Integrand size = 18, antiderivative size = 496 \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\frac {b^2 c^3 d \arctan (c x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) (a+b \arctan (c x))^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}-\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {b^2 c^2 e \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2} \]
[Out]
Time = 0.38 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4974, 4972, 720, 31, 649, 209, 266, 4966, 2449, 2352, 2497, 5104, 5004, 5040, 4964} \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\frac {c^2 (c d-e) (c d+e) (a+b \arctan (c x))^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {b c (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}-\frac {2 b c^3 d \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}+\frac {b^2 c^3 d \arctan (c x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b^2 c^2 e \log \left (c^2 x^2+1\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2} \]
[In]
[Out]
Rule 31
Rule 209
Rule 266
Rule 649
Rule 720
Rule 2352
Rule 2449
Rule 2497
Rule 4964
Rule 4966
Rule 4972
Rule 4974
Rule 5004
Rule 5040
Rule 5104
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}+\frac {(b c) \int \left (\frac {e^2 (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)^2}+\frac {2 c^2 d e^2 (a+b \arctan (c x))}{\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))}{\left (c^2 d^2+e^2\right )^2 \left (1+c^2 x^2\right )}\right ) \, dx}{e} \\ & = -\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}+\frac {(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))}{1+c^2 x^2} \, dx}{e \left (c^2 d^2+e^2\right )^2}+\frac {\left (2 b c^3 d e\right ) \int \frac {a+b \arctan (c x)}{d+e x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {(b c e) \int \frac {a+b \arctan (c x)}{(d+e x)^2} \, dx}{c^2 d^2+e^2} \\ & = -\frac {b c (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}-\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (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 {\left (2 b^2 c^4 d\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 (2 b^2 c^4 d\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 {(b c) \int \left (\frac {c^4 d^2 \left (1-\frac {e^2}{c^2 d^2}\right ) (a+b \arctan (c x))}{1+c^2 x^2}-\frac {2 c^4 d e x (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{e \left (c^2 d^2+e^2\right )^2}+\frac {\left (b^2 c^2\right ) \int \frac {1}{(d+e x) \left (1+c^2 x^2\right )} \, dx}{c^2 d^2+e^2} \\ & = -\frac {b c (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}-\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (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 {i b^2 c^3 d \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 {\left (b^2 c^2\right ) \int \frac {c^2 d-c^2 e x}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (2 i b^2 c^3 d\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 (2 b c^5 d\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 {\left (b^2 c^2 e^2\right ) \int \frac {1}{d+e x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (b c^3 (c d-e) (c d+e)\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{e \left (c^2 d^2+e^2\right )^2} \\ & = -\frac {b c (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) (a+b \arctan (c x))^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}-\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (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 {i b^2 c^3 d \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^3 d \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 {\left (2 b c^4 d\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (b^2 c^4 d\right ) \int \frac {1}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (b^2 c^4 e\right ) \int \frac {x}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2} \\ & = \frac {b^2 c^3 d \arctan (c x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) (a+b \arctan (c x))^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}-\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (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 {b^2 c^2 e \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \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^3 d \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 {\left (2 b^2 c^4 d\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 {b^2 c^3 d \arctan (c x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) (a+b \arctan (c x))^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}-\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (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 {b^2 c^2 e \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \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^3 d \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 {\left (2 i b^2 c^3 d\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 {b^2 c^3 d \arctan (c x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) (a+b \arctan (c x))^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}-\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (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 {b^2 c^2 e \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \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^3 d \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^3 d \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} \\ \end{align*}
Time = 6.54 (sec) , antiderivative size = 479, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=-\frac {a^2}{2 e (d+e x)^2}+\frac {a b \left (\left (-e^3+c^4 d^2 x (2 d+e x)-c^2 e \left (3 d^2+2 d e x+e^2 x^2\right )\right ) \arctan (c x)+c (d+e x) \left (-c^2 d^2-e^2+2 c^2 d (d+e x) \log (c (d+e x))-c^2 d (d+e x) \log \left (1+c^2 x^2\right )\right )\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)^2}+\frac {b^2 c^2 \left (-\frac {2 e^{i \arctan \left (\frac {c d}{e}\right )} \arctan (c x)^2}{\sqrt {1+\frac {c^2 d^2}{e^2}} e}-\frac {e \left (1+c^2 x^2\right ) \arctan (c x)^2}{c^2 (d+e x)^2}+\frac {2 x \arctan (c x) (e+c d \arctan (c x))}{c d (d+e x)}+\frac {-2 e^2 \arctan (c x)+2 c d e \log \left (\frac {c (d+e x)}{\sqrt {1+c^2 x^2}}\right )}{c^3 d^3+c d e^2}-\frac {2 c d \left (-i \left (\pi -2 \arctan \left (\frac {c d}{e}\right )\right ) \arctan (c x)-\pi \log \left (1+e^{-2 i \arctan (c x)}\right )-2 \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right ) \log \left (1-e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )-\frac {1}{2} \pi \log \left (1+c^2 x^2\right )+2 \arctan \left (\frac {c d}{e}\right ) \log \left (\sin \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )\right )}{c^2 d^2+e^2}\right )}{2 \left (c^2 d^2+e^2\right )} \]
[In]
[Out]
Time = 28.13 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.47
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} c^{3}}{2 \left (c e x +c d \right )^{2} e}+b^{2} c^{3} \left (-\frac {\arctan \left (c x \right )^{2}}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {\arctan \left (c x \right ) e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 \arctan \left (c x \right ) e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {\arctan \left (c x \right )^{2} c^{2} d^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d e}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right )^{2} e^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )^{2}}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e^{2} \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {e^{2} \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e d c \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e c d \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {2 e^{2} c d \left (\frac {i \ln \left (c e x +c d \right ) \left (-\ln \left (\frac {-c e x +i e}{c d +i e}\right )+\ln \left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-c e x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{e}\right )+2 a b \,c^{3} \left (-\frac {\arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {-c d e \ln \left (c^{2} x^{2}+1\right )+\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{2 e}\right )}{c}\) | \(729\) |
default | \(\frac {-\frac {a^{2} c^{3}}{2 \left (c e x +c d \right )^{2} e}+b^{2} c^{3} \left (-\frac {\arctan \left (c x \right )^{2}}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {\arctan \left (c x \right ) e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 \arctan \left (c x \right ) e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {\arctan \left (c x \right )^{2} c^{2} d^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d e}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right )^{2} e^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )^{2}}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e^{2} \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {e^{2} \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e d c \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e c d \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {2 e^{2} c d \left (\frac {i \ln \left (c e x +c d \right ) \left (-\ln \left (\frac {-c e x +i e}{c d +i e}\right )+\ln \left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-c e x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{e}\right )+2 a b \,c^{3} \left (-\frac {\arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {-c d e \ln \left (c^{2} x^{2}+1\right )+\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{2 e}\right )}{c}\) | \(729\) |
parts | \(-\frac {a^{2}}{2 \left (e x +d \right )^{2} e}+\frac {b^{2} \left (-\frac {c^{3} \arctan \left (c x \right )^{2}}{2 \left (c e x +c d \right )^{2} e}+\frac {c^{3} \left (-\frac {\arctan \left (c x \right ) e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 \arctan \left (c x \right ) e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {\arctan \left (c x \right )^{2} c^{2} d^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d e}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right )^{2} e^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )^{2}}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e^{2} \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {e^{2} \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e d c \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e c d \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {2 c d \,e^{2} \left (-\frac {i \ln \left (c e x +c d \right ) \left (\ln \left (\frac {-c e x +i e}{c d +i e}\right )-\ln \left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-c e x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}\right )}{e}\right )}{c}+\frac {2 a b \left (-\frac {c^{3} \arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {c^{3} \left (-\frac {e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {-c d e \ln \left (c^{2} x^{2}+1\right )+\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}\right )}{2 e}\right )}{c}\) | \(731\) |
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\[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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