Integrand size = 20, antiderivative size = 567 \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\frac {i b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 a b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {2 a b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {2 b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {2 b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {2 b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {a b d \log \left (1+(c+d x)^2\right )}{f^2+(d e-c f)^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2} \]
[Out]
Time = 1.03 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {5154, 2007, 719, 31, 648, 632, 210, 642, 6873, 5166, 720, 649, 209, 266, 6820, 12, 6857, 4967, 2449, 2352, 2497, 5105, 5005, 5041, 4965} \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=-\frac {2 a b d \arctan (c+d x) (d e-c f)}{f \left ((d e-c f)^2+f^2\right )}-\frac {2 a b d \log (e+f x)}{(d e-c f)^2+f^2}+\frac {a b d \log \left ((c+d x)^2+1\right )}{(d e-c f)^2+f^2}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {i b^2 d \cot ^{-1}(c+d x)^2}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}+\frac {2 b^2 d \log \left (\frac {2}{1-i (c+d x)}\right ) \cot ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac {2 b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (d e+(-c+i) f)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac {2 b^2 d \log \left (\frac {2}{1+i (c+d x)}\right ) \cot ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2} \]
[In]
[Out]
Rule 12
Rule 31
Rule 209
Rule 210
Rule 266
Rule 632
Rule 642
Rule 648
Rule 649
Rule 719
Rule 720
Rule 2007
Rule 2352
Rule 2449
Rule 2497
Rule 4965
Rule 4967
Rule 5005
Rule 5041
Rule 5105
Rule 5154
Rule 5166
Rule 6820
Rule 6857
Rule 6873
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {(2 b d) \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x) \left (1+(c+d x)^2\right )} \, dx}{f} \\ & = -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {(2 b d) \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x) \left (1+c^2+2 c d x+d^2 x^2\right )} \, dx}{f} \\ & = -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {(2 b) \text {Subst}\left (\int \frac {a+b \cot ^{-1}(x)}{\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 {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {(2 b) \text {Subst}\left (\int \frac {d \left (a+b \cot ^{-1}(x)\right )}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f} \\ & = -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {(2 b d) \text {Subst}\left (\int \frac {a+b \cot ^{-1}(x)}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f} \\ & = -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {(2 b d) \text {Subst}\left (\int \left (\frac {a}{(d e-c f+f x) \left (1+x^2\right )}+\frac {b \cot ^{-1}(x)}{(d e-c f+f x) \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f} \\ & = -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {(2 a b d) \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 (2 b^2 d\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(x)}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f} \\ & = -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {\left (2 b^2 d\right ) \text {Subst}\left (\int \left (\frac {f^2 \cot ^{-1}(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) \cot ^{-1}(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 {(2 a b d) \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 {(2 a b d f) \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 {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 a b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {\left (2 b^2 d\right ) \text {Subst}\left (\int \frac {(d e-c f-f x) \cot ^{-1}(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 (2 b^2 d f\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(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 {(2 a b d) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{f^2+(d e-c f)^2}-\frac {(2 a b d (d e-c f)) \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 {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 a b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {2 a b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {2 b^2 d \cot ^{-1}(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 {2 b^2 d \cot ^{-1}(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 {a b d \log \left (1+(c+d x)^2\right )}{f^2+(d e-c f)^2}+\frac {\left (2 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 (2 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 (2 b^2 d\right ) \text {Subst}\left (\int \left (\frac {d e \left (1-\frac {c f}{d e}\right ) \cot ^{-1}(x)}{1+x^2}-\frac {f x \cot ^{-1}(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 (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 a b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {2 a b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {2 b^2 d \cot ^{-1}(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 {2 b^2 d \cot ^{-1}(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 {a b d \log \left (1+(c+d x)^2\right )}{f^2+(d e-c f)^2}-\frac {i 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 {\left (2 i 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 (2 b^2 d\right ) \text {Subst}\left (\int \frac {x \cot ^{-1}(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 (2 b^2 d (d e-c f)\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(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 {i b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 a b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {2 a b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {2 b^2 d \cot ^{-1}(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 {2 b^2 d \cot ^{-1}(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 {a b d \log \left (1+(c+d x)^2\right )}{f^2+(d e-c f)^2}+\frac {i 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 {i 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 {\left (2 b^2 d\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(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 {i b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 a b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {2 a b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {2 b^2 d \cot ^{-1}(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 {2 b^2 d \cot ^{-1}(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 {2 b^2 d \cot ^{-1}(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 {a b d \log \left (1+(c+d x)^2\right )}{f^2+(d e-c f)^2}+\frac {i 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 {i 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 {\left (2 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 {i b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 a b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {2 a b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {2 b^2 d \cot ^{-1}(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 {2 b^2 d \cot ^{-1}(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 {2 b^2 d \cot ^{-1}(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 {a b d \log \left (1+(c+d x)^2\right )}{f^2+(d e-c f)^2}+\frac {i 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 {i 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 {\left (2 i 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 {i b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 a b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {2 a b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {2 b^2 d \cot ^{-1}(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 {2 b^2 d \cot ^{-1}(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 {2 b^2 d \cot ^{-1}(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 {a b d \log \left (1+(c+d x)^2\right )}{f^2+(d e-c f)^2}+\frac {i 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 {i 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 {i 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} \\ \end{align*}
Time = 6.12 (sec) , antiderivative size = 454, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=-\frac {a^2+\frac {2 a b f \left (\left (-c d e+f+c^2 f-d^2 e x+c d f x\right ) \cot ^{-1}(c+d x)+d (e+f x) \log \left (-\frac {d (e+f x)}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^2 d (e+f x) \left (1+(c+d x)^2\right ) \left (\frac {e^{i \arctan \left (\frac {f}{d e-c f}\right )} \cot ^{-1}(c+d x)^2}{(-d e+c f) \sqrt {1+\frac {f^2}{(d e-c f)^2}}}+\frac {\cot ^{-1}(c+d x)^2}{d e+d f x}+\frac {f \left (i \pi \cot ^{-1}(c+d x)+\pi \log \left (1+e^{-2 i \cot ^{-1}(c+d x)}\right )+2 \cot ^{-1}(c+d x) \log \left (1-e^{2 i \left (\cot ^{-1}(c+d x)+\arctan \left (\frac {f}{d e-c f}\right )\right )}\right )-\pi \log \left (\frac {1}{\sqrt {1+\frac {1}{(c+d x)^2}}}\right )+2 \arctan \left (\frac {f}{-d e+c f}\right ) \left (i \cot ^{-1}(c+d x)-\log \left (1-e^{2 i \left (\cot ^{-1}(c+d x)+\arctan \left (\frac {f}{d e-c f}\right )\right )}\right )+\log \left (\sin \left (\cot ^{-1}(c+d x)+\arctan \left (\frac {f}{d e-c f}\right )\right )\right )\right )-i \operatorname {PolyLog}\left (2,e^{2 i \left (\cot ^{-1}(c+d x)+\arctan \left (\frac {f}{d e-c f}\right )\right )}\right )\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}\right )}{(c+d x)^2 \left (1+\frac {1}{(c+d x)^2}\right )}}{f (e+f x)} \]
[In]
[Out]
Time = 4.18 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.38
method | result | size |
parts | \(-\frac {a^{2}}{\left (f x +e \right ) f}+\frac {b^{2} \left (-\frac {d^{2} \operatorname {arccot}\left (d x +c \right )^{2}}{\left (f \left (d x +c \right )-c f +d e \right ) f}-\frac {2 d^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right ) f \ln \left (f \left (d x +c \right )-c f +d e \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {\operatorname {arccot}\left (d x +c \right ) f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}-\frac {\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) c f}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) d e}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {f^{2} \left (-\frac {i \ln \left (f \left (d x +c \right )-c f +d e \right ) \left (\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )-\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )-\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )\right )}{2 f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {f \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}-\frac {\left (c f -d e \right ) \arctan \left (d x +c \right )^{2}}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}\right )}{f}\right )}{d}+\frac {2 a b \left (-\frac {d^{2} \operatorname {arccot}\left (d x +c \right )}{\left (f \left (d x +c \right )-c f +d e \right ) f}-\frac {d^{2} \left (\frac {f \ln \left (f \left (d x +c \right )-c f +d e \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {-\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (-c f +d e \right ) \arctan \left (d x +c \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}\right )}{f}\right )}{d}\) | \(784\) |
derivativedivides | \(\frac {\frac {a^{2} d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+b^{2} d^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right )^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {-\frac {2 \,\operatorname {arccot}\left (d x +c \right ) f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {2 \,\operatorname {arccot}\left (d x +c \right ) f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 \,\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) c f}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {2 \,\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) d e}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {2 \left (c f -d e \right ) \arctan \left (d x +c \right )^{2}}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 f \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 f^{2} \left (-\frac {i \ln \left (c f -d e -f \left (d x +c \right )\right ) \left (\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}}{f}\right )+2 a b \,d^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {\frac {\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c f -d e \right ) \arctan \left (d x +c \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}}{f}\right )}{d}\) | \(793\) |
default | \(\frac {\frac {a^{2} d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+b^{2} d^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right )^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {-\frac {2 \,\operatorname {arccot}\left (d x +c \right ) f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {2 \,\operatorname {arccot}\left (d x +c \right ) f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 \,\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) c f}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {2 \,\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) d e}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {2 \left (c f -d e \right ) \arctan \left (d x +c \right )^{2}}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 f \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 f^{2} \left (-\frac {i \ln \left (c f -d e -f \left (d x +c \right )\right ) \left (\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}}{f}\right )+2 a b \,d^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {\frac {\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c f -d e \right ) \arctan \left (d x +c \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}}{f}\right )}{d}\) | \(793\) |
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\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^2}{{\left (e+f\,x\right )}^2} \,d x \]
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