Optimal. Leaf size=567 \[ \frac{i b^2 d \text{PolyLog}\left (2,1-\frac{2}{1-i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac{i b^2 d \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac{i b^2 d \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac{2 a b d \log (e+f x)}{(d e-c f)^2+f^2}+\frac{a b d \log \left ((c+d x)^2+1\right )}{(d e-c f)^2+f^2}-\frac{2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left ((d e-c f)^2+f^2\right )}-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac{i b^2 d \cot ^{-1}(c+d x)^2}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac{b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}+\frac{2 b^2 d \log \left (\frac{2}{1-i (c+d x)}\right ) \cot ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac{2 b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac{2 b^2 d \log \left (\frac{2}{1+i (c+d x)}\right ) \cot ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.3864, antiderivative size = 567, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 25, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.25, Rules used = {5046, 1982, 705, 31, 634, 618, 204, 628, 6741, 5058, 706, 635, 203, 260, 6688, 12, 6725, 4857, 2402, 2315, 2447, 4985, 4885, 4921, 4855} \[ \frac{i b^2 d \text{PolyLog}\left (2,1-\frac{2}{1-i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac{i b^2 d \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (d e+(-c+i) f)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac{i b^2 d \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac{2 a b d \log (e+f x)}{(d e-c f)^2+f^2}+\frac{a b d \log \left ((c+d x)^2+1\right )}{(d e-c f)^2+f^2}-\frac{2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left ((d e-c f)^2+f^2\right )}-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac{i b^2 d \cot ^{-1}(c+d x)^2}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac{b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}+\frac{2 b^2 d \log \left (\frac{2}{1-i (c+d x)}\right ) \cot ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac{2 b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(1-i (c+d x)) (d e+(-c+i) f)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac{2 b^2 d \log \left (\frac{2}{1+i (c+d x)}\right ) \cot ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5046
Rule 1982
Rule 705
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rule 6741
Rule 5058
Rule 706
Rule 635
Rule 203
Rule 260
Rule 6688
Rule 12
Rule 6725
Rule 4857
Rule 2402
Rule 2315
Rule 2447
Rule 4985
Rule 4885
Rule 4921
Rule 4855
Rubi steps
\begin{align*} \int \frac{\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx &=-\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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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) \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{\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 ) \operatorname{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) \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 \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 ) \operatorname{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 ) \operatorname{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) \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{\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\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{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 ) \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 ) \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) \tan ^{-1}(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 \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 \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 ) \operatorname{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) \tan ^{-1}(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 \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{\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) \tan ^{-1}(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 \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{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) \tan ^{-1}(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 \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{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) \tan ^{-1}(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 \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*}
Mathematica [A] time = 8.94061, size = 454, normalized size = 0.8 \[ -\frac{\frac{b^2 d \left ((c+d x)^2+1\right ) (e+f x) \left (\frac{f \left (-i \text{PolyLog}\left (2,\exp \left (2 i \left (\tan ^{-1}\left (\frac{f}{d e-c f}\right )+\cot ^{-1}(c+d x)\right )\right )\right )+2 \cot ^{-1}(c+d x) \log \left (1-\exp \left (2 i \left (\tan ^{-1}\left (\frac{f}{d e-c f}\right )+\cot ^{-1}(c+d x)\right )\right )\right )+2 \tan ^{-1}\left (\frac{f}{c f-d e}\right ) \left (-\log \left (1-\exp \left (2 i \left (\tan ^{-1}\left (\frac{f}{d e-c f}\right )+\cot ^{-1}(c+d x)\right )\right )\right )+\log \left (\sin \left (\tan ^{-1}\left (\frac{f}{d e-c f}\right )+\cot ^{-1}(c+d x)\right )\right )+i \cot ^{-1}(c+d x)\right )-\pi \log \left (\frac{1}{\sqrt{\frac{1}{(c+d x)^2}+1}}\right )+i \pi \cot ^{-1}(c+d x)+\pi \log \left (1+e^{-2 i \cot ^{-1}(c+d x)}\right )\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac{\cot ^{-1}(c+d x)^2 e^{i \tan ^{-1}\left (\frac{f}{d e-c f}\right )}}{(c f-d e) \sqrt{\frac{f^2}{(d e-c f)^2}+1}}+\frac{\cot ^{-1}(c+d x)^2}{d e+d f x}\right )}{(c+d x)^2 \left (\frac{1}{(c+d x)^2}+1\right )}+a^2+\frac{2 a b f \left (\cot ^{-1}(c+d x) \left (c^2 f-c d e+c d f x-d^2 e x+f\right )+d (e+f x) \log \left (-\frac{d (e+f x)}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right )\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}{f (e+f x)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.114, size = 1180, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -{\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 \, \operatorname{arccot}\left (d x + c\right )}{f^{2} x + e f}\right )} a b - \frac{\frac{1}{4} \,{\left (28 \, \arctan \left (1, d x + c\right )^{2} - 4 \,{\left (f^{2} x + e f\right )} \int \frac{36 \, d^{2} f x^{2} \arctan \left (1, d x + c\right )^{2} + 8 \,{\left (9 \, c \arctan \left (1, d x + c\right )^{2} - 7 \, \arctan \left (1, d x + c\right )\right )} d f x - 56 \, d e \arctan \left (1, d x + c\right ) + 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} + 36 \,{\left (c^{2} \arctan \left (1, d x + c\right )^{2} + \arctan \left (1, d x + c\right )^{2}\right )} f - 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{arccot}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arccot}\left (d x + c\right ) + a^{2}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccot}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]