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} \]
[Out]
Time = 0.75 (sec) , antiderivative size = 936, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4974, 4966, 2449, 2352, 2497, 5104, 5004, 5040, 4964, 4968, 5114, 6745} \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\frac {3 i c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 i c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 c^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 c^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 c^3 d \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 c^2 e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}+\frac {3 c^2 e (a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}+\frac {3 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 ) b^2}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i c^3 d (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i c^3 d (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i 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 ) b^2}{\left (c^2 d^2+e^2\right )^2}-\frac {3 c (a+b \arctan (c x))^2 b}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {3 c^3 d (a+b \arctan (c x))^2 b}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i c^2 e (a+b \arctan (c x))^2 b}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 c^3 d (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right ) b}{\left (c^2 d^2+e^2\right )^2}+\frac {3 c^3 d (a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right ) b}{\left (c^2 d^2+e^2\right )^2}+\frac {3 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 ) b}{\left (c^2 d^2+e^2\right )^2}+\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} \]
[In]
[Out]
Rule 2352
Rule 2449
Rule 2497
Rule 4964
Rule 4966
Rule 4968
Rule 4974
Rule 5004
Rule 5040
Rule 5104
Rule 5114
Rule 6745
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arctan (c x))^3}{2 e (d+e x)^2}+\frac {(3 b c) \int \left (\frac {e^2 (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right ) (d+e x)^2}+\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 {\left (c^4 d^2-c^2 e^2-2 c^4 d e x\right ) (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 e} \\ & = -\frac {(a+b \arctan (c x))^3}{2 e (d+e x)^2}+\frac {(3 b c) \int \frac {\left (c^4 d^2-c^2 e^2-2 c^4 d e x\right ) (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{2 e \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b c^3 d e\right ) \int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {(3 b c e) \int \frac {(a+b \arctan (c x))^2}{(d+e x)^2} \, dx}{2 \left (c^2 d^2+e^2\right )} \\ & = -\frac {3 b c (a+b \arctan (c x))^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {(a+b \arctan (c x))^3}{2 e (d+e x)^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 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^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^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 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 b c) \int \left (\frac {c^4 d^2 \left (1-\frac {e^2}{c^2 d^2}\right ) (a+b \arctan (c x))^2}{1+c^2 x^2}-\frac {2 c^4 d e x (a+b \arctan (c x))^2}{1+c^2 x^2}\right ) \, dx}{2 e \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^2\right ) \int \left (\frac {e^2 (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {c^2 (d-e x) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{c^2 d^2+e^2} \\ & = -\frac {3 b c (a+b \arctan (c x))^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {(a+b \arctan (c x))^3}{2 e (d+e x)^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 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^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^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 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^4\right ) \int \frac {(d-e x) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (3 b c^5 d\right ) \int \frac {x (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^2 e^2\right ) \int \frac {a+b \arctan (c x)}{d+e x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b c^3 (c d-e) (c d+e)\right ) \int \frac {(a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{2 e \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 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^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^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 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^4\right ) \int \left (\frac {d (a+b \arctan (c x))}{1+c^2 x^2}-\frac {e x (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b c^4 d\right ) \int \frac {(a+b \arctan (c x))^2}{i-c x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^3 c^3 e\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (3 b^3 c^3 e\right ) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{\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 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^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 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^4 d\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (6 b^2 c^4 d\right ) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 i b^3 c^2 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (3 b^2 c^4 e\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2} \\ & = \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 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^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 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}-\frac {\left (3 i b^3 c^4 d\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^3 e\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{\left (c^2 d^2+e^2\right )^2} \\ & = \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^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}-\frac {\left (3 b^3 c^3 e\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2} \\ & = \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^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}+\frac {\left (3 i b^3 c^2 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{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}{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} \\ \end{align*}
\[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx \]
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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 | \(\text {Expression too large to display}\) | \(40263\) |
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\[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{{\left (d+e\,x\right )}^3} \,d x \]
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