Optimal. Leaf size=321 \[ \frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac {4 c^2 \text {Li}_3\left (1-\frac {2}{i a x+1}\right )}{35 a^3}-\frac {8 i c^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)}{35 a^3}-\frac {1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac {8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac {c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac {8 c^2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{35 a^3}+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac {c^2 x \tan ^{-1}(a x)}{70 a^2}+\frac {c^2 \log \left (a^2 x^2+1\right )}{30 a^3}-\frac {1}{140} a c^2 x^4-\frac {27}{140} a c^2 x^4 \tan ^{-1}(a x)^2+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {17}{210} c^2 x^3 \tan ^{-1}(a x)-\frac {11 c^2 x^2}{420 a}-\frac {4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a} \]
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Rubi [A] time = 1.80, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 73, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {4948, 4852, 4916, 4846, 260, 4884, 4920, 4854, 4994, 6610, 266, 43} \[ -\frac {4 c^2 \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{35 a^3}-\frac {8 i c^2 \tan ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^3}+\frac {c^2 \log \left (a^2 x^2+1\right )}{30 a^3}+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac {1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac {c^2 x \tan ^{-1}(a x)}{70 a^2}-\frac {8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac {c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac {8 c^2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{35 a^3}-\frac {1}{140} a c^2 x^4-\frac {11 c^2 x^2}{420 a}-\frac {27}{140} a c^2 x^4 \tan ^{-1}(a x)^2+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {17}{210} c^2 x^3 \tan ^{-1}(a x)-\frac {4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 4846
Rule 4852
Rule 4854
Rule 4884
Rule 4916
Rule 4920
Rule 4948
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx &=\int \left (c^2 x^2 \tan ^{-1}(a x)^3+2 a^2 c^2 x^4 \tan ^{-1}(a x)^3+a^4 c^2 x^6 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^2 \int x^2 \tan ^{-1}(a x)^3 \, dx+\left (2 a^2 c^2\right ) \int x^4 \tan ^{-1}(a x)^3 \, dx+\left (a^4 c^2\right ) \int x^6 \tan ^{-1}(a x)^3 \, dx\\ &=\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\left (a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{5} \left (6 a^3 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{7} \left (3 a^5 c^2\right ) \int \frac {x^7 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac {c^2 \int x \tan ^{-1}(a x)^2 \, dx}{a}+\frac {c^2 \int \frac {x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a}-\frac {1}{5} \left (6 a c^2\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx+\frac {1}{5} \left (6 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{7} \left (3 a^3 c^2\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx+\frac {1}{7} \left (3 a^3 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 x^2 \tan ^{-1}(a x)^2}{2 a}-\frac {3}{10} a c^2 x^4 \tan ^{-1}(a x)^2-\frac {1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac {i c^2 \tan ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3+c^2 \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {c^2 \int \frac {\tan ^{-1}(a x)^2}{i-a x} \, dx}{a^2}+\frac {\left (6 c^2\right ) \int x \tan ^{-1}(a x)^2 \, dx}{5 a}-\frac {\left (6 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{7} \left (3 a c^2\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx-\frac {1}{7} \left (3 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac {1}{5} \left (3 a^2 c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{7} \left (a^4 c^2\right ) \int \frac {x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {c^2 x^2 \tan ^{-1}(a x)^2}{10 a}-\frac {27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac {1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2+\frac {i c^2 \tan ^{-1}(a x)^3}{15 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac {c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3}+\frac {1}{5} \left (3 c^2\right ) \int x^2 \tan ^{-1}(a x) \, dx-\frac {1}{5} \left (3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (6 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {c^2 \int \tan ^{-1}(a x) \, dx}{a^2}-\frac {c^2 \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}+\frac {\left (6 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{i-a x} \, dx}{5 a^2}+\frac {\left (2 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}-\frac {\left (3 c^2\right ) \int x \tan ^{-1}(a x)^2 \, dx}{7 a}+\frac {\left (3 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{7 a}+\frac {1}{7} \left (a^2 c^2\right ) \int x^4 \tan ^{-1}(a x) \, dx-\frac {1}{7} \left (a^2 c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{14} \left (3 a^2 c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {c^2 x \tan ^{-1}(a x)}{a^2}+\frac {1}{5} c^2 x^3 \tan ^{-1}(a x)+\frac {1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{2 a^3}-\frac {4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac {1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac {8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3+\frac {c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^3}-\frac {i c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a^3}-\frac {1}{7} c^2 \int x^2 \tan ^{-1}(a x) \, dx+\frac {1}{7} c^2 \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{14} \left (3 c^2\right ) \int x^2 \tan ^{-1}(a x) \, dx+\frac {1}{14} \left (3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{7} \left (3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {\left (i c^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}-\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{i-a x} \, dx}{7 a^2}-\frac {\left (3 c^2\right ) \int \tan ^{-1}(a x) \, dx}{5 a^2}+\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (6 c^2\right ) \int \tan ^{-1}(a x) \, dx}{5 a^2}+\frac {\left (6 c^2\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (12 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}-\frac {c^2 \int \frac {x}{1+a^2 x^2} \, dx}{a}-\frac {1}{5} \left (a c^2\right ) \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {1}{35} \left (a^3 c^2\right ) \int \frac {x^5}{1+a^2 x^2} \, dx\\ &=-\frac {4 c^2 x \tan ^{-1}(a x)}{5 a^2}+\frac {17}{210} c^2 x^3 \tan ^{-1}(a x)+\frac {1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac {2 c^2 \tan ^{-1}(a x)^2}{5 a^3}-\frac {4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac {1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac {8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac {8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}-\frac {c^2 \log \left (1+a^2 x^2\right )}{2 a^3}+\frac {i c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{2 a^3}-\frac {\left (6 i c^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}+\frac {c^2 \int \tan ^{-1}(a x) \, dx}{7 a^2}-\frac {c^2 \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (3 c^2\right ) \int \tan ^{-1}(a x) \, dx}{14 a^2}-\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{14 a^2}+\frac {\left (3 c^2\right ) \int \tan ^{-1}(a x) \, dx}{7 a^2}-\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (6 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (3 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx}{5 a}+\frac {\left (6 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{21} \left (a c^2\right ) \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{14} \left (a c^2\right ) \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {1}{10} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{70} \left (a^3 c^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {c^2 x \tan ^{-1}(a x)}{70 a^2}+\frac {17}{210} c^2 x^3 \tan ^{-1}(a x)+\frac {1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac {c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac {4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac {1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac {8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac {8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}+\frac {2 c^2 \log \left (1+a^2 x^2\right )}{5 a^3}-\frac {8 i c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{35 a^3}+\frac {c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{10 a^3}+\frac {\left (3 i c^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}-\frac {c^2 \int \frac {x}{1+a^2 x^2} \, dx}{7 a}-\frac {\left (3 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx}{14 a}-\frac {\left (3 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx}{7 a}+\frac {1}{42} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{28} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (a c^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{70} \left (a^3 c^2\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^4}+\frac {x}{a^2}+\frac {1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 c^2 x^2}{35 a}-\frac {1}{140} a c^2 x^4-\frac {c^2 x \tan ^{-1}(a x)}{70 a^2}+\frac {17}{210} c^2 x^3 \tan ^{-1}(a x)+\frac {1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac {c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac {4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac {1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac {8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac {8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}+\frac {13 c^2 \log \left (1+a^2 x^2\right )}{140 a^3}-\frac {8 i c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{35 a^3}-\frac {4 c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{35 a^3}+\frac {1}{42} \left (a c^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {1}{28} \left (a c^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {11 c^2 x^2}{420 a}-\frac {1}{140} a c^2 x^4-\frac {c^2 x \tan ^{-1}(a x)}{70 a^2}+\frac {17}{210} c^2 x^3 \tan ^{-1}(a x)+\frac {1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac {c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac {4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac {1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac {8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac {8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}+\frac {c^2 \log \left (1+a^2 x^2\right )}{30 a^3}-\frac {8 i c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{35 a^3}-\frac {4 c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{35 a^3}\\ \end {align*}
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Mathematica [A] time = 1.19, size = 233, normalized size = 0.73 \[ \frac {c^2 \left (60 a^7 x^7 \tan ^{-1}(a x)^3-30 a^6 x^6 \tan ^{-1}(a x)^2+168 a^5 x^5 \tan ^{-1}(a x)^3+12 a^5 x^5 \tan ^{-1}(a x)-3 a^4 x^4-81 a^4 x^4 \tan ^{-1}(a x)^2+140 a^3 x^3 \tan ^{-1}(a x)^3+34 a^3 x^3 \tan ^{-1}(a x)-11 a^2 x^2+14 \log \left (a^2 x^2+1\right )-48 a^2 x^2 \tan ^{-1}(a x)^2+96 i \tan ^{-1}(a x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )-48 \text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )-6 a x \tan ^{-1}(a x)+32 i \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2-96 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-8\right )}{420 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{3}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 16.69, size = 1121, normalized size = 3.49 \[ \text {result too large to display} \]
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 \[ \frac {1}{840} \, {\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \arctan \left (a x\right )^{3} - \frac {1}{1120} \, {\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac {980 \, {\left (a^{6} c^{2} x^{8} + 3 \, a^{4} c^{2} x^{6} + 3 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{3} - 4 \, {\left (15 \, a^{5} c^{2} x^{7} + 42 \, a^{3} c^{2} x^{5} + 35 \, a c^{2} x^{3}\right )} \arctan \left (a x\right )^{2} + 4 \, {\left (15 \, a^{6} c^{2} x^{8} + 42 \, a^{4} c^{2} x^{6} + 35 \, a^{2} c^{2} x^{4}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) + {\left (15 \, a^{5} c^{2} x^{7} + 42 \, a^{3} c^{2} x^{5} + 35 \, a c^{2} x^{3} + 105 \, {\left (a^{6} c^{2} x^{8} + 3 \, a^{4} c^{2} x^{6} + 3 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{1120 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
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 \[ \int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]
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 \[ c^{2} \left (\int x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{6} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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