Integrand size = 22, antiderivative size = 382 \[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=-3 a b^2 d^3 x+\frac {1}{4} i b^3 d^3 x-\frac {i b^3 d^3 \arctan (c x)}{4 c}-3 b^3 d^3 x \arctan (c x)-\frac {1}{4} i b^2 c d^3 x^2 (a+b \arctan (c x))+\frac {7 b d^3 (a+b \arctan (c x))^2}{c}-\frac {21}{4} i b d^3 x (a+b \arctan (c x))^2+\frac {3}{2} b c d^3 x^2 (a+b \arctan (c x))^2+\frac {1}{4} i b c^2 d^3 x^3 (a+b \arctan (c x))^2-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {6 b d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {11 i b^2 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 d^3 \log \left (1+c^2 x^2\right )}{2 c}-\frac {6 i b^2 d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}+\frac {11 b^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c}+\frac {3 b^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{c} \]
-3*a*b^2*d^3*x+1/4*I*b*c^2*d^3*x^3*(a+b*arctan(c*x))^2-1/4*I*b^2*c*d^3*x^2 *(a+b*arctan(c*x))-3*b^3*d^3*x*arctan(c*x)-21/4*I*b*d^3*x*(a+b*arctan(c*x) )^2+7*b*d^3*(a+b*arctan(c*x))^2/c-1/4*I*b^3*d^3*arctan(c*x)/c+3/2*b*c*d^3* x^2*(a+b*arctan(c*x))^2+1/4*I*b^3*d^3*x-6*I*b^2*d^3*(a+b*arctan(c*x))*poly log(2,1-2/(1-I*c*x))/c+6*b*d^3*(a+b*arctan(c*x))^2*ln(2/(1-I*c*x))/c-11*I* b^2*d^3*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c+3/2*b^3*d^3*ln(c^2*x^2+1)/c-1/ 4*I*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^3/c+11/2*b^3*d^3*polylog(2,1-2/(1+I* c*x))/c+3*b^3*d^3*polylog(3,1-2/(1-I*c*x))/c
Time = 2.14 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.81 \[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=-\frac {i d^3 \left (a b^2+4 i a^3 c x+21 a^2 b c x-12 i a b^2 c x-b^3 c x-6 a^3 c^2 x^2+6 i a^2 b c^2 x^2+a b^2 c^2 x^2-4 i a^3 c^3 x^3-a^2 b c^3 x^3+a^3 c^4 x^4-21 a^2 b \arctan (c x)+12 i a b^2 \arctan (c x)+b^3 \arctan (c x)+12 i a^2 b c x \arctan (c x)+42 a b^2 c x \arctan (c x)-12 i b^3 c x \arctan (c x)-18 a^2 b c^2 x^2 \arctan (c x)+12 i a b^2 c^2 x^2 \arctan (c x)+b^3 c^2 x^2 \arctan (c x)-12 i a^2 b c^3 x^3 \arctan (c x)-2 a b^2 c^3 x^3 \arctan (c x)+3 a^2 b c^4 x^4 \arctan (c x)+3 a b^2 \arctan (c x)^2-16 i b^3 \arctan (c x)^2+12 i a b^2 c x \arctan (c x)^2+21 b^3 c x \arctan (c x)^2-18 a b^2 c^2 x^2 \arctan (c x)^2+6 i b^3 c^2 x^2 \arctan (c x)^2-12 i a b^2 c^3 x^3 \arctan (c x)^2-b^3 c^3 x^3 \arctan (c x)^2+3 a b^2 c^4 x^4 \arctan (c x)^2+b^3 \arctan (c x)^3+4 i b^3 c x \arctan (c x)^3-6 b^3 c^2 x^2 \arctan (c x)^3-4 i b^3 c^3 x^3 \arctan (c x)^3+b^3 c^4 x^4 \arctan (c x)^3+48 i a b^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+44 b^3 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+24 i b^3 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-12 i a^2 b \log \left (1+c^2 x^2\right )-22 a b^2 \log \left (1+c^2 x^2\right )+6 i b^3 \log \left (1+c^2 x^2\right )+2 b^2 (12 a-11 i b+12 b \arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+12 i b^3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{4 c} \]
((-1/4*I)*d^3*(a*b^2 + (4*I)*a^3*c*x + 21*a^2*b*c*x - (12*I)*a*b^2*c*x - b ^3*c*x - 6*a^3*c^2*x^2 + (6*I)*a^2*b*c^2*x^2 + a*b^2*c^2*x^2 - (4*I)*a^3*c ^3*x^3 - a^2*b*c^3*x^3 + a^3*c^4*x^4 - 21*a^2*b*ArcTan[c*x] + (12*I)*a*b^2 *ArcTan[c*x] + b^3*ArcTan[c*x] + (12*I)*a^2*b*c*x*ArcTan[c*x] + 42*a*b^2*c *x*ArcTan[c*x] - (12*I)*b^3*c*x*ArcTan[c*x] - 18*a^2*b*c^2*x^2*ArcTan[c*x] + (12*I)*a*b^2*c^2*x^2*ArcTan[c*x] + b^3*c^2*x^2*ArcTan[c*x] - (12*I)*a^2 *b*c^3*x^3*ArcTan[c*x] - 2*a*b^2*c^3*x^3*ArcTan[c*x] + 3*a^2*b*c^4*x^4*Arc Tan[c*x] + 3*a*b^2*ArcTan[c*x]^2 - (16*I)*b^3*ArcTan[c*x]^2 + (12*I)*a*b^2 *c*x*ArcTan[c*x]^2 + 21*b^3*c*x*ArcTan[c*x]^2 - 18*a*b^2*c^2*x^2*ArcTan[c* x]^2 + (6*I)*b^3*c^2*x^2*ArcTan[c*x]^2 - (12*I)*a*b^2*c^3*x^3*ArcTan[c*x]^ 2 - b^3*c^3*x^3*ArcTan[c*x]^2 + 3*a*b^2*c^4*x^4*ArcTan[c*x]^2 + b^3*ArcTan [c*x]^3 + (4*I)*b^3*c*x*ArcTan[c*x]^3 - 6*b^3*c^2*x^2*ArcTan[c*x]^3 - (4*I )*b^3*c^3*x^3*ArcTan[c*x]^3 + b^3*c^4*x^4*ArcTan[c*x]^3 + (48*I)*a*b^2*Arc Tan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 44*b^3*ArcTan[c*x]*Log[1 + E^((2 *I)*ArcTan[c*x])] + (24*I)*b^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x]) ] - (12*I)*a^2*b*Log[1 + c^2*x^2] - 22*a*b^2*Log[1 + c^2*x^2] + (6*I)*b^3* Log[1 + c^2*x^2] + 2*b^2*(12*a - (11*I)*b + 12*b*ArcTan[c*x])*PolyLog[2, - E^((2*I)*ArcTan[c*x])] + (12*I)*b^3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/c
Time = 0.88 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.99, 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.
| \(\displaystyle \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {3 i b \int \left (c^2 x^2 (a+b \arctan (c x))^2 d^4-4 i c x (a+b \arctan (c x))^2 d^4-\frac {8 i (i-c x) (a+b \arctan (c x))^2 d^4}{c^2 x^2+1}-7 (a+b \arctan (c x))^2 d^4\right )dx}{4 d}-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 i b \left (\frac {1}{3} c^2 d^4 x^3 (a+b \arctan (c x))^2-\frac {8 b d^4 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{c}-2 i c d^4 x^2 (a+b \arctan (c x))^2-\frac {1}{3} b c d^4 x^2 (a+b \arctan (c x))-7 d^4 x (a+b \arctan (c x))^2-\frac {28 i d^4 (a+b \arctan (c x))^2}{3 c}-\frac {8 i d^4 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{c}-\frac {44 b d^4 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c}+4 i a b d^4 x+4 i b^2 d^4 x \arctan (c x)-\frac {b^2 d^4 \arctan (c x)}{3 c}-\frac {2 i b^2 d^4 \log \left (c^2 x^2+1\right )}{c}-\frac {22 i b^2 d^4 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c}-\frac {4 i b^2 d^4 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{c}+\frac {1}{3} b^2 d^4 x\right )}{4 d}-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}\) |
((-1/4*I)*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^3)/c + (((3*I)/4)*b*((4*I) *a*b*d^4*x + (b^2*d^4*x)/3 - (b^2*d^4*ArcTan[c*x])/(3*c) + (4*I)*b^2*d^4*x *ArcTan[c*x] - (b*c*d^4*x^2*(a + b*ArcTan[c*x]))/3 - (((28*I)/3)*d^4*(a + b*ArcTan[c*x])^2)/c - 7*d^4*x*(a + b*ArcTan[c*x])^2 - (2*I)*c*d^4*x^2*(a + b*ArcTan[c*x])^2 + (c^2*d^4*x^3*(a + b*ArcTan[c*x])^2)/3 - ((8*I)*d^4*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/c - (44*b*d^4*(a + b*ArcTan[c*x])*L og[2/(1 + I*c*x)])/(3*c) - ((2*I)*b^2*d^4*Log[1 + c^2*x^2])/c - (8*b*d^4*( a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/c - (((22*I)/3)*b^2*d^4* PolyLog[2, 1 - 2/(1 + I*c*x)])/c - ((4*I)*b^2*d^4*PolyLog[3, 1 - 2/(1 - I* c*x)])/c))/d
3.2.20.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 61.01 (sec) , antiderivative size = 1513, normalized size of antiderivative = 3.96
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1513\) |
| default | \(\text {Expression too large to display}\) | \(1513\) |
| parts | \(\text {Expression too large to display}\) | \(1521\) |
1/c*(-1/4*I*d^3*a^3*(1+I*c*x)^4+d^3*b^3*(-1/4*I*arctan(c*x)^3*c^4*x^4-arct an(c*x)^3*c^3*x^3+3/2*I*arctan(c*x)^3*c^2*x^2+arctan(c*x)^3*c*x-1/4*I*arct an(c*x)^3+3/4*I*(1/3*I+1/3*c*x+1/3*c^3*x^3*arctan(c*x)^2-7*arctan(c*x)^2*c *x-8*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-2/3*arctan(c*x)*(c*x- I)*(c*x+I)+1/3*arctan(c*x)*(c*x-I)^2-44/3*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^ 2*x^2+1)^(1/2))-44/3*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*Pi* csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^3*arctan(c*x)^2-2*Pi*csgn(I*(1+I*c*x )^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^3*arctan(c*x)^2-2*Pi*csgn(I *(1+I*c*x)^2/(c^2*x^2+1))^3*arctan(c*x)^2+2*Pi*csgn(I*(1+(1+I*c*x)^2/(c^2* x^2+1)))^2*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*arctan(c*x)^2-2*Pi*csgn(I *(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*arctan(c*x )^2+2*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2 +1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2-4*Pi*csgn(I*(1+(1+I*c*x )^2/(c^2*x^2+1)))*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2+2* Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I* c*x)^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2+4*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^( 1/2))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*arctan(c*x)^2-2*Pi*csgn(I/(1+(1+I* c*x)^2/(c^2*x^2+1))^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/ (c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*arctan(c*x)^2-2*I*arctan(c*x)^2 *c^2*x^2+4*I*ln(1+(1+I*c*x)^2/(c^2*x^2+1))+16/3*I*arctan(c*x)^2-4*I*pol...
\[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
-1/32*(b^3*c^3*d^3*x^4 - 4*I*b^3*c^2*d^3*x^3 - 6*b^3*c*d^3*x^2 + 4*I*b^3*d ^3*x)*log(-(c*x + I)/(c*x - I))^3 + integral(1/16*(-16*I*a^3*c^5*d^3*x^5 - 48*a^3*c^4*d^3*x^4 + 32*I*a^3*c^3*d^3*x^3 - 32*a^3*c^2*d^3*x^2 + 48*I*a^3 *c*d^3*x + 16*a^3*d^3 - 3*(-4*I*a*b^2*c^5*d^3*x^5 - (12*a*b^2 - I*b^3)*c^4 *d^3*x^4 + 4*(2*I*a*b^2 + b^3)*c^3*d^3*x^3 - 2*(4*a*b^2 + 3*I*b^3)*c^2*d^3 *x^2 + 4*a*b^2*d^3 + 4*(3*I*a*b^2 - b^3)*c*d^3*x)*log(-(c*x + I)/(c*x - I) )^2 + 24*(a^2*b*c^5*d^3*x^5 - 3*I*a^2*b*c^4*d^3*x^4 - 2*a^2*b*c^3*d^3*x^3 - 2*I*a^2*b*c^2*d^3*x^2 - 3*a^2*b*c*d^3*x + I*a^2*b*d^3)*log(-(c*x + I)/(c *x - I)))/(c^2*x^2 + 1), x)
Timed out. \[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=\text {Timed out} \]
\[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
-1/4*I*a^3*c^3*d^3*x^4 - 24*b^3*c^5*d^3*integrate(1/128*x^5*arctan(c*x)^2* log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 2*b^3*c^5*d^3*integrate(1/128*x^5*log (c^2*x^2 + 1)^3/(c^2*x^2 + 1), x) - 12*b^3*c^5*d^3*integrate(1/128*x^5*arc tan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3*c^5*d^3*integrate(1/128*x^5*log(c^2*x ^2 + 1)^2/(c^2*x^2 + 1), x) - a^3*c^2*d^3*x^3 - 336*b^3*c^4*d^3*integrate( 1/128*x^4*arctan(c*x)^3/(c^2*x^2 + 1), x) - 36*b^3*c^4*d^3*integrate(1/128 *x^4*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 1152*a*b^2*c^4*d^3 *integrate(1/128*x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) - 60*b^3*c^4*d^3*inte grate(1/128*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - 1/4*I*(3* x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a^2*b*c^3*d ^3 + 48*b^3*c^3*d^3*integrate(1/128*x^3*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^ 2*x^2 + 1), x) - 4*b^3*c^3*d^3*integrate(1/128*x^3*log(c^2*x^2 + 1)^3/(c^2 *x^2 + 1), x) + 120*b^3*c^3*d^3*integrate(1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) - 30*b^3*c^3*d^3*integrate(1/128*x^3*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 3/2*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a ^2*b*c^2*d^3 + 3/2*I*a^3*c*d^3*x^2 + 7/32*b^3*d^3*arctan(c*x)^4/c - 224*b^ 3*c^2*d^3*integrate(1/128*x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) - 24*b^3*c^2 *d^3*integrate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 768*a*b^2*c^2*d^3*integrate(1/128*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 120*b^3*c^2*d^3*integrate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x...
\[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3 \,d x \]