Optimal. Leaf size=124 \[ \frac{a b x^3}{6 c^3}-\frac{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{12 c^4}+\frac{1}{12} x^{12} \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2-\frac{b x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{18 c}+\frac{b^2 x^6}{36 c^2}-\frac{b^2 \log \left (c^2 x^6+1\right )}{9 c^4}+\frac{b^2 x^3 \tan ^{-1}\left (c x^3\right )}{6 c^3} \]
[Out]
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Rubi [C] time = 1.65455, antiderivative size = 731, normalized size of antiderivative = 5.9, 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 = {5035, 2454, 2398, 2411, 43, 2334, 12, 14, 2301, 2395, 2439, 2416, 2389, 2295, 2394, 2393, 2391, 2410, 2390} \[ -\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-i c x^3\right )\right )}{24 c^4}-\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^4}+\frac{a b x^3}{12 c^3}-\frac{b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c^2}+\frac{1}{288} i b \left (-\frac{3 \left (1-i c x^3\right )^4}{c^4}+\frac{16 \left (1-i c x^3\right )^3}{c^4}-\frac{36 \left (1-i c x^3\right )^2}{c^4}+\frac{48 \left (1-i c x^3\right )}{c^4}-\frac{12 \log \left (1-i c x^3\right )}{c^4}\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )+\frac{b \log \left (\frac{1}{2} \left (1+i c x^3\right )\right ) \left (2 i a-b \log \left (1-i c x^3\right )\right )}{24 c^4}+\frac{1}{48} x^{12} \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{96} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right )-\frac{1}{24} b x^{12} \log \left (1+i c x^3\right ) \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{72 c}+\frac{b^2 x^6}{192 c^2}-\frac{23 i b^2 x^3}{288 c^3}-\frac{b^2 \left (1-i c x^3\right )^4}{384 c^4}+\frac{b^2 \left (1-i c x^3\right )^3}{54 c^4}-\frac{b^2 \left (1-i c x^3\right )^2}{16 c^4}-\frac{b^2 \log ^2\left (1-i c x^3\right )}{48 c^4}+\frac{b^2 \log ^2\left (1+i c x^3\right )}{48 c^4}-\frac{b^2 \log \left (-c x^3+i\right )}{36 c^4}-\frac{b^2 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{24 c^4}-\frac{b^2 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{12 c^4}-\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c^4}+\frac{5 b^2 \log \left (c x^3+i\right )}{288 c^4}-\frac{7 i b^2 x^9}{864 c}-\frac{1}{48} b^2 x^{12} \log ^2\left (1+i c x^3\right )+\frac{i b^2 x^9 \log \left (1+i c x^3\right )}{36 c}+\frac{b^2 x^{12}}{384} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 5035
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^{11} \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^{11} \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{2} b x^{11} \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{4} b^2 x^{11} \log ^2\left (1+i c x^3\right )\right ) \, dx\\ &=\frac{1}{4} \int x^{11} \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \, dx+\frac{1}{2} b \int x^{11} \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right ) \, dx-\frac{1}{4} b^2 \int x^{11} \log ^2\left (1+i c x^3\right ) \, dx\\ &=\frac{1}{12} \operatorname{Subst}\left (\int x^3 (2 a+i b \log (1-i c x))^2 \, dx,x,x^3\right )+\frac{1}{6} b \operatorname{Subst}\left (\int x^3 (-2 i a+b \log (1-i c x)) \log (1+i c x) \, dx,x,x^3\right )-\frac{1}{12} b^2 \operatorname{Subst}\left (\int x^3 \log ^2(1+i c x) \, dx,x,x^3\right )\\ &=\frac{1}{48} x^{12} \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{1}{24} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{48} b^2 x^{12} \log ^2\left (1+i c x^3\right )-\frac{1}{24} (i b c) \operatorname{Subst}\left (\int \frac{x^4 (-2 i a+b \log (1-i c x))}{1+i c x} \, dx,x,x^3\right )-\frac{1}{24} (b c) \operatorname{Subst}\left (\int \frac{x^4 (2 a+i b \log (1-i c x))}{1-i c x} \, dx,x,x^3\right )+\frac{1}{24} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4 \log (1+i c x)}{1-i c x} \, dx,x,x^3\right )+\frac{1}{24} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4 \log (1+i c x)}{1+i c x} \, dx,x,x^3\right )\\ &=\frac{1}{48} x^{12} \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{1}{24} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{48} b^2 x^{12} \log ^2\left (1+i c x^3\right )-\frac{1}{24} (i b) \operatorname{Subst}\left (\int \frac{\left (-\frac{i}{c}+\frac{i x}{c}\right )^4 (2 a+i b \log (x))}{x} \, dx,x,1-i c x^3\right )-\frac{1}{24} (i b c) \operatorname{Subst}\left (\int \left (-\frac{-2 i a+b \log (1-i c x)}{c^4}+\frac{i x (-2 i a+b \log (1-i c x))}{c^3}+\frac{x^2 (-2 i a+b \log (1-i c x))}{c^2}-\frac{i x^3 (-2 i a+b \log (1-i c x))}{c}-\frac{i (-2 i a+b \log (1-i c x))}{c^4 (-i+c x)}\right ) \, dx,x,x^3\right )+\frac{1}{24} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (1+i c x)}{c^4}+\frac{i x \log (1+i c x)}{c^3}+\frac{x^2 \log (1+i c x)}{c^2}-\frac{i x^3 \log (1+i c x)}{c}-\frac{i \log (1+i c x)}{c^4 (-i+c x)}\right ) \, dx,x,x^3\right )+\frac{1}{24} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (1+i c x)}{c^4}-\frac{i x \log (1+i c x)}{c^3}+\frac{x^2 \log (1+i c x)}{c^2}+\frac{i x^3 \log (1+i c x)}{c}+\frac{i \log (1+i c x)}{c^4 (i+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac{1}{48} x^{12} \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{288} i b \left (2 a+i b \log \left (1-i c x^3\right )\right ) \left (\frac{48 \left (1-i c x^3\right )}{c^4}-\frac{36 \left (1-i c x^3\right )^2}{c^4}+\frac{16 \left (1-i c x^3\right )^3}{c^4}-\frac{3 \left (1-i c x^3\right )^4}{c^4}-\frac{12 \log \left (1-i c x^3\right )}{c^4}\right )-\frac{1}{24} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{48} b^2 x^{12} \log ^2\left (1+i c x^3\right )-\frac{1}{24} b \operatorname{Subst}\left (\int x^3 (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )-\frac{1}{24} 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-i c x^3\right )+\frac{(i b) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )}{24 c^3}-\frac{b \operatorname{Subst}\left (\int \frac{-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^3\right )}{24 c^3}-2 \frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log (1+i c x) \, dx,x,x^3\right )}{24 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{-i+c x} \, dx,x,x^3\right )}{24 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{i+c x} \, dx,x,x^3\right )}{24 c^3}+\frac{b \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )}{24 c^2}-\frac{(i b) \operatorname{Subst}\left (\int x^2 (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )}{24 c}+2 \frac{\left (i b^2\right ) \operatorname{Subst}\left (\int x^2 \log (1+i c x) \, dx,x,x^3\right )}{24 c}\\ &=\frac{a b x^3}{12 c^3}-\frac{b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c^2}+\frac{i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{72 c}+\frac{1}{96} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{1}{48} x^{12} \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{288} i b \left (2 a+i b \log \left (1-i c x^3\right )\right ) \left (\frac{48 \left (1-i c x^3\right )}{c^4}-\frac{36 \left (1-i c x^3\right )^2}{c^4}+\frac{16 \left (1-i c x^3\right )^3}{c^4}-\frac{3 \left (1-i c x^3\right )^4}{c^4}-\frac{12 \log \left (1-i c x^3\right )}{c^4}\right )+\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^4}-\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c^4}-\frac{1}{24} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{48} b^2 x^{12} \log ^2\left (1+i c x^3\right )+\frac{1}{72} b^2 \operatorname{Subst}\left (\int \frac{x^3}{1-i c x} \, dx,x,x^3\right )+2 \left (\frac{i b^2 x^9 \log \left (1+i c x^3\right )}{72 c}+\frac{1}{72} b^2 \operatorname{Subst}\left (\int \frac{x^3}{1+i c x} \, dx,x,x^3\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-i c x^3\right )}{288 c^4}-2 \frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^3\right )}{24 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+i c x^3\right )}{24 c^4}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log (1-i c x) \, dx,x,x^3\right )}{24 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^3\right )}{24 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^3\right )}{24 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-i c x} \, dx,x,x^3\right )}{48 c}-\frac{1}{96} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4}{1-i c x} \, dx,x,x^3\right )\\ &=\frac{a b x^3}{12 c^3}-\frac{b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c^2}+\frac{i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{72 c}+\frac{1}{96} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{1}{48} x^{12} \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{288} i b \left (2 a+i b \log \left (1-i c x^3\right )\right ) \left (\frac{48 \left (1-i c x^3\right )}{c^4}-\frac{36 \left (1-i c x^3\right )^2}{c^4}+\frac{16 \left (1-i c x^3\right )^3}{c^4}-\frac{3 \left (1-i c x^3\right )^4}{c^4}-\frac{12 \log \left (1-i c x^3\right )}{c^4}\right )+\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^4}-\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c^4}-\frac{1}{24} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac{b^2 \log ^2\left (1+i c x^3\right )}{48 c^4}-\frac{1}{48} b^2 x^{12} \log ^2\left (1+i c x^3\right )-2 \left (-\frac{i b^2 x^3}{24 c^3}+\frac{b^2 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{24 c^4}\right )+2 \left (\frac{i b^2 x^9 \log \left (1+i c x^3\right )}{72 c}+\frac{1}{72} b^2 \operatorname{Subst}\left (\int \left (\frac{i}{c^3}+\frac{x}{c^2}-\frac{i x^2}{c}-\frac{1}{c^3 (-i+c x)}\right ) \, dx,x,x^3\right )\right )+\frac{1}{72} b^2 \operatorname{Subst}\left (\int \left (-\frac{i}{c^3}+\frac{x}{c^2}+\frac{i x^2}{c}-\frac{1}{c^3 (i+c x)}\right ) \, dx,x,x^3\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-i c x^3\right )}{288 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-i c x^3\right )}{24 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+i c x^3\right )}{24 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^3\right )}{24 c^4}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{i x}{c}-\frac{i}{c^2 (i+c x)}\right ) \, dx,x,x^3\right )}{48 c}-\frac{1}{96} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{i x}{c^3}+\frac{x^2}{c^2}+\frac{i x^3}{c}+\frac{i}{c^4 (i+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac{a b x^3}{12 c^3}-\frac{55 i b^2 x^3}{288 c^3}-\frac{5 b^2 x^6}{576 c^2}+\frac{i b^2 x^9}{864 c}+\frac{b^2 x^{12}}{384}-\frac{b^2 \left (1-i c x^3\right )^2}{16 c^4}+\frac{b^2 \left (1-i c x^3\right )^3}{54 c^4}-\frac{b^2 \left (1-i c x^3\right )^4}{384 c^4}-\frac{b^2 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{24 c^4}-\frac{b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c^2}+\frac{i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{72 c}+\frac{1}{96} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{1}{48} x^{12} \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{288} i b \left (2 a+i b \log \left (1-i c x^3\right )\right ) \left (\frac{48 \left (1-i c x^3\right )}{c^4}-\frac{36 \left (1-i c x^3\right )^2}{c^4}+\frac{16 \left (1-i c x^3\right )^3}{c^4}-\frac{3 \left (1-i c x^3\right )^4}{c^4}-\frac{12 \log \left (1-i c x^3\right )}{c^4}\right )+\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^4}-\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c^4}-\frac{1}{24} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac{b^2 \log ^2\left (1+i c x^3\right )}{48 c^4}-\frac{1}{48} b^2 x^{12} \log ^2\left (1+i c x^3\right )+2 \left (\frac{i b^2 x^3}{72 c^3}+\frac{b^2 x^6}{144 c^2}-\frac{i b^2 x^9}{216 c}-\frac{b^2 \log \left (i-c x^3\right )}{72 c^4}+\frac{i b^2 x^9 \log \left (1+i c x^3\right )}{72 c}\right )-2 \left (-\frac{i b^2 x^3}{24 c^3}+\frac{b^2 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{24 c^4}\right )+\frac{5 b^2 \log \left (i+c x^3\right )}{288 c^4}-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{24 c^4}-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-i c x^3\right )}{24 c^4}\\ &=\frac{a b x^3}{12 c^3}-\frac{55 i b^2 x^3}{288 c^3}-\frac{5 b^2 x^6}{576 c^2}+\frac{i b^2 x^9}{864 c}+\frac{b^2 x^{12}}{384}-\frac{b^2 \left (1-i c x^3\right )^2}{16 c^4}+\frac{b^2 \left (1-i c x^3\right )^3}{54 c^4}-\frac{b^2 \left (1-i c x^3\right )^4}{384 c^4}-\frac{b^2 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{24 c^4}-\frac{b^2 \log ^2\left (1-i c x^3\right )}{48 c^4}-\frac{b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c^2}+\frac{i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{72 c}+\frac{1}{96} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{1}{48} x^{12} \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{288} i b \left (2 a+i b \log \left (1-i c x^3\right )\right ) \left (\frac{48 \left (1-i c x^3\right )}{c^4}-\frac{36 \left (1-i c x^3\right )^2}{c^4}+\frac{16 \left (1-i c x^3\right )^3}{c^4}-\frac{3 \left (1-i c x^3\right )^4}{c^4}-\frac{12 \log \left (1-i c x^3\right )}{c^4}\right )+\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^4}-\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c^4}-\frac{1}{24} b x^{12} \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac{b^2 \log ^2\left (1+i c x^3\right )}{48 c^4}-\frac{1}{48} b^2 x^{12} \log ^2\left (1+i c x^3\right )+2 \left (\frac{i b^2 x^3}{72 c^3}+\frac{b^2 x^6}{144 c^2}-\frac{i b^2 x^9}{216 c}-\frac{b^2 \log \left (i-c x^3\right )}{72 c^4}+\frac{i b^2 x^9 \log \left (1+i c x^3\right )}{72 c}\right )-2 \left (-\frac{i b^2 x^3}{24 c^3}+\frac{b^2 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{24 c^4}\right )+\frac{5 b^2 \log \left (i+c x^3\right )}{288 c^4}-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{24 c^4}-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^4}\\ \end{align*}
Mathematica [A] time = 0.0954633, size = 121, normalized size = 0.98 \[ \frac{c x^3 \left (3 a^2 c^3 x^9-2 a b c^2 x^6+6 a b+b^2 c x^3\right )-2 b \tan ^{-1}\left (c x^3\right ) \left (a \left (3-3 c^4 x^{12}\right )+b c x^3 \left (c^2 x^6-3\right )\right )-4 b^2 \log \left (c^2 x^6+1\right )+3 b^2 \left (c^4 x^{12}-1\right ) \tan ^{-1}\left (c x^3\right )^2}{36 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 151, normalized size = 1.2 \begin{align*}{\frac{{x}^{12}{a}^{2}}{12}}+{\frac{{b}^{2}{x}^{12} \left ( \arctan \left ( c{x}^{3} \right ) \right ) ^{2}}{12}}-{\frac{{b}^{2}\arctan \left ( c{x}^{3} \right ){x}^{9}}{18\,c}}+{\frac{{b}^{2}{x}^{3}\arctan \left ( c{x}^{3} \right ) }{6\,{c}^{3}}}-{\frac{{b}^{2} \left ( \arctan \left ( c{x}^{3} \right ) \right ) ^{2}}{12\,{c}^{4}}}+{\frac{{b}^{2}{x}^{6}}{36\,{c}^{2}}}-{\frac{{b}^{2}\ln \left ({c}^{2}{x}^{6}+1 \right ) }{9\,{c}^{4}}}+{\frac{ab{x}^{12}\arctan \left ( c{x}^{3} \right ) }{6}}-{\frac{ab{x}^{9}}{18\,c}}+{\frac{ab{x}^{3}}{6\,{c}^{3}}}-{\frac{ab\arctan \left ( c{x}^{3} \right ) }{6\,{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.92842, size = 228, normalized size = 1.84 \begin{align*} \frac{1}{12} \, b^{2} x^{12} \arctan \left (c x^{3}\right )^{2} + \frac{1}{12} \, a^{2} x^{12} + \frac{1}{18} \,{\left (3 \, x^{12} \arctan \left (c x^{3}\right ) - c{\left (\frac{c^{2} x^{9} - 3 \, x^{3}}{c^{4}} + \frac{3 \, \arctan \left (c x^{3}\right )}{c^{5}}\right )}\right )} a b - \frac{1}{36} \,{\left (2 \, c{\left (\frac{c^{2} x^{9} - 3 \, x^{3}}{c^{4}} + \frac{3 \, \arctan \left (c x^{3}\right )}{c^{5}}\right )} \arctan \left (c x^{3}\right ) - \frac{c^{2} x^{6} + 3 \, \arctan \left (c x^{3}\right )^{2} - 3 \, \log \left (18 \, c^{7} x^{6} + 18 \, c^{5}\right ) - \log \left (c^{2} x^{6} + 1\right )}{c^{4}}\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.71134, size = 281, normalized size = 2.27 \begin{align*} \frac{3 \, a^{2} c^{4} x^{12} - 2 \, a b c^{3} x^{9} + b^{2} c^{2} x^{6} + 6 \, a b c x^{3} + 3 \,{\left (b^{2} c^{4} x^{12} - b^{2}\right )} \arctan \left (c x^{3}\right )^{2} - 4 \, b^{2} \log \left (c^{2} x^{6} + 1\right ) + 2 \,{\left (3 \, a b c^{4} x^{12} - b^{2} c^{3} x^{9} + 3 \, b^{2} c x^{3} - 3 \, a b\right )} \arctan \left (c x^{3}\right )}{36 \, c^{4}} \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 [A] time = 1.14039, size = 196, normalized size = 1.58 \begin{align*} \frac{3 \, a^{2} c x^{12} + 2 \,{\left (3 \, c x^{12} \arctan \left (c x^{3}\right ) - \frac{3 \, \arctan \left (c x^{3}\right )}{c^{3}} - \frac{c^{9} x^{9} - 3 \, c^{7} x^{3}}{c^{9}}\right )} a b +{\left (3 \, c x^{12} \arctan \left (c x^{3}\right )^{2} - \frac{2 \, c^{3} x^{9} \arctan \left (c x^{3}\right ) - c^{2} x^{6} - 6 \, c x^{3} \arctan \left (c x^{3}\right ) + 3 \, \arctan \left (c x^{3}\right )^{2} + 4 \, \log \left (c^{2} x^{6} + 1\right )}{c^{3}}\right )} b^{2}}{36 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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