Integrand size = 18, antiderivative size = 936 \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\frac {3 b c^3 d (a+b \arctan (c x))^2}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b c^2 e (a+b \arctan (c x))^2}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 b c (a+b \arctan (c x))^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d (a+b \arctan (c x))^3}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) (a+b \arctan (c x))^3}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {(a+b \arctan (c x))^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b c^3 d (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^3 c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^3 c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i b^3 c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 i b^2 c^3 d (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b^3 c^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2} \]
3/2*b*c^3*d*(a+b*arctan(c*x))^2/(c^2*d^2+e^2)^2+3/2*I*b^3*c^2*e*polylog(2, 1-2/(1-I*c*x))/(c^2*d^2+e^2)^2-3/2*b*c*(a+b*arctan(c*x))^2/(c^2*d^2+e^2)/( e*x+d)-3/2*I*b^3*c^2*e*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d ^2+e^2)^2+1/2*c^2*(c*d-e)*(c*d+e)*(a+b*arctan(c*x))^3/e/(c^2*d^2+e^2)^2-1/ 2*(a+b*arctan(c*x))^3/e/(e*x+d)^2-3*b^2*c^2*e*(a+b*arctan(c*x))*ln(2/(1-I* c*x))/(c^2*d^2+e^2)^2-3*b*c^3*d*(a+b*arctan(c*x))^2*ln(2/(1-I*c*x))/(c^2*d ^2+e^2)^2+3*b^2*c^2*e*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/(c^2*d^2+e^2)^2+3* b*c^3*d*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/(c^2*d^2+e^2)^2+3*b^2*c^2*e*(a +b*arctan(c*x))*ln(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d^2+e^2)^2+3*b*c^ 3*d*(a+b*arctan(c*x))^2*ln(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d^2+e^2)^ 2+I*c^3*d*(a+b*arctan(c*x))^3/(c^2*d^2+e^2)^2+3/2*I*b^3*c^2*e*polylog(2,1- 2/(1+I*c*x))/(c^2*d^2+e^2)^2-3*I*b^2*c^3*d*(a+b*arctan(c*x))*polylog(2,1-2 *c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d^2+e^2)^2+3/2*I*b*c^2*e*(a+b*arctan( c*x))^2/(c^2*d^2+e^2)^2+3*I*b^2*c^3*d*(a+b*arctan(c*x))*polylog(2,1-2/(1+I *c*x))/(c^2*d^2+e^2)^2+3*I*b^2*c^3*d*(a+b*arctan(c*x))*polylog(2,1-2/(1-I* c*x))/(c^2*d^2+e^2)^2-3/2*b^3*c^3*d*polylog(3,1-2/(1-I*c*x))/(c^2*d^2+e^2) ^2+3/2*b^3*c^3*d*polylog(3,1-2/(1+I*c*x))/(c^2*d^2+e^2)^2+3/2*b^3*c^3*d*po lylog(3,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d^2+e^2)^2
\[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx \]
Time = 1.36 (sec) , antiderivative size = 921, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {3 b c \int \left (\frac {2 c^2 d e^2 (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {c^2 \left (d^2 c^2-2 d e x c^2-e^2\right ) (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2 \left (c^2 x^2+1\right )}+\frac {e^2 (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right ) (d+e x)^2}\right )dx}{2 e}-\frac {(a+b \arctan (c x))^3}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b c \left (\frac {2 i c^2 d e (a+b \arctan (c x))^3}{3 b \left (c^2 d^2+e^2\right )^2}+\frac {c (c d-e) (c d+e) (a+b \arctan (c x))^3}{3 b \left (c^2 d^2+e^2\right )^2}-\frac {2 c^2 d e \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}+\frac {2 c^2 d e \log \left (\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}+\frac {2 c^2 d e \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right ) (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}-\frac {e (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c e^2 (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 d e (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}-\frac {2 b c e^2 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c e^2 \log \left (\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c e^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right ) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right )^2}+\frac {2 i b c^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right )^2}+\frac {2 i b c^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right )^2}-\frac {2 i b c^2 d e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right ) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c e^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {b^2 c^2 d e \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 d e \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 d e \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}\right )}{2 e}-\frac {(a+b \arctan (c x))^3}{2 e (d+e x)^2}\) |
-1/2*(a + b*ArcTan[c*x])^3/(e*(d + e*x)^2) + (3*b*c*((c^2*d*e*(a + b*ArcTa n[c*x])^2)/(c^2*d^2 + e^2)^2 + (I*c*e^2*(a + b*ArcTan[c*x])^2)/(c^2*d^2 + e^2)^2 - (e*(a + b*ArcTan[c*x])^2)/((c^2*d^2 + e^2)*(d + e*x)) + (((2*I)/3 )*c^2*d*e*(a + b*ArcTan[c*x])^3)/(b*(c^2*d^2 + e^2)^2) + (c*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])^3)/(3*b*(c^2*d^2 + e^2)^2) - (2*b*c*e^2*(a + b*A rcTan[c*x])*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 - (2*c^2*d*e*(a + b*ArcT an[c*x])^2*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 + (2*b*c*e^2*(a + b*ArcTa n[c*x])*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2)^2 + (2*c^2*d*e*(a + b*ArcTan[c *x])^2*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2)^2 + (2*b*c*e^2*(a + b*ArcTan[c* x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2)^2 + (2 *c^2*d*e*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x ))])/(c^2*d^2 + e^2)^2 + (I*b^2*c*e^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2* d^2 + e^2)^2 + ((2*I)*b*c^2*d*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 + (I*b^2*c*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/( c^2*d^2 + e^2)^2 + ((2*I)*b*c^2*d*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/( 1 + I*c*x)])/(c^2*d^2 + e^2)^2 - (I*b^2*c*e^2*PolyLog[2, 1 - (2*c*(d + e*x ))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2)^2 - ((2*I)*b*c^2*d*e*(a + b *ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/( c^2*d^2 + e^2)^2 - (b^2*c^2*d*e*PolyLog[3, 1 - 2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 + (b^2*c^2*d*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^2*d^2 + e^2)^2 ...
3.1.20.3.1 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 = 93.31 (sec) , antiderivative size = 40258, normalized size of antiderivative = 43.01
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(40258\) |
| default | \(\text {Expression too large to display}\) | \(40258\) |
| parts | \(\text {Expression too large to display}\) | \(40263\) |
\[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
integral((b^3*arctan(c*x)^3 + 3*a*b^2*arctan(c*x)^2 + 3*a^2*b*arctan(c*x) + a^3)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{{\left (d+e\,x\right )}^3} \,d x \]