Optimal. Leaf size=568 \[ \frac {2 a b d \log (e+f x)}{(d e-c f)^2+f^2}-\frac {a b d \log \left ((c+d x)^2+1\right )}{(d e-c f)^2+f^2}+\frac {2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left ((d e-c f)^2+f^2\right )}-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac {i b^2 d \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {i b^2 d \tan ^{-1}(c+d x)^2}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {b^2 d (d e-c f) \tan ^{-1}(c+d x)^2}{f \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}-\frac {2 b^2 d \log \left (\frac {2}{1-i (c+d x)}\right ) \tan ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {2 b^2 d \tan ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {2 b^2 d \log \left (\frac {2}{1+i (c+d x)}\right ) \tan ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2} \]
[Out]
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Rubi [A] time = 1.35, antiderivative size = 568, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 25, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {5045, 1982, 705, 31, 634, 618, 204, 628, 6741, 5057, 706, 635, 203, 260, 6688, 12, 6725, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854} \[ \frac {i b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac {i b^2 d \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(1-i (c+d x)) (d e+(-c+i) f)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {i b^2 d \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {2 a b d \log (e+f x)}{(d e-c f)^2+f^2}-\frac {a b d \log \left ((c+d x)^2+1\right )}{(d e-c f)^2+f^2}+\frac {2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left ((d e-c f)^2+f^2\right )}-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {i b^2 d \tan ^{-1}(c+d x)^2}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {b^2 d (d e-c f) \tan ^{-1}(c+d x)^2}{f \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}-\frac {2 b^2 d \log \left (\frac {2}{1-i (c+d x)}\right ) \tan ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {2 b^2 d \tan ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (d e+(-c+i) f)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {2 b^2 d \log \left (\frac {2}{1+i (c+d x)}\right ) \tan ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 203
Rule 204
Rule 260
Rule 618
Rule 628
Rule 634
Rule 635
Rule 705
Rule 706
Rule 1982
Rule 2315
Rule 2402
Rule 2447
Rule 4854
Rule 4856
Rule 4884
Rule 4920
Rule 4984
Rule 5045
Rule 5057
Rule 6688
Rule 6725
Rule 6741
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx &=-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {(2 b d) \int \frac {a+b \tan ^{-1}(c+d x)}{(e+f x) \left (1+(c+d x)^2\right )} \, dx}{f}\\ &=-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {(2 b d) \int \frac {a+b \tan ^{-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 \tan ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {a+b \tan ^{-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 \tan ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {d \left (a+b \tan ^{-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 \tan ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {(2 b d) \operatorname {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {(2 b d) \operatorname {Subst}\left (\int \left (\frac {a}{(d e-c f+f x) \left (1+x^2\right )}+\frac {b \tan ^{-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 \tan ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {(2 a b d) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}(x)}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {\left (2 b^2 d\right ) \operatorname {Subst}\left (\int \left (\frac {f^2 \tan ^{-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) \tan ^{-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) \operatorname {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) \operatorname {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 \tan ^{-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 ) \operatorname {Subst}\left (\int \frac {(d e-c f-f x) \tan ^{-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 ) \operatorname {Subst}\left (\int \frac {\tan ^{-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) \operatorname {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)) \operatorname {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 {2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {\left (a+b \tan ^{-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 {2 b^2 d \tan ^{-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 \tan ^{-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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \left (\frac {d e \left (1-\frac {c f}{d e}\right ) \tan ^{-1}(x)}{1+x^2}-\frac {f x \tan ^{-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 {2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {\left (a+b \tan ^{-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 {2 b^2 d \tan ^{-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 \tan ^{-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 \text {Li}_2\left (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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {x \tan ^{-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 ) \operatorname {Subst}\left (\int \frac {\tan ^{-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 {2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {i b^2 d \tan ^{-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) \tan ^{-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 \tan ^{-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 {2 b^2 d \tan ^{-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 \tan ^{-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 \text {Li}_2\left (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 \text {Li}_2\left (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 ) \operatorname {Subst}\left (\int \frac {\tan ^{-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 {2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {i b^2 d \tan ^{-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) \tan ^{-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 \tan ^{-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 {2 b^2 d \tan ^{-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 \tan ^{-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 \tan ^{-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 \text {Li}_2\left (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 \text {Li}_2\left (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 ) \operatorname {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 {2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {i b^2 d \tan ^{-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) \tan ^{-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 \tan ^{-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 {2 b^2 d \tan ^{-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 \tan ^{-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 \tan ^{-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 \text {Li}_2\left (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 \text {Li}_2\left (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 ) \operatorname {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 {2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {i b^2 d \tan ^{-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) \tan ^{-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 \tan ^{-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 {2 b^2 d \tan ^{-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 \tan ^{-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 \tan ^{-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 \text {Li}_2\left (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 \text {Li}_2\left (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 \text {Li}_2\left (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*}
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Mathematica [A] time = 7.49, size = 419, normalized size = 0.74 \[ \frac {-\frac {a^2}{f}+\frac {2 a b \left (d (e+f x) \log \left (\frac {d (e+f x)}{\sqrt {(c+d x)^2+1}}\right )-\tan ^{-1}(c+d x) \left (c^2 f-c d e+c d f x-d^2 e x+f\right )\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {b^2 d (e+f x) \left (-\frac {(d e-c f) \left (i \text {Li}_2\left (\exp \left (2 i \left (\tan ^{-1}\left (\frac {d e-c f}{f}\right )+\tan ^{-1}(c+d x)\right )\right )\right )-2 \left (\tan ^{-1}\left (\frac {d e-c f}{f}\right )+\tan ^{-1}(c+d x)\right ) \log \left (1-\exp \left (2 i \left (\tan ^{-1}\left (\frac {d e-c f}{f}\right )+\tan ^{-1}(c+d x)\right )\right )\right )-i \tan ^{-1}(c+d x) \left (\pi -2 \tan ^{-1}\left (\frac {d e-c f}{f}\right )\right )+2 \tan ^{-1}\left (\frac {d e-c f}{f}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac {d e-c f}{f}\right )+\tan ^{-1}(c+d x)\right )\right )+\pi \log \left (\frac {1}{\sqrt {(c+d x)^2+1}}\right )-\pi \log \left (1+e^{-2 i \tan ^{-1}(c+d x)}\right )\right )}{f^2 \left (\frac {(d e-c f)^2}{f^2}+1\right )}-\frac {\tan ^{-1}(c+d x)^2 e^{i \tan ^{-1}\left (\frac {d e-c f}{f}\right )}}{f \sqrt {\frac {(d e-c f)^2}{f^2}+1}}+\frac {(c+d x) \tan ^{-1}(c+d x)^2}{d (e+f x)}\right )}{d e-c f}}{e+f x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arctan \left (d x + c\right )^{2} + 2 \, a b \arctan \left (d x + c\right ) + a^{2}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 1087, normalized size = 1.91 \[ -\frac {d \,a^{2}}{\left (d f x +d e \right ) f}-\frac {d \,b^{2} \arctan \left (d x +c \right )^{2}}{\left (d f x +d e \right ) f}+\frac {2 d \,b^{2} \arctan \left (d x +c \right ) \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 {d \,b^{2} \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {d \,b^{2} \arctan \left (d x +c \right )^{2} c}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {d^{2} b^{2} \arctan \left (d x +c \right )^{2} e}{f \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}+\frac {i d \,b^{2} \ln \left (d x +c -i\right )^{2}}{4 c^{2} f^{2}-8 c d e f +4 d^{2} e^{2}+4 f^{2}}-\frac {i d \,b^{2} \ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}+\frac {i d \,b^{2} \ln \left (f \left (d x +c \right )-c f +d e \right ) \ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {i d \,b^{2} \ln \left (d x +c -i\right ) \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 {i d \,b^{2} \dilog \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {i d \,b^{2} \dilog \left (-\frac {i \left (d x +c +i\right )}{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {i d \,b^{2} \ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}-\frac {i d \,b^{2} \dilog \left (\frac {i \left (d x +c -i\right )}{2}\right )}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}-\frac {i d \,b^{2} \ln \left (f \left (d x +c \right )-c f +d e \right ) \ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {i d \,b^{2} \ln \left (d x +c +i\right )^{2}}{4 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}+\frac {i d \,b^{2} \ln \left (d x +c +i\right ) \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 {i d \,b^{2} \dilog \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {2 d a b \arctan \left (d x +c \right )}{\left (d f x +d e \right ) f}+\frac {2 d a b \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 {d a b \ln \left (1+\left (d x +c \right )^{2}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {2 d a b \arctan \left (d x +c \right ) c}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {2 d^{2} a b \arctan \left (d x +c \right ) e}{f \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (d {\left (\frac {2 \, {\left (d^{2} e - c d f\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{{\left (d^{2} e^{2} f - 2 \, c d e f^{2} + {\left (c^{2} + 1\right )} f^{3}\right )} d} - \frac {\log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{2} e^{2} - 2 \, c d e f + {\left (c^{2} + 1\right )} f^{2}} + \frac {2 \, \log \left (f x + e\right )}{d^{2} e^{2} - 2 \, c d e f + {\left (c^{2} + 1\right )} f^{2}}\right )} - \frac {2 \, \arctan \left (d x + c\right )}{f^{2} x + e f}\right )} a b - \frac {\frac {1}{4} \, {\left (28 \, \arctan \left (d x + c\right )^{2} - 4 \, {\left (f^{2} x + e f\right )} \int \frac {36 \, {\left (d^{2} f x^{2} + 2 \, c d f x + {\left (c^{2} + 1\right )} f\right )} \arctan \left (d x + c\right )^{2} + 3 \, {\left (d^{2} f x^{2} + 2 \, c d f x + {\left (c^{2} + 1\right )} f\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + 56 \, {\left (d f x + d e\right )} \arctan \left (d x + c\right ) - 12 \, {\left (d^{2} f x^{2} + c d e + {\left (d^{2} e + c d f\right )} x\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{4 \, {\left (d^{2} f^{3} x^{4} + {\left (c^{2} + 1\right )} e^{2} f + 2 \, {\left (d^{2} e f^{2} + c d f^{3}\right )} x^{3} + {\left (d^{2} e^{2} f + 4 \, c d e f^{2} + {\left (c^{2} + 1\right )} f^{3}\right )} x^{2} + 2 \, {\left (c d e^{2} f + {\left (c^{2} + 1\right )} e f^{2}\right )} x\right )}}\,{d x} - 3 \, \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2}\right )} b^{2}}{16 \, {\left (f^{2} x + e f\right )}} - \frac {a^{2}}{f^{2} x + e f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^2}{{\left (e+f\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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