Integrand size = 20, antiderivative size = 1233 \[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\frac {3 a^2 b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \arctan (c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \arctan (c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \arctan (c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \arctan (c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \]
[Out]
Time = 1.58 (sec) , antiderivative size = 1233, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5153, 6873, 5165, 6820, 12, 6857, 720, 31, 649, 209, 266, 4966, 2449, 2352, 2497, 5104, 5004, 5040, 4964, 4968, 5114, 6745} \[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\frac {i d \arctan (c+d x)^3 b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {d (d e-c f) \arctan (c+d x)^3 b^3}{f \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}-\frac {3 d \arctan (c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 d \arctan (c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 d \arctan (c+d x)^2 \log \left (\frac {2}{i (c+d x)+1}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 i d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {3 i d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 i d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right ) b^3}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac {3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right ) b^3}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac {3 d \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right ) b^3}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac {3 i a d \arctan (c+d x)^2 b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 a d (d e-c f) \arctan (c+d x)^2 b^2}{f \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}-\frac {6 a d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {6 a d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {6 a d \arctan (c+d x) \log \left (\frac {2}{i (c+d x)+1}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 i a d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {3 i a d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 i a d \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 a^2 d (d e-c f) \arctan (c+d x) b}{f \left (f^2+(d e-c f)^2\right )}+\frac {3 a^2 d \log (e+f x) b}{f^2+(d e-c f)^2}-\frac {3 a^2 d \log \left ((c+d x)^2+1\right ) b}{2 \left (f^2+(d e-c f)^2\right )}-\frac {(a+b \arctan (c+d x))^3}{f (e+f x)} \]
[In]
[Out]
Rule 12
Rule 31
Rule 209
Rule 266
Rule 649
Rule 720
Rule 2352
Rule 2449
Rule 2497
Rule 4964
Rule 4966
Rule 4968
Rule 5004
Rule 5040
Rule 5104
Rule 5114
Rule 5153
Rule 5165
Rule 6745
Rule 6820
Rule 6857
Rule 6873
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {(3 b d) \int \frac {(a+b \arctan (c+d x))^2}{(e+f x) \left (1+(c+d x)^2\right )} \, dx}{f} \\ & = -\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {(3 b d) \int \frac {(a+b \arctan (c+d x))^2}{(e+f x) \left (1+c^2+2 c d x+d^2 x^2\right )} \, dx}{f} \\ & = -\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {(3 b) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2}{\left (\frac {d e-c f}{d}+\frac {f x}{d}\right ) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f} \\ & = -\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {(3 b) \text {Subst}\left (\int \frac {d (a+b \arctan (x))^2}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f} \\ & = -\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {(3 b d) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f} \\ & = -\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {(3 b d) \text {Subst}\left (\int \left (\frac {a^2}{(d e-c f+f x) \left (1+x^2\right )}+\frac {2 a b \arctan (x)}{(d e-c f+f x) \left (1+x^2\right )}+\frac {b^2 \arctan (x)^2}{(d e-c f+f x) \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f} \\ & = -\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {\left (3 a^2 b d\right ) \text {Subst}\left (\int \frac {1}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}+\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\arctan (x)}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}+\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\arctan (x)^2}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f} \\ & = -\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \left (\frac {f^2 \arctan (x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (d e-c f+f x)}+\frac {(d e-c f-f x) \arctan (x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}+\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \left (\frac {f^2 \arctan (x)^2}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (d e-c f+f x)}+\frac {(d e-c f-f x) \arctan (x)^2}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}+\frac {\left (3 a^2 b d\right ) \text {Subst}\left (\int \frac {d e-c f-f x}{1+x^2} \, dx,x,c+d x\right )}{f \left (f^2+(d e-c f)^2\right )}+\frac {\left (3 a^2 b d f\right ) \text {Subst}\left (\int \frac {1}{d e-c f+f x} \, dx,x,c+d x\right )}{f^2+(d e-c f)^2} \\ & = -\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {(d e-c f-f x) \arctan (x)}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {(d e-c f-f x) \arctan (x)^2}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (6 a b^2 d f\right ) \text {Subst}\left (\int \frac {\arctan (x)}{d e-c f+f x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {\left (3 b^3 d f\right ) \text {Subst}\left (\int \frac {\arctan (x)^2}{d e-c f+f x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (3 a^2 b d\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{f^2+(d e-c f)^2}+\frac {\left (3 a^2 b d (d e-c f)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{f \left (f^2+(d e-c f)^2\right )} \\ & = \frac {3 a^2 b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2 (d e-c f+f x)}{(d e+i f-c f) (1-i x)}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \left (\frac {d e \left (1-\frac {c f}{d e}\right ) \arctan (x)}{1+x^2}-\frac {f x \arctan (x)}{1+x^2}\right ) \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \left (\frac {d e \left (1-\frac {c f}{d e}\right ) \arctan (x)^2}{1+x^2}-\frac {f x \arctan (x)^2}{1+x^2}\right ) \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \\ & = \frac {3 a^2 b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (6 i a b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {x \arctan (x)}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {x \arctan (x)^2}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {\left (6 a b^2 d (d e-c f)\right ) \text {Subst}\left (\int \frac {\arctan (x)}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (3 b^3 d (d e-c f)\right ) \text {Subst}\left (\int \frac {\arctan (x)^2}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \\ & = \frac {3 a^2 b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \arctan (c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \arctan (c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \arctan (c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \arctan (c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\arctan (x)}{i-x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\arctan (x)^2}{i-x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2} \\ & = \frac {3 a^2 b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \arctan (c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \arctan (c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \arctan (c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \arctan (c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (6 b^3 d\right ) \text {Subst}\left (\int \frac {\arctan (x) \log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2} \\ & = \frac {3 a^2 b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \arctan (c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \arctan (c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \arctan (c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \arctan (c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (6 i a b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (3 i b^3 d\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2} \\ & = \frac {3 a^2 b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \arctan (c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \arctan (c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \arctan (c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \arctan (c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \\ \end{align*}
\[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.29 (sec) , antiderivative size = 3708, normalized size of antiderivative = 3.01
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(3708\) |
default | \(\text {Expression too large to display}\) | \(3708\) |
parts | \(\text {Expression too large to display}\) | \(3834\) |
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\[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3}{{\left (e+f\,x\right )}^2} \,d x \]
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