Optimal. Leaf size=125 \[ \frac{a b x^2}{4 c^3}-\frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{8 c^4}+\frac{1}{8} x^8 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2+\frac{b x^6 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{12 c}+\frac{b^2 x^4}{24 c^2}+\frac{b^2 \log \left (1-c^2 x^4\right )}{6 c^4}+\frac{b^2 x^2 \tanh ^{-1}\left (c x^2\right )}{4 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 1.54731, antiderivative size = 636, normalized size of antiderivative = 5.09, number of steps used = 62, number of rules used = 19, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.187, Rules used = {6099, 2454, 2398, 2411, 43, 2334, 12, 14, 2301, 2395, 2439, 2416, 2389, 2295, 2394, 2393, 2391, 2410, 2390} \[ \frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-c x^2\right )\right )}{16 c^4}+\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (c x^2+1\right )\right )}{16 c^4}+\frac{a b x^2}{8 c^3}-\frac{b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )}{32 c^2}-\frac{1}{192} b \left (-\frac{3 \left (1-c x^2\right )^4}{c^4}+\frac{16 \left (1-c x^2\right )^3}{c^4}-\frac{36 \left (1-c x^2\right )^2}{c^4}+\frac{48 \left (1-c x^2\right )}{c^4}-\frac{12 \log \left (1-c x^2\right )}{c^4}\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \log \left (\frac{1}{2} \left (c x^2+1\right )\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{16 c^4}+\frac{1}{32} x^8 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{1}{64} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right )+\frac{1}{16} b x^8 \log \left (c x^2+1\right ) \left (2 a-b \log \left (1-c x^2\right )\right )+\frac{b x^6 \left (2 a-b \log \left (1-c x^2\right )\right )}{48 c}+\frac{b^2 x^4}{128 c^2}+\frac{23 b^2 x^2}{192 c^3}+\frac{b^2 \left (1-c x^2\right )^4}{256 c^4}-\frac{b^2 \left (1-c x^2\right )^3}{36 c^4}+\frac{3 b^2 \left (1-c x^2\right )^2}{32 c^4}+\frac{b^2 \log ^2\left (1-c x^2\right )}{32 c^4}-\frac{b^2 \log ^2\left (c x^2+1\right )}{32 c^4}+\frac{b^2 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{16 c^4}-\frac{5 b^2 \log \left (1-c x^2\right )}{192 c^4}+\frac{b^2 \left (c x^2+1\right ) \log \left (c x^2+1\right )}{8 c^4}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log \left (c x^2+1\right )}{16 c^4}+\frac{b^2 \log \left (c x^2+1\right )}{24 c^4}-\frac{7 b^2 x^6}{576 c}+\frac{1}{32} b^2 x^8 \log ^2\left (c x^2+1\right )+\frac{b^2 x^6 \log \left (c x^2+1\right )}{24 c}-\frac{1}{256} b^2 x^8 \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 6099
Rule 2454
Rule 2398
Rule 2411
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rule 2395
Rule 2439
Rule 2416
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rule 2410
Rule 2390
Rubi steps
\begin{align*} \int x^7 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^7 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{1}{2} b x^7 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x^7 \log ^2\left (1+c x^2\right )\right ) \, dx\\ &=\frac{1}{4} \int x^7 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \, dx-\frac{1}{2} b \int x^7 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right ) \, dx+\frac{1}{4} b^2 \int x^7 \log ^2\left (1+c x^2\right ) \, dx\\ &=\frac{1}{8} \operatorname{Subst}\left (\int x^3 (2 a-b \log (1-c x))^2 \, dx,x,x^2\right )-\frac{1}{4} b \operatorname{Subst}\left (\int x^3 (-2 a+b \log (1-c x)) \log (1+c x) \, dx,x,x^2\right )+\frac{1}{8} b^2 \operatorname{Subst}\left (\int x^3 \log ^2(1+c x) \, dx,x,x^2\right )\\ &=\frac{1}{32} x^8 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{1}{16} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )+\frac{1}{32} b^2 x^8 \log ^2\left (1+c x^2\right )-\frac{1}{16} (b c) \operatorname{Subst}\left (\int \frac{x^4 (2 a-b \log (1-c x))}{1-c x} \, dx,x,x^2\right )+\frac{1}{16} (b c) \operatorname{Subst}\left (\int \frac{x^4 (-2 a+b \log (1-c x))}{1+c x} \, dx,x,x^2\right )-\frac{1}{16} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4 \log (1+c x)}{1-c x} \, dx,x,x^2\right )-\frac{1}{16} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4 \log (1+c x)}{1+c x} \, dx,x,x^2\right )\\ &=\frac{1}{32} x^8 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{1}{16} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )+\frac{1}{32} b^2 x^8 \log ^2\left (1+c x^2\right )+\frac{1}{16} b \operatorname{Subst}\left (\int \frac{\left (\frac{1}{c}-\frac{x}{c}\right )^4 (2 a-b \log (x))}{x} \, dx,x,1-c x^2\right )+\frac{1}{16} (b c) \operatorname{Subst}\left (\int \left (-\frac{-2 a+b \log (1-c x)}{c^4}+\frac{x (-2 a+b \log (1-c x))}{c^3}-\frac{x^2 (-2 a+b \log (1-c x))}{c^2}+\frac{x^3 (-2 a+b \log (1-c x))}{c}+\frac{-2 a+b \log (1-c x)}{c^4 (1+c x)}\right ) \, dx,x,x^2\right )-\frac{1}{16} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (1+c x)}{c^4}-\frac{x \log (1+c x)}{c^3}-\frac{x^2 \log (1+c x)}{c^2}-\frac{x^3 \log (1+c x)}{c}-\frac{\log (1+c x)}{c^4 (-1+c x)}\right ) \, dx,x,x^2\right )-\frac{1}{16} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (1+c x)}{c^4}+\frac{x \log (1+c x)}{c^3}-\frac{x^2 \log (1+c x)}{c^2}+\frac{x^3 \log (1+c x)}{c}+\frac{\log (1+c x)}{c^4 (1+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{32} x^8 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{1}{192} b \left (2 a-b \log \left (1-c x^2\right )\right ) \left (\frac{48 \left (1-c x^2\right )}{c^4}-\frac{36 \left (1-c x^2\right )^2}{c^4}+\frac{16 \left (1-c x^2\right )^3}{c^4}-\frac{3 \left (1-c x^2\right )^4}{c^4}-\frac{12 \log \left (1-c x^2\right )}{c^4}\right )+\frac{1}{16} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )+\frac{1}{32} b^2 x^8 \log ^2\left (1+c x^2\right )+\frac{1}{16} b \operatorname{Subst}\left (\int x^3 (-2 a+b \log (1-c x)) \, dx,x,x^2\right )+\frac{1}{16} b^2 \operatorname{Subst}\left (\int \frac{x \left (-48+36 x-16 x^2+3 x^3\right )+12 \log (x)}{12 c^4 x} \, dx,x,1-c x^2\right )-\frac{b \operatorname{Subst}\left (\int (-2 a+b \log (1-c x)) \, dx,x,x^2\right )}{16 c^3}+\frac{b \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^2\right )}{16 c^3}+2 \frac{b^2 \operatorname{Subst}\left (\int \log (1+c x) \, dx,x,x^2\right )}{16 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+c x)}{-1+c x} \, dx,x,x^2\right )}{16 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,x^2\right )}{16 c^3}+\frac{b \operatorname{Subst}\left (\int x (-2 a+b \log (1-c x)) \, dx,x,x^2\right )}{16 c^2}-\frac{b \operatorname{Subst}\left (\int x^2 (-2 a+b \log (1-c x)) \, dx,x,x^2\right )}{16 c}+2 \frac{b^2 \operatorname{Subst}\left (\int x^2 \log (1+c x) \, dx,x,x^2\right )}{16 c}\\ &=\frac{a b x^2}{8 c^3}-\frac{b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )}{32 c^2}+\frac{b x^6 \left (2 a-b \log \left (1-c x^2\right )\right )}{48 c}-\frac{1}{64} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right )+\frac{1}{32} x^8 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{1}{192} b \left (2 a-b \log \left (1-c x^2\right )\right ) \left (\frac{48 \left (1-c x^2\right )}{c^4}-\frac{36 \left (1-c x^2\right )^2}{c^4}+\frac{16 \left (1-c x^2\right )^3}{c^4}-\frac{3 \left (1-c x^2\right )^4}{c^4}-\frac{12 \log \left (1-c x^2\right )}{c^4}\right )-\frac{b \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^4}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c^4}+\frac{1}{16} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )+\frac{1}{32} b^2 x^8 \log ^2\left (1+c x^2\right )-\frac{1}{48} b^2 \operatorname{Subst}\left (\int \frac{x^3}{1-c x} \, dx,x,x^2\right )+2 \left (\frac{b^2 x^6 \log \left (1+c x^2\right )}{48 c}-\frac{1}{48} b^2 \operatorname{Subst}\left (\int \frac{x^3}{1+c x} \, dx,x,x^2\right )\right )+\frac{b^2 \operatorname{Subst}\left (\int \frac{x \left (-48+36 x-16 x^2+3 x^3\right )+12 \log (x)}{x} \, dx,x,1-c x^2\right )}{192 c^4}+2 \frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{16 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+c x^2\right )}{16 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^2\right )}{16 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \log (1-c x) \, dx,x,x^2\right )}{16 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^2\right )}{16 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^2}{1-c x} \, dx,x,x^2\right )}{32 c}+\frac{1}{64} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4}{1-c x} \, dx,x,x^2\right )\\ &=\frac{a b x^2}{8 c^3}-\frac{b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )}{32 c^2}+\frac{b x^6 \left (2 a-b \log \left (1-c x^2\right )\right )}{48 c}-\frac{1}{64} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right )+\frac{1}{32} x^8 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{1}{192} b \left (2 a-b \log \left (1-c x^2\right )\right ) \left (\frac{48 \left (1-c x^2\right )}{c^4}-\frac{36 \left (1-c x^2\right )^2}{c^4}+\frac{16 \left (1-c x^2\right )^3}{c^4}-\frac{3 \left (1-c x^2\right )^4}{c^4}-\frac{12 \log \left (1-c x^2\right )}{c^4}\right )-\frac{b \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^4}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c^4}+\frac{1}{16} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )-\frac{b^2 \log ^2\left (1+c x^2\right )}{32 c^4}+\frac{1}{32} b^2 x^8 \log ^2\left (1+c x^2\right )+2 \left (-\frac{b^2 x^2}{16 c^3}+\frac{b^2 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^4}\right )-\frac{1}{48} b^2 \operatorname{Subst}\left (\int \left (-\frac{1}{c^3}-\frac{x}{c^2}-\frac{x^2}{c}-\frac{1}{c^3 (-1+c x)}\right ) \, dx,x,x^2\right )+2 \left (\frac{b^2 x^6 \log \left (1+c x^2\right )}{48 c}-\frac{1}{48} b^2 \operatorname{Subst}\left (\int \left (\frac{1}{c^3}-\frac{x}{c^2}+\frac{x^2}{c}-\frac{1}{c^3 (1+c x)}\right ) \, dx,x,x^2\right )\right )+\frac{b^2 \operatorname{Subst}\left (\int \left (-48+36 x-16 x^2+3 x^3+\frac{12 \log (x)}{x}\right ) \, dx,x,1-c x^2\right )}{192 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-c x^2\right )}{16 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+c x^2\right )}{16 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{16 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx,x,x^2\right )}{32 c}+\frac{1}{64} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x}{c^3}-\frac{x^2}{c^2}-\frac{x^3}{c}-\frac{1}{c^4 (-1+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{a b x^2}{8 c^3}+\frac{55 b^2 x^2}{192 c^3}-\frac{5 b^2 x^4}{384 c^2}+\frac{b^2 x^6}{576 c}-\frac{b^2 x^8}{256}+\frac{3 b^2 \left (1-c x^2\right )^2}{32 c^4}-\frac{b^2 \left (1-c x^2\right )^3}{36 c^4}+\frac{b^2 \left (1-c x^2\right )^4}{256 c^4}-\frac{5 b^2 \log \left (1-c x^2\right )}{192 c^4}+\frac{b^2 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{16 c^4}-\frac{b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )}{32 c^2}+\frac{b x^6 \left (2 a-b \log \left (1-c x^2\right )\right )}{48 c}-\frac{1}{64} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right )+\frac{1}{32} x^8 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{1}{192} b \left (2 a-b \log \left (1-c x^2\right )\right ) \left (\frac{48 \left (1-c x^2\right )}{c^4}-\frac{36 \left (1-c x^2\right )^2}{c^4}+\frac{16 \left (1-c x^2\right )^3}{c^4}-\frac{3 \left (1-c x^2\right )^4}{c^4}-\frac{12 \log \left (1-c x^2\right )}{c^4}\right )-\frac{b \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^4}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c^4}+\frac{1}{16} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )-\frac{b^2 \log ^2\left (1+c x^2\right )}{32 c^4}+\frac{1}{32} b^2 x^8 \log ^2\left (1+c x^2\right )+2 \left (-\frac{b^2 x^2}{48 c^3}+\frac{b^2 x^4}{96 c^2}-\frac{b^2 x^6}{144 c}+\frac{b^2 \log \left (1+c x^2\right )}{48 c^4}+\frac{b^2 x^6 \log \left (1+c x^2\right )}{48 c}\right )+2 \left (-\frac{b^2 x^2}{16 c^3}+\frac{b^2 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^4}\right )+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-c x^2\right )\right )}{16 c^4}+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x^2\right )}{16 c^4}\\ &=\frac{a b x^2}{8 c^3}+\frac{55 b^2 x^2}{192 c^3}-\frac{5 b^2 x^4}{384 c^2}+\frac{b^2 x^6}{576 c}-\frac{b^2 x^8}{256}+\frac{3 b^2 \left (1-c x^2\right )^2}{32 c^4}-\frac{b^2 \left (1-c x^2\right )^3}{36 c^4}+\frac{b^2 \left (1-c x^2\right )^4}{256 c^4}-\frac{5 b^2 \log \left (1-c x^2\right )}{192 c^4}+\frac{b^2 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{16 c^4}+\frac{b^2 \log ^2\left (1-c x^2\right )}{32 c^4}-\frac{b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )}{32 c^2}+\frac{b x^6 \left (2 a-b \log \left (1-c x^2\right )\right )}{48 c}-\frac{1}{64} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right )+\frac{1}{32} x^8 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{1}{192} b \left (2 a-b \log \left (1-c x^2\right )\right ) \left (\frac{48 \left (1-c x^2\right )}{c^4}-\frac{36 \left (1-c x^2\right )^2}{c^4}+\frac{16 \left (1-c x^2\right )^3}{c^4}-\frac{3 \left (1-c x^2\right )^4}{c^4}-\frac{12 \log \left (1-c x^2\right )}{c^4}\right )-\frac{b \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^4}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c^4}+\frac{1}{16} b x^8 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )-\frac{b^2 \log ^2\left (1+c x^2\right )}{32 c^4}+\frac{1}{32} b^2 x^8 \log ^2\left (1+c x^2\right )+2 \left (-\frac{b^2 x^2}{48 c^3}+\frac{b^2 x^4}{96 c^2}-\frac{b^2 x^6}{144 c}+\frac{b^2 \log \left (1+c x^2\right )}{48 c^4}+\frac{b^2 x^6 \log \left (1+c x^2\right )}{48 c}\right )+2 \left (-\frac{b^2 x^2}{16 c^3}+\frac{b^2 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^4}\right )+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-c x^2\right )\right )}{16 c^4}+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^4}\\ \end{align*}
Mathematica [A] time = 0.0734534, size = 146, normalized size = 1.17 \[ \frac{3 a^2 c^4 x^8+2 a b c^3 x^6+2 b c x^2 \tanh ^{-1}\left (c x^2\right ) \left (3 a c^3 x^6+b \left (c^2 x^4+3\right )\right )+6 a b c x^2+b (3 a+4 b) \log \left (1-c x^2\right )-3 a b \log \left (c x^2+1\right )+b^2 c^2 x^4+3 b^2 \left (c^4 x^8-1\right ) \tanh ^{-1}\left (c x^2\right )^2+4 b^2 \log \left (c x^2+1\right )}{24 c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{x}^{7} \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00696, size = 293, normalized size = 2.34 \begin{align*} \frac{1}{8} \, b^{2} x^{8} \operatorname{artanh}\left (c x^{2}\right )^{2} + \frac{1}{8} \, a^{2} x^{8} + \frac{1}{24} \,{\left (6 \, x^{8} \operatorname{artanh}\left (c x^{2}\right ) + c{\left (\frac{2 \,{\left (c^{2} x^{6} + 3 \, x^{2}\right )}}{c^{4}} - \frac{3 \, \log \left (c x^{2} + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x^{2} - 1\right )}{c^{5}}\right )}\right )} a b + \frac{1}{96} \,{\left (4 \, c{\left (\frac{2 \,{\left (c^{2} x^{6} + 3 \, x^{2}\right )}}{c^{4}} - \frac{3 \, \log \left (c x^{2} + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x^{2} - 1\right )}{c^{5}}\right )} \operatorname{artanh}\left (c x^{2}\right ) + \frac{4 \, c^{2} x^{4} - 2 \,{\left (3 \, \log \left (c x^{2} - 1\right ) - 8\right )} \log \left (c x^{2} + 1\right ) + 3 \, \log \left (c x^{2} + 1\right )^{2} + 3 \, \log \left (c x^{2} - 1\right )^{2} + 16 \, \log \left (c x^{2} - 1\right )}{c^{4}}\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.0428, size = 375, normalized size = 3. \begin{align*} \frac{12 \, a^{2} c^{4} x^{8} + 8 \, a b c^{3} x^{6} + 4 \, b^{2} c^{2} x^{4} + 24 \, a b c x^{2} + 3 \,{\left (b^{2} c^{4} x^{8} - b^{2}\right )} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right )^{2} - 4 \,{\left (3 \, a b - 4 \, b^{2}\right )} \log \left (c x^{2} + 1\right ) + 4 \,{\left (3 \, a b + 4 \, b^{2}\right )} \log \left (c x^{2} - 1\right ) + 4 \,{\left (3 \, a b c^{4} x^{8} + b^{2} c^{3} x^{6} + 3 \, b^{2} c x^{2}\right )} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right )}{96 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 39.731, size = 206, normalized size = 1.65 \begin{align*} \begin{cases} \frac{a^{2} x^{8}}{8} + \frac{a b x^{8} \operatorname{atanh}{\left (c x^{2} \right )}}{4} + \frac{a b x^{6}}{12 c} + \frac{a b x^{2}}{4 c^{3}} - \frac{a b \operatorname{atanh}{\left (c x^{2} \right )}}{4 c^{4}} + \frac{b^{2} x^{8} \operatorname{atanh}^{2}{\left (c x^{2} \right )}}{8} + \frac{b^{2} x^{6} \operatorname{atanh}{\left (c x^{2} \right )}}{12 c} + \frac{b^{2} x^{4}}{24 c^{2}} + \frac{b^{2} x^{2} \operatorname{atanh}{\left (c x^{2} \right )}}{4 c^{3}} + \frac{b^{2} \log{\left (x - i \sqrt{\frac{1}{c}} \right )}}{3 c^{4}} + \frac{b^{2} \log{\left (x + i \sqrt{\frac{1}{c}} \right )}}{3 c^{4}} - \frac{b^{2} \operatorname{atanh}^{2}{\left (c x^{2} \right )}}{8 c^{4}} - \frac{b^{2} \operatorname{atanh}{\left (c x^{2} \right )}}{3 c^{4}} & \text{for}\: c \neq 0 \\\frac{a^{2} x^{8}}{8} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.50282, size = 236, normalized size = 1.89 \begin{align*} \frac{1}{8} \, a^{2} x^{8} + \frac{a b x^{6}}{12 \, c} + \frac{b^{2} x^{4}}{24 \, c^{2}} + \frac{1}{32} \,{\left (b^{2} x^{8} - \frac{b^{2}}{c^{4}}\right )} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right )^{2} + \frac{1}{24} \,{\left (3 \, a b x^{8} + \frac{b^{2} x^{6}}{c} + \frac{3 \, b^{2} x^{2}}{c^{3}}\right )} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + \frac{a b x^{2}}{4 \, c^{3}} - \frac{{\left (3 \, a b - 4 \, b^{2}\right )} \log \left (c x^{2} + 1\right )}{24 \, c^{4}} + \frac{{\left (3 \, a b + 4 \, b^{2}\right )} \log \left (c x^{2} - 1\right )}{24 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]