Optimal. Leaf size=154 \[ -\frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x^2}\right )}{6 c^3}-\frac{i \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{6 c^3}-\frac{b \log \left (\frac{2}{1+i c x^2}\right ) \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{3 c^3}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2-\frac{b x^4 \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{6 c}+\frac{b^2 x^2}{6 c^2}-\frac{b^2 \tan ^{-1}\left (c x^2\right )}{6 c^3} \]
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Rubi [B] time = 1.36568, antiderivative size = 647, normalized size of antiderivative = 4.2, number of steps used = 53, 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} \[ \frac{i b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-i c x^2\right )\right )}{12 c^3}-\frac{i b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1+i c x^2\right )\right )}{12 c^3}-\frac{i a b x^2}{6 c^2}+\frac{1}{72} i b \left (\frac{2 i \left (1-i c x^2\right )^3}{c^3}-\frac{9 i \left (1-i c x^2\right )^2}{c^3}+\frac{18 i \left (1-i c x^2\right )}{c^3}-\frac{6 i \log \left (1-i c x^2\right )}{c^3}\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{i b \log \left (\frac{1}{2} \left (1+i c x^2\right )\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )}{12 c^3}+\frac{1}{24} x^6 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{36} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right )-\frac{1}{12} b x^6 \log \left (1+i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{24 c}+\frac{19 b^2 x^2}{72 c^2}+\frac{i b^2 \left (1-i c x^2\right )^3}{108 c^3}-\frac{i b^2 \left (1-i c x^2\right )^2}{16 c^3}-\frac{i b^2 \log ^2\left (1-i c x^2\right )}{24 c^3}-\frac{i b^2 \log ^2\left (1+i c x^2\right )}{24 c^3}+\frac{i b^2 \log \left (-c x^2+i\right )}{12 c^3}+\frac{i b^2 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{12 c^3}-\frac{i b^2 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{12 c^3}-\frac{i b^2 \log \left (c x^2+i\right )}{72 c^3}-\frac{5 i b^2 x^4}{144 c}-\frac{1}{24} b^2 x^6 \log ^2\left (1+i c x^2\right )+\frac{i b^2 x^4 \log \left (1+i c x^2\right )}{12 c}+\frac{b^2 x^6}{108} \]
Warning: Unable to verify antiderivative.
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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^5 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^5 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{2} b x^5 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 x^5 \log ^2\left (1+i c x^2\right )\right ) \, dx\\ &=\frac{1}{4} \int x^5 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \, dx+\frac{1}{2} b \int x^5 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right ) \, dx-\frac{1}{4} b^2 \int x^5 \log ^2\left (1+i c x^2\right ) \, dx\\ &=\frac{1}{8} \operatorname{Subst}\left (\int x^2 (2 a+i b \log (1-i c x))^2 \, dx,x,x^2\right )+\frac{1}{4} b \operatorname{Subst}\left (\int x^2 (-2 i a+b \log (1-i c x)) \log (1+i c x) \, dx,x,x^2\right )-\frac{1}{8} b^2 \operatorname{Subst}\left (\int x^2 \log ^2(1+i c x) \, dx,x,x^2\right )\\ &=\frac{1}{24} x^6 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{24} b^2 x^6 \log ^2\left (1+i c x^2\right )-\frac{1}{12} (i b c) \operatorname{Subst}\left (\int \frac{x^3 (-2 i a+b \log (1-i c x))}{1+i c x} \, dx,x,x^2\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{x^3 (2 a+i b \log (1-i c x))}{1-i c x} \, dx,x,x^2\right )+\frac{1}{12} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3 \log (1+i c x)}{1-i c x} \, dx,x,x^2\right )+\frac{1}{12} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3 \log (1+i c x)}{1+i c x} \, dx,x,x^2\right )\\ &=\frac{1}{24} x^6 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{24} b^2 x^6 \log ^2\left (1+i c x^2\right )-\frac{1}{12} (i b) \operatorname{Subst}\left (\int \frac{\left (-\frac{i}{c}+\frac{i x}{c}\right )^3 (2 a+i b \log (x))}{x} \, dx,x,1-i c x^2\right )-\frac{1}{12} (i b c) \operatorname{Subst}\left (\int \left (\frac{i (-2 i a+b \log (1-i c x))}{c^3}+\frac{x (-2 i a+b \log (1-i c x))}{c^2}-\frac{i x^2 (-2 i a+b \log (1-i c x))}{c}-\frac{-2 i a+b \log (1-i c x)}{c^3 (-i+c x)}\right ) \, dx,x,x^2\right )+\frac{1}{12} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{i \log (1+i c x)}{c^3}+\frac{x \log (1+i c x)}{c^2}-\frac{i x^2 \log (1+i c x)}{c}-\frac{\log (1+i c x)}{c^3 (-i+c x)}\right ) \, dx,x,x^2\right )+\frac{1}{12} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{i \log (1+i c x)}{c^3}+\frac{x \log (1+i c x)}{c^2}+\frac{i x^2 \log (1+i c x)}{c}-\frac{\log (1+i c x)}{c^3 (i+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{24} x^6 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{72} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) \left (\frac{18 i \left (1-i c x^2\right )}{c^3}-\frac{9 i \left (1-i c x^2\right )^2}{c^3}-\frac{2 \left (i+c x^2\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^2\right )}{c^3}\right )-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{24} b^2 x^6 \log ^2\left (1+i c x^2\right )-\frac{1}{12} b \operatorname{Subst}\left (\int x^2 (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )-\frac{1}{12} b^2 \operatorname{Subst}\left (\int -\frac{i \left (x \left (18-9 x+2 x^2\right )-6 \log (x)\right )}{6 c^3 x} \, dx,x,1-i c x^2\right )+\frac{(i b) \operatorname{Subst}\left (\int \frac{-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^2\right )}{12 c^2}+\frac{b \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )}{12 c^2}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{-i+c x} \, dx,x,x^2\right )}{12 c^2}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{i+c x} \, dx,x,x^2\right )}{12 c^2}-\frac{(i b) \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )}{12 c}+2 \frac{\left (i b^2\right ) \operatorname{Subst}\left (\int x \log (1+i c x) \, dx,x,x^2\right )}{12 c}\\ &=-\frac{i a b x^2}{6 c^2}+\frac{i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{24 c}+\frac{1}{36} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{1}{24} x^6 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{72} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) \left (\frac{18 i \left (1-i c x^2\right )}{c^3}-\frac{9 i \left (1-i c x^2\right )^2}{c^3}-\frac{2 \left (i+c x^2\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^2\right )}{c^3}\right )-\frac{i b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{12 c^3}-\frac{i b^2 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{12 c^3}-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{24} b^2 x^6 \log ^2\left (1+i c x^2\right )+\frac{1}{24} b^2 \operatorname{Subst}\left (\int \frac{x^2}{1-i c x} \, dx,x,x^2\right )+2 \left (\frac{i b^2 x^4 \log \left (1+i c x^2\right )}{24 c}+\frac{1}{24} b^2 \operatorname{Subst}\left (\int \frac{x^2}{1+i c x} \, dx,x,x^2\right )\right )+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{x \left (18-9 x+2 x^2\right )-6 \log (x)}{x} \, dx,x,1-i c x^2\right )}{72 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+i c x^2\right )}{12 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \log (1-i c x) \, dx,x,x^2\right )}{12 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^2\right )}{12 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (-\frac{1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^2\right )}{12 c^2}-\frac{1}{36} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3}{1-i c x} \, dx,x,x^2\right )\\ &=-\frac{i a b x^2}{6 c^2}+\frac{i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{24 c}+\frac{1}{36} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{1}{24} x^6 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{72} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) \left (\frac{18 i \left (1-i c x^2\right )}{c^3}-\frac{9 i \left (1-i c x^2\right )^2}{c^3}-\frac{2 \left (i+c x^2\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^2\right )}{c^3}\right )-\frac{i b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{12 c^3}-\frac{i b^2 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{12 c^3}-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{i b^2 \log ^2\left (1+i c x^2\right )}{24 c^3}-\frac{1}{24} b^2 x^6 \log ^2\left (1+i c x^2\right )+2 \left (\frac{i b^2 x^4 \log \left (1+i c x^2\right )}{24 c}+\frac{1}{24} b^2 \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{i x}{c}+\frac{i}{c^2 (-i+c x)}\right ) \, dx,x,x^2\right )\right )+\frac{1}{24} b^2 \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{i x}{c}-\frac{i}{c^2 (i+c x)}\right ) \, dx,x,x^2\right )+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \left (18-9 x+2 x^2-\frac{6 \log (x)}{x}\right ) \, dx,x,1-i c x^2\right )}{72 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-i c x^2\right )}{12 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+i c x^2\right )}{12 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{12 c^3}-\frac{1}{36} \left (i b^2 c\right ) \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^2\right )\\ &=-\frac{i a b x^2}{6 c^2}+\frac{13 b^2 x^2}{72 c^2}+\frac{i b^2 x^4}{144 c}+\frac{b^2 x^6}{108}-\frac{i b^2 \left (1-i c x^2\right )^2}{16 c^3}+\frac{i b^2 \left (1-i c x^2\right )^3}{108 c^3}+\frac{i b^2 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{12 c^3}+\frac{i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{24 c}+\frac{1}{36} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{1}{24} x^6 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{72} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) \left (\frac{18 i \left (1-i c x^2\right )}{c^3}-\frac{9 i \left (1-i c x^2\right )^2}{c^3}-\frac{2 \left (i+c x^2\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^2\right )}{c^3}\right )-\frac{i b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{12 c^3}-\frac{i b^2 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{12 c^3}-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{i b^2 \log ^2\left (1+i c x^2\right )}{24 c^3}-\frac{1}{24} b^2 x^6 \log ^2\left (1+i c x^2\right )+2 \left (\frac{b^2 x^2}{24 c^2}-\frac{i b^2 x^4}{48 c}+\frac{i b^2 \log \left (i-c x^2\right )}{24 c^3}+\frac{i b^2 x^4 \log \left (1+i c x^2\right )}{24 c}\right )-\frac{i b^2 \log \left (i+c x^2\right )}{72 c^3}+\frac{i b^2 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{12 c^3}-\frac{i b^2 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{12 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-i c x^2\right )}{12 c^3}\\ &=-\frac{i a b x^2}{6 c^2}+\frac{13 b^2 x^2}{72 c^2}+\frac{i b^2 x^4}{144 c}+\frac{b^2 x^6}{108}-\frac{i b^2 \left (1-i c x^2\right )^2}{16 c^3}+\frac{i b^2 \left (1-i c x^2\right )^3}{108 c^3}+\frac{i b^2 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{12 c^3}-\frac{i b^2 \log ^2\left (1-i c x^2\right )}{24 c^3}+\frac{i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{24 c}+\frac{1}{36} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{1}{24} x^6 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{72} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) \left (\frac{18 i \left (1-i c x^2\right )}{c^3}-\frac{9 i \left (1-i c x^2\right )^2}{c^3}-\frac{2 \left (i+c x^2\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^2\right )}{c^3}\right )-\frac{i b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{12 c^3}-\frac{i b^2 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{12 c^3}-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{i b^2 \log ^2\left (1+i c x^2\right )}{24 c^3}-\frac{1}{24} b^2 x^6 \log ^2\left (1+i c x^2\right )+2 \left (\frac{b^2 x^2}{24 c^2}-\frac{i b^2 x^4}{48 c}+\frac{i b^2 \log \left (i-c x^2\right )}{24 c^3}+\frac{i b^2 x^4 \log \left (1+i c x^2\right )}{24 c}\right )-\frac{i b^2 \log \left (i+c x^2\right )}{72 c^3}+\frac{i b^2 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{12 c^3}-\frac{i b^2 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{12 c^3}\\ \end{align*}
Mathematica [A] time = 0.288996, size = 141, normalized size = 0.92 \[ \frac{i b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}\left (c x^2\right )}\right )+a^2 c^3 x^6-a b c^2 x^4+a b \log \left (c^2 x^4+1\right )-b \tan ^{-1}\left (c x^2\right ) \left (-2 a c^3 x^6+b c^2 x^4+2 b \log \left (1+e^{2 i \tan ^{-1}\left (c x^2\right )}\right )+b\right )+b^2 \left (c^3 x^6+i\right ) \tan ^{-1}\left (c x^2\right )^2+b^2 c x^2}{6 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ( a+b\arctan \left ( c{x}^{2} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (b^{2} x^{5} \arctan \left (c x^{2}\right )^{2} + 2 \, a b x^{5} \arctan \left (c x^{2}\right ) + a^{2} x^{5}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \left (a + b \operatorname{atan}{\left (c x^{2} \right )}\right )^{2}\, dx \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{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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