Optimal. Leaf size=750 \[ -\frac {a b d^2 \log (-c-d x+1)}{2 f (-c f+d e+f)^2}+\frac {a b d^2 \log (c+d x+1)}{2 f (-c f+d e-f)^2}-\frac {2 a b d^2 (d e-c f) \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac {a b d}{(e+f x) \left (f^2-(d e-c f)^2\right )}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac {b^2 d^2 \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{4 f (-c f+d e+f)^2}+\frac {b^2 d^2 \text {Li}_2\left (1-\frac {2}{c+d x+1}\right )}{4 f (-c f+d e-f)^2}-\frac {b^2 d^2 (d e-c f) \text {Li}_2\left (1-\frac {2}{c+d x+1}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac {b^2 d^2 (d e-c f) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac {b^2 d^2 \log (-c-d x+1)}{2 (-c f+d e+f)^2 (d e-(c+1) f)}-\frac {b^2 d^2 \log (c+d x+1)}{2 (-c f+d e+f) (d e-(c+1) f)^2}+\frac {b^2 d^2 f \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac {b^2 d^2 \log \left (\frac {2}{-c-d x+1}\right ) \tanh ^{-1}(c+d x)}{2 f (-c f+d e+f)^2}-\frac {b^2 d^2 \log \left (\frac {2}{c+d x+1}\right ) \tanh ^{-1}(c+d x)}{2 f (-c f+d e-f)^2}+\frac {2 b^2 d^2 (d e-c f) \log \left (\frac {2}{c+d x+1}\right ) \tanh ^{-1}(c+d x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac {2 b^2 d^2 (d e-c f) \tanh ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac {b^2 d \tanh ^{-1}(c+d x)}{(e+f x) (-c f+d e+f) (d e-(c+1) f)} \]
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Rubi [A] time = 2.13, antiderivative size = 750, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 18, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, 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 31
Rule 633
Rule 706
Rule 709
Rule 710
Rule 800
Rule 801
Rule 1982
Rule 2315
Rule 2402
Rule 2447
Rule 5918
Rule 5920
Rule 5926
Rule 6109
Rule 6121
Rule 6725
Rule 6741
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*}
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Mathematica [C] time = 14.79, size = 1968, normalized size = 2.62 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {artanh}\left (d x + c\right ) + a^{2}}{f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 1428, normalized size = 1.90 \[ \text {result too large to display} \]
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 \[ \frac {1}{2} \, {\left (d {\left (\frac {d \log \left (d x + c + 1\right )}{d^{2} e^{2} f - 2 \, {\left (c + 1\right )} d e f^{2} + {\left (c^{2} + 2 \, c + 1\right )} f^{3}} - \frac {d \log \left (d x + c - 1\right )}{d^{2} e^{2} f - 2 \, {\left (c - 1\right )} d e f^{2} + {\left (c^{2} - 2 \, c + 1\right )} f^{3}} - \frac {4 \, {\left (d^{2} e - c d f\right )} \log \left (f x + e\right )}{d^{4} e^{4} - 4 \, c d^{3} e^{3} f + 2 \, {\left (3 \, c^{2} - 1\right )} d^{2} e^{2} f^{2} - 4 \, {\left (c^{3} - c\right )} d e f^{3} + {\left (c^{4} - 2 \, c^{2} + 1\right )} f^{4}} + \frac {2}{d^{2} e^{3} - 2 \, c d e^{2} f + {\left (c^{2} - 1\right )} e f^{2} + {\left (d^{2} e^{2} f - 2 \, c d e f^{2} + {\left (c^{2} - 1\right )} f^{3}\right )} x}\right )} - \frac {2 \, \operatorname {artanh}\left (d x + c\right )}{f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f}\right )} a b - \frac {1}{8} \, b^{2} {\left (\frac {\log \left (-d x - c + 1\right )^{2}}{f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f} + 2 \, \int -\frac {{\left (d f x + c f - f\right )} \log \left (d x + c + 1\right )^{2} + {\left (d f x + d e - 2 \, {\left (d f x + c f - f\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{d f^{4} x^{4} + c e^{3} f - e^{3} f + {\left (3 \, d e f^{3} + c f^{4} - f^{4}\right )} x^{3} + 3 \, {\left (d e^{2} f^{2} + c e f^{3} - e f^{3}\right )} x^{2} + {\left (d e^{3} f + 3 \, c e^{2} f^{2} - 3 \, e^{2} f^{2}\right )} x}\,{d x}\right )} - \frac {a^{2}}{2 \, {\left (f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f\right )}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2}{{\left (e+f\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c + d x \right )}\right )^{2}}{\left (e + f x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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