Optimal. Leaf size=750 \[ \frac{b^2 d^2 \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{4 f (-c f+d e+f)^2}+\frac{b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right )}{4 f (-c f+d e-f)^2}-\frac{b^2 d^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b^2 d^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac{a b d^2 \log (-c-d x+1)}{2 f (-c f+d e+f)^2}+\frac{a b d^2 \log (c+d x+1)}{2 f (-c f+d e-f)^2}-\frac{2 a b d^2 (d e-c f) \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac{a b d}{(e+f x) \left (f^2-(d e-c f)^2\right )}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b^2 d^2 \log (-c-d x+1)}{2 (-c f+d e+f)^2 (d e-(c+1) f)}-\frac{b^2 d^2 \log (c+d x+1)}{2 (-c f+d e+f) (d e-(c+1) f)^2}+\frac{b^2 d^2 f \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b^2 d^2 \log \left (\frac{2}{-c-d x+1}\right ) \tanh ^{-1}(c+d x)}{2 f (-c f+d e+f)^2}-\frac{b^2 d^2 \log \left (\frac{2}{c+d x+1}\right ) \tanh ^{-1}(c+d x)}{2 f (-c f+d e-f)^2}+\frac{2 b^2 d^2 (d e-c f) \log \left (\frac{2}{c+d x+1}\right ) \tanh ^{-1}(c+d x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac{2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b^2 d \tanh ^{-1}(c+d x)}{(e+f x) (-c f+d e+f) (d e-(c+1) f)} \]
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Rubi [A] time = 2.12567, antiderivative size = 750, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 18, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {6109, 1982, 709, 800, 6741, 6121, 710, 801, 6725, 5918, 2402, 2315, 5926, 706, 31, 633, 5920, 2447} \[ \frac{b^2 d^2 \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{4 f (-c f+d e+f)^2}+\frac{b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right )}{4 f (-c f+d e-f)^2}-\frac{b^2 d^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b^2 d^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac{a b d^2 \log (-c-d x+1)}{2 f (-c f+d e+f)^2}+\frac{a b d^2 \log (c+d x+1)}{2 f (-c f+d e-f)^2}-\frac{2 a b d^2 (d e-c f) \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac{a b d}{(e+f x) \left (f^2-(d e-c f)^2\right )}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b^2 d^2 \log (-c-d x+1)}{2 (-c f+d e+f)^2 (d e-(c+1) f)}-\frac{b^2 d^2 \log (c+d x+1)}{2 (-c f+d e+f) (d e-(c+1) f)^2}+\frac{b^2 d^2 f \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b^2 d^2 \log \left (\frac{2}{-c-d x+1}\right ) \tanh ^{-1}(c+d x)}{2 f (-c f+d e+f)^2}-\frac{b^2 d^2 \log \left (\frac{2}{c+d x+1}\right ) \tanh ^{-1}(c+d x)}{2 f (-c f+d e-f)^2}+\frac{2 b^2 d^2 (d e-c f) \log \left (\frac{2}{c+d x+1}\right ) \tanh ^{-1}(c+d x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac{2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b^2 d \tanh ^{-1}(c+d x)}{(e+f x) (-c f+d e+f) (d e-(c+1) f)} \]
Antiderivative was successfully verified.
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Rule 6109
Rule 1982
Rule 709
Rule 800
Rule 6741
Rule 6121
Rule 710
Rule 801
Rule 6725
Rule 5918
Rule 2402
Rule 2315
Rule 5926
Rule 706
Rule 31
Rule 633
Rule 5920
Rule 2447
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{(b d) \int \frac{a+b \tanh ^{-1}(c+d x)}{(e+f x)^2 \left (1-(c+d x)^2\right )} \, dx}{f}\\ &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{(b d) \int \frac{a+b \tanh ^{-1}(c+d x)}{(e+f x)^2 \left (1-c^2-2 c d x-d^2 x^2\right )} \, dx}{f}\\ &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(x)}{\left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^2 \left (1-x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b \operatorname{Subst}\left (\int \left (-\frac{a d^2}{(d e-c f+f x)^2 \left (-1+x^2\right )}-\frac{b d^2 \tanh ^{-1}(x)}{(d e-c f+f x)^2 \left (-1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}\\ &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}-\frac{\left (a b d^2\right ) \operatorname{Subst}\left (\int \frac{1}{(d e-c f+f x)^2 \left (-1+x^2\right )} \, dx,x,c+d x\right )}{f}-\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{(d e-c f+f x)^2 \left (-1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac{a b d}{\left (f^2-(d e-c f)^2\right ) (e+f x)}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}-\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{\tanh ^{-1}(x)}{2 (d e-(1+c) f)^2 (-1-x)}+\frac{\tanh ^{-1}(x)}{2 (d e+f-c f)^2 (-1+x)}+\frac{f^2 \tanh ^{-1}(x)}{(d e+(1-c) f) (d e-f-c f) (d e-c f+f x)^2}+\frac{2 f^2 (d e-c f) \tanh ^{-1}(x)}{(d e+(1-c) f)^2 (d e-f-c f)^2 (d e-c f+f x)}\right ) \, dx,x,c+d x\right )}{f}+\frac{\left (a b d^2\right ) \operatorname{Subst}\left (\int \frac{d e-c f-f x}{(d e-c f+f x) \left (-1+x^2\right )} \, dx,x,c+d x\right )}{f \left (f^2-(d e-c f)^2\right )}\\ &=-\frac{a b d}{\left (f^2-(d e-c f)^2\right ) (e+f x)}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}-\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{-1-x} \, dx,x,c+d x\right )}{2 f (d e-f-c f)^2}-\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{-1+x} \, dx,x,c+d x\right )}{2 f (d e+f-c f)^2}-\frac{\left (2 b^2 d^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{d e-c f+f x} \, dx,x,c+d x\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac{\left (b^2 d^2 f\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{(d e-c f+f x)^2} \, dx,x,c+d x\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac{\left (a b d^2\right ) \operatorname{Subst}\left (\int \left (\frac{-d e+(1+c) f}{2 (d e+f-c f) (1-x)}+\frac{-d e-(1-c) f}{2 (d e-(1+c) f) (1+x)}+\frac{2 f^2 (d e-c f)}{(d e+(1-c) f) (d e-f-c f) (d e-c f+f x)}\right ) \, dx,x,c+d x\right )}{f \left (f^2-(d e-c f)^2\right )}\\ &=-\frac{a b d}{\left (f^2-(d e-c f)^2\right ) (e+f x)}+\frac{b^2 d \tanh ^{-1}(c+d x)}{(d e+f-c f) (d e-(1+c) f) (e+f x)}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b^2 d^2 \tanh ^{-1}(c+d x) \log \left (\frac{2}{1-c-d x}\right )}{2 f (d e+f-c f)^2}-\frac{a b d^2 \log (1-c-d x)}{2 f (d e+f-c f)^2}-\frac{b^2 d^2 \tanh ^{-1}(c+d x) \log \left (\frac{2}{1+c+d x}\right )}{2 f (d e-f-c f)^2}+\frac{2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac{2}{1+c+d x}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{a b d^2 \log (1+c+d x)}{2 f (d e-f-c f)^2}-\frac{2 a b d^2 (d e-c f) \log (e+f x)}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac{2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{2 f (d e-f-c f)^2}-\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{2 f (d e+f-c f)^2}-\frac{\left (2 b^2 d^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{\left (2 b^2 d^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2 (d e-c f+f x)}{(d e+f-c f) (1+x)}\right )}{1-x^2} \, dx,x,c+d x\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{(d e-c f+f x) \left (1-x^2\right )} \, dx,x,c+d x\right )}{(d e+f-c f) (d e-(1+c) f)}\\ &=-\frac{a b d}{\left (f^2-(d e-c f)^2\right ) (e+f x)}+\frac{b^2 d \tanh ^{-1}(c+d x)}{(d e+f-c f) (d e-(1+c) f) (e+f x)}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b^2 d^2 \tanh ^{-1}(c+d x) \log \left (\frac{2}{1-c-d x}\right )}{2 f (d e+f-c f)^2}-\frac{a b d^2 \log (1-c-d x)}{2 f (d e+f-c f)^2}-\frac{b^2 d^2 \tanh ^{-1}(c+d x) \log \left (\frac{2}{1+c+d x}\right )}{2 f (d e-f-c f)^2}+\frac{2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac{2}{1+c+d x}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{a b d^2 \log (1+c+d x)}{2 f (d e-f-c f)^2}-\frac{2 a b d^2 (d e-c f) \log (e+f x)}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac{2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{b^2 d^2 (d e-c f) \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c+d x}\right )}{2 f (d e-f-c f)^2}+\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c-d x}\right )}{2 f (d e+f-c f)^2}+\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{-d e+c f+f x}{1-x^2} \, dx,x,c+d x\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{\left (b^2 d^2 f^2\right ) \operatorname{Subst}\left (\int \frac{1}{d e-c f+f x} \, dx,x,c+d x\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac{\left (2 b^2 d^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c+d x}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}\\ &=-\frac{a b d}{\left (f^2-(d e-c f)^2\right ) (e+f x)}+\frac{b^2 d \tanh ^{-1}(c+d x)}{(d e+f-c f) (d e-(1+c) f) (e+f x)}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b^2 d^2 \tanh ^{-1}(c+d x) \log \left (\frac{2}{1-c-d x}\right )}{2 f (d e+f-c f)^2}-\frac{a b d^2 \log (1-c-d x)}{2 f (d e+f-c f)^2}-\frac{b^2 d^2 \tanh ^{-1}(c+d x) \log \left (\frac{2}{1+c+d x}\right )}{2 f (d e-f-c f)^2}+\frac{2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac{2}{1+c+d x}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{a b d^2 \log (1+c+d x)}{2 f (d e-f-c f)^2}+\frac{b^2 d^2 f \log (e+f x)}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac{2 a b d^2 (d e-c f) \log (e+f x)}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac{2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{b^2 d^2 \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{4 f (d e+f-c f)^2}+\frac{b^2 d^2 \text{Li}_2\left (1-\frac{2}{1+c+d x}\right )}{4 f (d e-f-c f)^2}-\frac{b^2 d^2 (d e-c f) \text{Li}_2\left (1-\frac{2}{1+c+d x}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{b^2 d^2 (d e-c f) \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x} \, dx,x,c+d x\right )}{2 (d e+f-c f) (d e-(1+c) f)^2}-\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,c+d x\right )}{2 (d e+f-c f)^2 (d e-(1+c) f)}\\ &=-\frac{a b d}{\left (f^2-(d e-c f)^2\right ) (e+f x)}+\frac{b^2 d \tanh ^{-1}(c+d x)}{(d e+f-c f) (d e-(1+c) f) (e+f x)}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b^2 d^2 \tanh ^{-1}(c+d x) \log \left (\frac{2}{1-c-d x}\right )}{2 f (d e+f-c f)^2}-\frac{a b d^2 \log (1-c-d x)}{2 f (d e+f-c f)^2}+\frac{b^2 d^2 \log (1-c-d x)}{2 (d e+f-c f)^2 (d e-(1+c) f)}-\frac{b^2 d^2 \tanh ^{-1}(c+d x) \log \left (\frac{2}{1+c+d x}\right )}{2 f (d e-f-c f)^2}+\frac{2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac{2}{1+c+d x}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{a b d^2 \log (1+c+d x)}{2 f (d e-f-c f)^2}-\frac{b^2 d^2 \log (1+c+d x)}{2 (d e+f-c f) (d e-(1+c) f)^2}+\frac{b^2 d^2 f \log (e+f x)}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac{2 a b d^2 (d e-c f) \log (e+f x)}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac{2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{b^2 d^2 \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{4 f (d e+f-c f)^2}+\frac{b^2 d^2 \text{Li}_2\left (1-\frac{2}{1+c+d x}\right )}{4 f (d e-f-c f)^2}-\frac{b^2 d^2 (d e-c f) \text{Li}_2\left (1-\frac{2}{1+c+d x}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac{b^2 d^2 (d e-c f) \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}\\ \end{align*}
Mathematica [C] time = 14.6627, size = 1970, normalized size = 2.63 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.179, size = 1428, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (d{\left (\frac{d \log \left (d x + c + 1\right )}{d^{2} e^{2} f - 2 \,{\left (c + 1\right )} d e f^{2} +{\left (c^{2} + 2 \, c + 1\right )} f^{3}} - \frac{d \log \left (d x + c - 1\right )}{d^{2} e^{2} f - 2 \,{\left (c - 1\right )} d e f^{2} +{\left (c^{2} - 2 \, c + 1\right )} f^{3}} - \frac{4 \,{\left (d^{2} e - c d f\right )} \log \left (f x + e\right )}{d^{4} e^{4} - 4 \, c d^{3} e^{3} f + 2 \,{\left (3 \, c^{2} - 1\right )} d^{2} e^{2} f^{2} - 4 \,{\left (c^{3} - c\right )} d e f^{3} +{\left (c^{4} - 2 \, c^{2} + 1\right )} f^{4}} + \frac{2}{d^{2} e^{3} - 2 \, c d e^{2} f +{\left (c^{2} - 1\right )} e f^{2} +{\left (d^{2} e^{2} f - 2 \, c d e f^{2} +{\left (c^{2} - 1\right )} f^{3}\right )} x}\right )} - \frac{2 \, \operatorname{artanh}\left (d x + c\right )}{f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f}\right )} a b - \frac{1}{8} \, b^{2}{\left (\frac{\log \left (-d x - c + 1\right )^{2}}{f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f} + 2 \, \int -\frac{{\left (d f x + c f - f\right )} \log \left (d x + c + 1\right )^{2} +{\left (d f x + d e - 2 \,{\left (d f x + c f - f\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{d f^{4} x^{4} + c e^{3} f - e^{3} f +{\left (3 \, d e f^{3} + c f^{4} - f^{4}\right )} x^{3} + 3 \,{\left (d e^{2} f^{2} + c e f^{3} - e f^{3}\right )} x^{2} +{\left (d e^{3} f + 3 \, c e^{2} f^{2} - 3 \, e^{2} f^{2}\right )} x}\,{d x}\right )} - \frac{a^{2}}{2 \,{\left (f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{artanh}\left (d x + c\right )^{2} + 2 \, a b \operatorname{artanh}\left (d x + c\right ) + a^{2}}{f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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