i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c} \]
-7/2*I*a*b*d^3*x-b^2*d^3*x-1/12*I*b^2*c*d^3*x^2+b^2*d^3*arctan(c*x)/c-7/2* I*b^2*d^3*x*arctan(c*x)+b*c*d^3*x^2*(a+b*arctan(c*x))+1/6*I*b*c^2*d^3*x^3* (a+b*arctan(c*x))-1/4*I*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/c+4*b*d^3*(a+b *arctan(c*x))*ln(2/(1-I*c*x))/c+11/6*I*b^2*d^3*ln(c^2*x^2+1)/c-2*I*b^2*d^3 *polylog(2,1-2/(1-I*c*x))/c
Time = 1.65 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.18 \[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=-\frac {i d^3 \left (b^2+12 i a^2 c x+42 a b c x-12 i b^2 c x-18 a^2 c^2 x^2+12 i a b c^2 x^2+b^2 c^2 x^2-12 i a^2 c^3 x^3-2 a b c^3 x^3+3 a^2 c^4 x^4+3 b^2 (-i+c x)^4 \arctan (c x)^2+2 b \arctan (c x) \left (b \left (6 i+21 c x+6 i c^2 x^2-c^3 x^3\right )+3 a \left (-7+4 i c x-6 c^2 x^2-4 i c^3 x^3+c^4 x^4\right )+24 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-24 i a b \log \left (1+c^2 x^2\right )-22 b^2 \log \left (1+c^2 x^2\right )+24 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{12 c} \]
((-1/12*I)*d^3*(b^2 + (12*I)*a^2*c*x + 42*a*b*c*x - (12*I)*b^2*c*x - 18*a^ 2*c^2*x^2 + (12*I)*a*b*c^2*x^2 + b^2*c^2*x^2 - (12*I)*a^2*c^3*x^3 - 2*a*b* c^3*x^3 + 3*a^2*c^4*x^4 + 3*b^2*(-I + c*x)^4*ArcTan[c*x]^2 + 2*b*ArcTan[c* x]*(b*(6*I + 21*c*x + (6*I)*c^2*x^2 - c^3*x^3) + 3*a*(-7 + (4*I)*c*x - 6*c ^2*x^2 - (4*I)*c^3*x^3 + c^4*x^4) + (24*I)*b*Log[1 + E^((2*I)*ArcTan[c*x]) ]) - (24*I)*a*b*Log[1 + c^2*x^2] - 22*b^2*Log[1 + c^2*x^2] + 24*b^2*PolyLo g[2, -E^((2*I)*ArcTan[c*x])]))/c
Time = 0.41 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.95, 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.
| \(\displaystyle \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {i b \int \left (c^2 x^2 (a+b \arctan (c x)) d^4-4 i c x (a+b \arctan (c x)) d^4-\frac {8 i (i-c x) (a+b \arctan (c x)) d^4}{c^2 x^2+1}-7 (a+b \arctan (c x)) d^4\right )dx}{2 d}-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i b \left (\frac {1}{3} c^2 d^4 x^3 (a+b \arctan (c x))-2 i c d^4 x^2 (a+b \arctan (c x))-\frac {8 i d^4 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{c}-7 a d^4 x-7 b d^4 x \arctan (c x)-\frac {2 i b d^4 \arctan (c x)}{c}+\frac {11 b d^4 \log \left (c^2 x^2+1\right )}{3 c}-\frac {4 b d^4 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}-\frac {1}{6} b c d^4 x^2+2 i b d^4 x\right )}{2 d}-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}\) |
((-1/4*I)*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/c + ((I/2)*b*(-7*a*d^4* x + (2*I)*b*d^4*x - (b*c*d^4*x^2)/6 - ((2*I)*b*d^4*ArcTan[c*x])/c - 7*b*d^ 4*x*ArcTan[c*x] - (2*I)*c*d^4*x^2*(a + b*ArcTan[c*x]) + (c^2*d^4*x^3*(a + b*ArcTan[c*x]))/3 - ((8*I)*d^4*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/c + (11*b*d^4*Log[1 + c^2*x^2])/(3*c) - (4*b*d^4*PolyLog[2, 1 - 2/(1 - I*c*x) ])/c))/d
3.1.87.3.1 Defintions of rubi rules used
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. 411 vs. \(2 (204 ) = 408\).
Time = 1.52 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.82
| method | result | size |
| derivativedivides | \(\frac {-\frac {i d^{3} a^{2} \left (i c x +1\right )^{4}}{4}+b^{2} d^{3} \left (-\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )^{2}+\frac {3 i \arctan \left (c x \right )^{2} c^{2} x^{2}}{2}+\arctan \left (c x \right )^{2} c x -\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \left (-7 c x \arctan \left (c x \right )+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+4 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-2 i \arctan \left (c x \right ) c^{2} x^{2}+4 \arctan \left (c x \right )^{2}-2 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right )^{2}-\ln \left (c x +i\right )^{2}-2 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+2 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-2 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-2 i \arctan \left (c x \right )-\frac {c^{2} x^{2}}{6}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{3}+2 i c x \right )}{2}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )+\frac {3 i \arctan \left (c x \right ) c^{2} x^{2}}{2}+c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right )}{4}+\frac {i \left (-7 c x +\frac {c^{3} x^{3}}{3}-2 i c^{2} x^{2}+4 i \ln \left (c^{2} x^{2}+1\right )+8 \arctan \left (c x \right )\right )}{4}\right )}{c}\) | \(412\) |
| default | \(\frac {-\frac {i d^{3} a^{2} \left (i c x +1\right )^{4}}{4}+b^{2} d^{3} \left (-\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )^{2}+\frac {3 i \arctan \left (c x \right )^{2} c^{2} x^{2}}{2}+\arctan \left (c x \right )^{2} c x -\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \left (-7 c x \arctan \left (c x \right )+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+4 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-2 i \arctan \left (c x \right ) c^{2} x^{2}+4 \arctan \left (c x \right )^{2}-2 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right )^{2}-\ln \left (c x +i\right )^{2}-2 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+2 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-2 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-2 i \arctan \left (c x \right )-\frac {c^{2} x^{2}}{6}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{3}+2 i c x \right )}{2}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )+\frac {3 i \arctan \left (c x \right ) c^{2} x^{2}}{2}+c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right )}{4}+\frac {i \left (-7 c x +\frac {c^{3} x^{3}}{3}-2 i c^{2} x^{2}+4 i \ln \left (c^{2} x^{2}+1\right )+8 \arctan \left (c x \right )\right )}{4}\right )}{c}\) | \(412\) |
| parts | \(-\frac {i d^{3} a^{2} \left (i c x +1\right )^{4}}{4 c}+\frac {b^{2} d^{3} \left (-\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )^{2}+\frac {3 i \arctan \left (c x \right )^{2} c^{2} x^{2}}{2}+\arctan \left (c x \right )^{2} c x -\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \left (-7 c x \arctan \left (c x \right )+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+4 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-2 i \arctan \left (c x \right ) c^{2} x^{2}+4 \arctan \left (c x \right )^{2}-2 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right )^{2}-\ln \left (c x +i\right )^{2}-2 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+2 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-2 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-2 i \arctan \left (c x \right )-\frac {c^{2} x^{2}}{6}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{3}+2 i c x \right )}{2}\right )}{c}+\frac {2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )+\frac {3 i \arctan \left (c x \right ) c^{2} x^{2}}{2}+c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right )}{4}+\frac {i \left (-7 c x +\frac {c^{3} x^{3}}{3}-2 i c^{2} x^{2}+4 i \ln \left (c^{2} x^{2}+1\right )+8 \arctan \left (c x \right )\right )}{4}\right )}{c}\) | \(417\) |
| risch | \(-b^{2} d^{3} x -\frac {2 b \ln \left (c^{2} x^{2}+1\right ) a \,d^{3}}{c}+\frac {47 b^{2} d^{3} \arctan \left (c x \right )}{32 c}+\frac {14 a b \,d^{3}}{3 c}+x \,d^{3} a^{2}-\frac {7 i a b \,d^{3} x}{2}+\frac {7 d^{3} b^{2} \ln \left (-i c x +1\right ) x}{4}+a b c \,d^{3} x^{2}-a^{2} c^{2} d^{3} x^{3}+i \ln \left (-i c x +1\right ) x a b \,d^{3}+\frac {i d^{3} \left (c x -i\right )^{4} b^{2} \ln \left (i c x +1\right )^{2}}{16 c}-\frac {\ln \left (-i c x +1\right )^{2} x \,b^{2} d^{3}}{4}-\frac {13 i b^{2} d^{3}}{12 c}+\frac {15 i d^{3} a^{2}}{4 c}+\frac {i d^{3} c^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}-\frac {3 i d^{3} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{8}+\frac {2 i b^{2} \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d^{3}}{c}-\frac {2 i b^{2} \ln \left (-i c x +1\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d^{3}}{c}+\frac {7 i b \arctan \left (c x \right ) a \,d^{3}}{2 c}-\frac {3 d^{3} c a b \ln \left (-i c x +1\right ) x^{2}}{2}+\frac {d^{3} c^{3} b a \ln \left (-i c x +1\right ) x^{4}}{4}+\frac {i d^{3} c^{2} a b \,x^{3}}{6}+\frac {i b^{2} d^{3} c \ln \left (-i c x +1\right ) x^{2}}{2}-i d^{3} c^{2} a b \ln \left (-i c x +1\right ) x^{3}+\left (-\frac {i d^{3} \left (c x -i\right )^{4} b^{2} \ln \left (-i c x +1\right )}{8 c}-\frac {b \,d^{3} \left (3 a \,c^{4} x^{4}-12 i a \,c^{3} x^{3}-b \,c^{3} x^{3}+6 i b \,c^{2} x^{2}-18 c^{2} x^{2} a +12 i x a c -24 i b \ln \left (-i c x +1\right )+21 x b c \right )}{12 c}\right ) \ln \left (i c x +1\right )-\frac {15 i d^{3} b^{2} \ln \left (-i c x +1\right )^{2}}{16 c}-\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right ) x^{3}}{12}+\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{4}-\frac {i a^{2} c^{3} d^{3} x^{4}}{4}+\frac {3 i d^{3} c \,x^{2} a^{2}}{2}+\frac {397 i b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )}{192 c}-\frac {15 i b^{2} \ln \left (-i c x +1\right ) d^{3}}{32 c}+\frac {2 i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right ) d^{3}}{c}-\frac {i b^{2} c \,d^{3} x^{2}}{12}\) | \(709\) |
1/c*(-1/4*I*d^3*a^2*(1+I*c*x)^4+b^2*d^3*(-1/4*I*arctan(c*x)^2*c^4*x^4-c^3* x^3*arctan(c*x)^2+3/2*I*arctan(c*x)^2*c^2*x^2+arctan(c*x)^2*c*x-1/4*I*arct an(c*x)^2+1/2*I*(-7*c*x*arctan(c*x)+1/3*c^3*x^3*arctan(c*x)+4*I*arctan(c*x )*ln(c^2*x^2+1)-2*I*arctan(c*x)*c^2*x^2+4*arctan(c*x)^2-2*ln(c*x-I)*ln(c^2 *x^2+1)+2*ln(c*x+I)*ln(c^2*x^2+1)+2*ln(c*x-I)*ln(-1/2*I*(c*x+I))+ln(c*x-I) ^2-ln(c*x+I)^2-2*ln(c*x+I)*ln(1/2*I*(c*x-I))+2*dilog(-1/2*I*(c*x+I))-2*dil og(1/2*I*(c*x-I))-2*I*arctan(c*x)-1/6*c^2*x^2+11/3*ln(c^2*x^2+1)+2*I*c*x)) +2*a*d^3*b*(-1/4*I*arctan(c*x)*c^4*x^4-c^3*x^3*arctan(c*x)+3/2*I*arctan(c* x)*c^2*x^2+c*x*arctan(c*x)-1/4*I*arctan(c*x)+1/4*I*(-7*c*x+1/3*c^3*x^3-2*I *c^2*x^2+4*I*ln(c^2*x^2+1)+8*arctan(c*x))))
\[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
1/16*(I*b^2*c^3*d^3*x^4 + 4*b^2*c^2*d^3*x^3 - 6*I*b^2*c*d^3*x^2 - 4*b^2*d^ 3*x)*log(-(c*x + I)/(c*x - I))^2 + integral(1/4*(-4*I*a^2*c^5*d^3*x^5 - 12 *a^2*c^4*d^3*x^4 + 8*I*a^2*c^3*d^3*x^3 - 8*a^2*c^2*d^3*x^2 + 12*I*a^2*c*d^ 3*x + 4*a^2*d^3 + (4*a*b*c^5*d^3*x^5 + (-12*I*a*b - b^2)*c^4*d^3*x^4 - 4*( 2*a*b - I*b^2)*c^3*d^3*x^3 - 2*(4*I*a*b - 3*b^2)*c^2*d^3*x^2 - 4*(3*a*b + I*b^2)*c*d^3*x + 4*I*a*b*d^3)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)
Timed out. \[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]
\[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
-1/4*I*a^2*c^3*d^3*x^4 - 4*b^2*c^5*d^3*integrate(1/16*x^5*arctan(c*x)*log( c^2*x^2 + 1)/(c^2*x^2 + 1), x) - 2*b^2*c^5*d^3*integrate(1/16*x^5*arctan(c *x)/(c^2*x^2 + 1), x) - a^2*c^2*d^3*x^3 - 36*b^2*c^4*d^3*integrate(1/16*x^ 4*arctan(c*x)^2/(c^2*x^2 + 1), x) - 3*b^2*c^4*d^3*integrate(1/16*x^4*log(c ^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 5*b^2*c^4*d^3*integrate(1/16*x^4*log(c^2 *x^2 + 1)/(c^2*x^2 + 1), x) - 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^3*d^3 + 8*b^2*c^3*d^3*integrate(1/16*x^3 *arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 20*b^2*c^3*d^3*integrate (1/16*x^3*arctan(c*x)/(c^2*x^2 + 1), x) - (2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*c^2*d^3 + 3/2*I*a^2*c*d^3*x^2 - 24*b^2*c^2*d^ 3*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) - 2*b^2*c^2*d^3*integ rate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 10*b^2*c^2*d^3*integr ate(1/16*x^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3*I*(x^2*arctan(c*x) - c *(x/c^2 - arctan(c*x)/c^3))*a*b*c*d^3 + 1/4*b^2*d^3*arctan(c*x)^3/c + 12*b ^2*c*d^3*integrate(1/16*x*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - 8*b^2*c*d^3*integrate(1/16*x*arctan(c*x)/(c^2*x^2 + 1), x) + a^2*d^3*x + b^2*d^3*integrate(1/16*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + (2*c*x*arcta n(c*x) - log(c^2*x^2 + 1))*a*b*d^3/c + 1/16*(-I*b^2*c^3*d^3*x^4 - 4*b^2*c^ 2*d^3*x^3 + 6*I*b^2*c*d^3*x^2 + 4*b^2*d^3*x)*arctan(c*x)^2 + 1/16*(b^2*c^3 *d^3*x^4 - 4*I*b^2*c^2*d^3*x^3 - 6*b^2*c*d^3*x^2 + 4*I*b^2*d^3*x)*arcta...
\[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3 \,d x \]