Integrand size = 18, antiderivative size = 411 \[ \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx=\frac {a b^2 e^2 x}{c^2}+\frac {b^3 e^2 x \arctan (c x)}{c^2}-\frac {3 i b d e (a+b \arctan (c x))^2}{c^2}-\frac {b e^2 (a+b \arctan (c x))^2}{2 c^3}-\frac {3 b d e x (a+b \arctan (c x))^2}{c}-\frac {b e^2 x^2 (a+b \arctan (c x))^2}{2 c}+\frac {i \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x))^3}{3 c^3}-\frac {d \left (d^2-\frac {3 e^2}{c^2}\right ) (a+b \arctan (c x))^3}{3 e}+\frac {(d+e x)^3 (a+b \arctan (c x))^3}{3 e}-\frac {6 b^2 d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^2}+\frac {b \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^3 e^2 \log \left (1+c^2 x^2\right )}{2 c^3}-\frac {3 i b^3 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2}+\frac {i b^2 \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3}+\frac {b^3 \left (3 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3} \]
a*b^2*e^2*x/c^2+b^3*e^2*x*arctan(c*x)/c^2-3*I*b*d*e*(a+b*arctan(c*x))^2/c^ 2-1/2*b*e^2*(a+b*arctan(c*x))^2/c^3-3*b*d*e*x*(a+b*arctan(c*x))^2/c-1/2*b* e^2*x^2*(a+b*arctan(c*x))^2/c+1/3*I*(3*c^2*d^2-e^2)*(a+b*arctan(c*x))^3/c^ 3-1/3*d*(d^2-3*e^2/c^2)*(a+b*arctan(c*x))^3/e+1/3*(e*x+d)^3*(a+b*arctan(c* x))^3/e-6*b^2*d*e*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^2+b*(3*c^2*d^2-e^2)* (a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c^3-1/2*b^3*e^2*ln(c^2*x^2+1)/c^3-3*I* b^3*d*e*polylog(2,1-2/(1+I*c*x))/c^2+I*b^2*(3*c^2*d^2-e^2)*(a+b*arctan(c*x ))*polylog(2,1-2/(1+I*c*x))/c^3+1/2*b^3*(3*c^2*d^2-e^2)*polylog(3,1-2/(1+I *c*x))/c^3
Time = 1.16 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.51 \[ \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx=\frac {6 a^2 c^2 d (a c d-3 b e) x+3 a^2 c^2 e (2 a c d-b e) x^2+2 a^3 c^3 e^2 x^3+18 a^2 b c d e \arctan (c x)+6 a^2 b c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \arctan (c x)-3 a^2 b \left (3 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )+18 a b^2 c d e \left (-2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2+\log \left (1+c^2 x^2\right )\right )+18 a b^2 c^2 d^2 \left (\arctan (c x) \left ((-i+c x) \arctan (c x)+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+6 a b^2 e^2 \left (c x+\left (i+c^3 x^3\right ) \arctan (c x)^2-\arctan (c x) \left (1+c^2 x^2+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+6 b^3 c d e \left (\arctan (c x) \left ((3 i-3 c x) \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2-6 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+3 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+b^3 e^2 \left (6 c x \arctan (c x)-3 \arctan (c x)^2-3 c^2 x^2 \arctan (c x)^2+2 i \arctan (c x)^3+2 c^3 x^3 \arctan (c x)^3-6 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-3 \log \left (1+c^2 x^2\right )+6 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )+3 b^3 c^2 d^2 \left (2 \arctan (c x)^2 \left ((-i+c x) \arctan (c x)+3 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-6 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{6 c^3} \]
(6*a^2*c^2*d*(a*c*d - 3*b*e)*x + 3*a^2*c^2*e*(2*a*c*d - b*e)*x^2 + 2*a^3*c ^3*e^2*x^3 + 18*a^2*b*c*d*e*ArcTan[c*x] + 6*a^2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTan[c*x] - 3*a^2*b*(3*c^2*d^2 - e^2)*Log[1 + c^2*x^2] + 18*a* b^2*c*d*e*(-2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2* x^2]) + 18*a*b^2*c^2*d^2*(ArcTan[c*x]*((-I + c*x)*ArcTan[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 6*a*b^2* e^2*(c*x + (I + c^3*x^3)*ArcTan[c*x]^2 - ArcTan[c*x]*(1 + c^2*x^2 + 2*Log[ 1 + E^((2*I)*ArcTan[c*x])]) + I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 6*b^ 3*c*d*e*(ArcTan[c*x]*((3*I - 3*c*x)*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x ]^2 - 6*Log[1 + E^((2*I)*ArcTan[c*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTa n[c*x])]) + b^3*e^2*(6*c*x*ArcTan[c*x] - 3*ArcTan[c*x]^2 - 3*c^2*x^2*ArcTa n[c*x]^2 + (2*I)*ArcTan[c*x]^3 + 2*c^3*x^3*ArcTan[c*x]^3 - 6*ArcTan[c*x]^2 *Log[1 + E^((2*I)*ArcTan[c*x])] - 3*Log[1 + c^2*x^2] + (6*I)*ArcTan[c*x]*P olyLog[2, -E^((2*I)*ArcTan[c*x])] - 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]) + 3*b^3*c^2*d^2*(2*ArcTan[c*x]^2*((-I + c*x)*ArcTan[c*x] + 3*Log[1 + E^((2 *I)*ArcTan[c*x])]) - (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(6*c^3)
Time = 0.95 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \arctan (c x))^3}{3 e}-\frac {b c \int \left (\frac {x (a+b \arctan (c x))^2 e^3}{c^2}+\frac {3 d (a+b \arctan (c x))^2 e^2}{c^2}+\frac {\left (d \left (c^2 d^2-3 e^2\right )+e \left (3 c^2 d^2-e^2\right ) x\right ) (a+b \arctan (c x))^2}{c^2 \left (c^2 x^2+1\right )}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \arctan (c x))^3}{3 e}-\frac {b c \left (\frac {e^3 (a+b \arctan (c x))^2}{2 c^4}+\frac {3 i d e^2 (a+b \arctan (c x))^2}{c^3}+\frac {6 b d e^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^3}+\frac {3 d e^2 x (a+b \arctan (c x))^2}{c^2}+\frac {e^3 x^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {i b e \left (3 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^4}-\frac {i e \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x))^3}{3 b c^4}-\frac {e \left (3 c^2 d^2-e^2\right ) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^4}+\frac {d \left (c^2 d^2-3 e^2\right ) (a+b \arctan (c x))^3}{3 b c^3}-\frac {a b e^3 x}{c^3}-\frac {b^2 e^3 x \arctan (c x)}{c^3}+\frac {3 i b^2 d e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^3}-\frac {b^2 e \left (3 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^4}+\frac {b^2 e^3 \log \left (c^2 x^2+1\right )}{2 c^4}\right )}{e}\) |
((d + e*x)^3*(a + b*ArcTan[c*x])^3)/(3*e) - (b*c*(-((a*b*e^3*x)/c^3) - (b^ 2*e^3*x*ArcTan[c*x])/c^3 + ((3*I)*d*e^2*(a + b*ArcTan[c*x])^2)/c^3 + (e^3* (a + b*ArcTan[c*x])^2)/(2*c^4) + (3*d*e^2*x*(a + b*ArcTan[c*x])^2)/c^2 + ( e^3*x^2*(a + b*ArcTan[c*x])^2)/(2*c^2) + (d*(c^2*d^2 - 3*e^2)*(a + b*ArcTa n[c*x])^3)/(3*b*c^3) - ((I/3)*e*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])^3)/( b*c^4) + (6*b*d*e^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^3 - (e*(3*c^ 2*d^2 - e^2)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^4 + (b^2*e^3*Log[ 1 + c^2*x^2])/(2*c^4) + ((3*I)*b^2*d*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^ 3 - (I*b*e*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c *x)])/c^4 - (b^2*e*(3*c^2*d^2 - e^2)*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^4 )))/e
3.1.16.3.1 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 = 32.97 (sec) , antiderivative size = 2633, normalized size of antiderivative = 6.41
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2633\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2647\) |
| default | \(\text {Expression too large to display}\) | \(2647\) |
1/3*a^3*(e*x+d)^3/e+b^3/c*(1/3*c*e^2*arctan(c*x)^3*x^3+c*e*arctan(c*x)^3*x ^2*d+arctan(c*x)^3*c*x*d^2+1/3*c/e*arctan(c*x)^3*d^3-1/c^2/e*(d^3*c^3*arct an(c*x)^3+1/2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))*e^3-e^3*ln(1+(1+I*c*x)^2 /(c^2*x^2+1))+1/2*e^3*arctan(c*x)^2+6*e^2*d*c*arctan(c*x)*ln(1+I*(1+I*c*x) /(c^2*x^2+1)^(1/2))+6*e^2*d*c*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/ 2))-3*e*d^2*c^2*ln(2)*arctan(c*x)^2-3*e*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))*c^ 2*d^2*arctan(c*x)^2+3/2*arctan(c*x)^2*ln(c^2*x^2+1)*e*c^2*d^2-6*I*e^2*d*c* dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/4*I*e^3*Pi*csgn(I*(1+(1+I*c*x)^2/ (c^2*x^2+1))^2)^3*arctan(c*x)^2-1/4*I*e^3*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1 ))^3*arctan(c*x)^2-1/4*I*e^3*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)^2-3*I*c*d*e^2*arctan(c*x)^2-6*I*e^2*d*c* dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*e*c^2*d^2*arctan(c*x)^3+3/2*I*e*d ^2*c^2*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+3/4*I*e*d^2*c^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)^2-3/2*I*e*d^2*c^2*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*arct an(c*x)^2-3/4*I*e*d^2*c^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)^2-3/4*I*e*d ^2*c^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)^2-3/4*I*e*d^2*c^2*Pi*...
\[ \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
integral(a^3*e^2*x^2 + 2*a^3*d*e*x + a^3*d^2 + (b^3*e^2*x^2 + 2*b^3*d*e*x + b^3*d^2)*arctan(c*x)^3 + 3*(a*b^2*e^2*x^2 + 2*a*b^2*d*e*x + a*b^2*d^2)*a rctan(c*x)^2 + 3*(a^2*b*e^2*x^2 + 2*a^2*b*d*e*x + a^2*b*d^2)*arctan(c*x), x)
\[ \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{2}\, dx \]
\[ \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
1/3*a^3*e^2*x^3 + 7/32*b^3*d^2*arctan(c*x)^4/c + 28*b^3*c^2*e^2*integrate( 1/32*x^4*arctan(c*x)^3/(c^2*x^2 + 1), x) + 3*b^3*c^2*e^2*integrate(1/32*x^ 4*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 96*a*b^2*c^2*e^2*inte grate(1/32*x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) + 56*b^3*c^2*d*e*integrate( 1/32*x^3*arctan(c*x)^3/(c^2*x^2 + 1), x) + 4*b^3*c^2*e^2*integrate(1/32*x^ 4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 6*b^3*c^2*d*e*integrate (1/32*x^3*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 192*a*b^2*c^2 *d*e*integrate(1/32*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + 28*b^3*c^2*d^2*i ntegrate(1/32*x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^3*c^2*d*e*integra te(1/32*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3*b^3*c^2*d^2 *integrate(1/32*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 96* a*b^2*c^2*d^2*integrate(1/32*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 12*b^3* c^2*d^2*integrate(1/32*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + a^3*d*e*x^2 + a*b^2*d^2*arctan(c*x)^3/c - 4*b^3*c*e^2*integrate(1/32*x^3 *arctan(c*x)^2/(c^2*x^2 + 1), x) + b^3*c*e^2*integrate(1/32*x^3*log(c^2*x^ 2 + 1)^2/(c^2*x^2 + 1), x) - 12*b^3*c*d*e*integrate(1/32*x^2*arctan(c*x)^2 /(c^2*x^2 + 1), x) + 3*b^3*c*d*e*integrate(1/32*x^2*log(c^2*x^2 + 1)^2/(c^ 2*x^2 + 1), x) - 12*b^3*c*d^2*integrate(1/32*x*arctan(c*x)^2/(c^2*x^2 + 1) , x) + 3*b^3*c*d^2*integrate(1/32*x*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a^2*b*d*e + 1/2*(2*x...
\[ \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3\,{\left (d+e\,x\right )}^2 \,d x \]