Integrand size = 22, antiderivative size = 240 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=-\frac {107 c^3 x^2}{12600 a^2}+\frac {53 c^3 x^4}{6300}+\frac {71 a^2 c^3 x^6}{7560}+\frac {1}{360} a^4 c^3 x^8+\frac {c^3 x \arctan (a x)}{20 a^3}-\frac {c^3 x^3 \arctan (a x)}{60 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {26 c^3 \log \left (1+a^2 x^2\right )}{1575 a^4} \]
[Out]
Time = 0.87 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 72, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5068, 4946, 5036, 272, 45, 4930, 266, 5004} \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{40 a^4}+\frac {1}{360} a^4 c^3 x^8-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)+\frac {c^3 x \arctan (a x)}{20 a^3}+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {71 a^2 c^3 x^6}{7560}-\frac {107 c^3 x^2}{12600 a^2}-\frac {26 c^3 \log \left (a^2 x^2+1\right )}{1575 a^4}-\frac {9}{100} a c^3 x^5 \arctan (a x)+\frac {1}{4} c^3 x^4 \arctan (a x)^2-\frac {c^3 x^3 \arctan (a x)}{60 a}+\frac {53 c^3 x^4}{6300} \]
[In]
[Out]
Rule 45
Rule 266
Rule 272
Rule 4930
Rule 4946
Rule 5004
Rule 5036
Rule 5068
Rubi steps \begin{align*} \text {integral}& = \int \left (c^3 x^3 \arctan (a x)^2+3 a^2 c^3 x^5 \arctan (a x)^2+3 a^4 c^3 x^7 \arctan (a x)^2+a^6 c^3 x^9 \arctan (a x)^2\right ) \, dx \\ & = c^3 \int x^3 \arctan (a x)^2 \, dx+\left (3 a^2 c^3\right ) \int x^5 \arctan (a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^7 \arctan (a x)^2 \, dx+\left (a^6 c^3\right ) \int x^9 \arctan (a x)^2 \, dx \\ & = \frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {1}{2} \left (a c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx-\left (a^3 c^3\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^5 c^3\right ) \int \frac {x^8 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (a^7 c^3\right ) \int \frac {x^{10} \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {c^3 \int x^2 \arctan (a x) \, dx}{2 a}+\frac {c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx}{2 a}-\left (a c^3\right ) \int x^4 \arctan (a x) \, dx+\left (a c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^3 c^3\right ) \int x^6 \arctan (a x) \, dx+\frac {1}{4} \left (3 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (a^5 c^3\right ) \int x^8 \arctan (a x) \, dx+\frac {1}{5} \left (a^5 c^3\right ) \int \frac {x^8 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {c^3 x^3 \arctan (a x)}{6 a}-\frac {1}{5} a c^3 x^5 \arctan (a x)-\frac {3}{28} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2+\frac {1}{6} c^3 \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {c^3 \int \arctan (a x) \, dx}{2 a^3}-\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{2 a^3}+\frac {c^3 \int x^2 \arctan (a x) \, dx}{a}-\frac {c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx}{a}+\frac {1}{4} \left (3 a c^3\right ) \int x^4 \arctan (a x) \, dx-\frac {1}{4} \left (3 a c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^2 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^3 c^3\right ) \int x^6 \arctan (a x) \, dx-\frac {1}{5} \left (a^3 c^3\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{28} \left (3 a^4 c^3\right ) \int \frac {x^7}{1+a^2 x^2} \, dx+\frac {1}{45} \left (a^6 c^3\right ) \int \frac {x^9}{1+a^2 x^2} \, dx \\ & = \frac {c^3 x \arctan (a x)}{2 a^3}+\frac {c^3 x^3 \arctan (a x)}{6 a}-\frac {1}{20} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{4 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2+\frac {1}{12} c^3 \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{3} c^3 \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {c^3 \int \arctan (a x) \, dx}{a^3}+\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{a^3}-\frac {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{2 a^2}-\frac {\left (3 c^3\right ) \int x^2 \arctan (a x) \, dx}{4 a}+\frac {\left (3 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx}{4 a}-\frac {1}{5} \left (a c^3\right ) \int x^4 \arctan (a x) \, dx+\frac {1}{5} \left (a c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{10} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{20} \left (3 a^2 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx-\frac {1}{35} \left (a^4 c^3\right ) \int \frac {x^7}{1+a^2 x^2} \, dx+\frac {1}{56} \left (3 a^4 c^3\right ) \text {Subst}\left (\int \frac {x^3}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{90} \left (a^6 c^3\right ) \text {Subst}\left (\int \frac {x^4}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {c^3 x \arctan (a x)}{2 a^3}-\frac {c^3 x^3 \arctan (a x)}{12 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)+\frac {c^3 \arctan (a x)^2}{4 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {c^3 \log \left (1+a^2 x^2\right )}{4 a^4}+\frac {1}{12} c^3 \text {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}{6} c^3 \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{4} c^3 \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {\left (3 c^3\right ) \int \arctan (a x) \, dx}{4 a^3}-\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{4 a^3}+\frac {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{a^2}+\frac {c^3 \int x^2 \arctan (a x) \, dx}{5 a}-\frac {c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{25} \left (a^2 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx-\frac {1}{40} \left (3 a^2 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{10} \left (a^2 c^3\right ) \text {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 {1}{70} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {x^3}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{56} \left (3 a^4 c^3\right ) \text {Subst}\left (\int \left (\frac {1}{a^6}-\frac {x}{a^4}+\frac {x^2}{a^2}-\frac {1}{a^6 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {1}{90} \left (a^6 c^3\right ) \text {Subst}\left (\int \left (-\frac {1}{a^8}+\frac {x}{a^6}-\frac {x^2}{a^4}+\frac {x^3}{a^2}+\frac {1}{a^8 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {13 c^3 x^2}{504 a^2}+\frac {29 c^3 x^4}{1008}+\frac {107 a^2 c^3 x^6}{7560}+\frac {1}{360} a^4 c^3 x^8+\frac {c^3 x \arctan (a x)}{4 a^3}-\frac {c^3 x^3 \arctan (a x)}{60 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{8 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2+\frac {113 c^3 \log \left (1+a^2 x^2\right )}{504 a^4}-\frac {1}{15} c^3 \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{8} c^3 \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{6} c^3 \text {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 {c^3 \int \arctan (a x) \, dx}{5 a^3}+\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{5 a^3}-\frac {\left (3 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx}{4 a^2}+\frac {1}{50} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{40} \left (3 a^2 c^3\right ) \text {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 {1}{70} \left (a^4 c^3\right ) \text {Subst}\left (\int \left (\frac {1}{a^6}-\frac {x}{a^4}+\frac {x^2}{a^2}-\frac {1}{a^6 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = -\frac {101 c^3 x^2}{1260 a^2}-\frac {c^3 x^4}{630}+\frac {71 a^2 c^3 x^6}{7560}+\frac {1}{360} a^4 c^3 x^8+\frac {c^3 x \arctan (a x)}{20 a^3}-\frac {c^3 x^3 \arctan (a x)}{60 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {113 c^3 \log \left (1+a^2 x^2\right )}{2520 a^4}-\frac {1}{30} c^3 \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{8} c^3 \text {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 {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{5 a^2}+\frac {1}{50} \left (a^2 c^3\right ) \text {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 {313 c^3 x^2}{12600 a^2}+\frac {53 c^3 x^4}{6300}+\frac {71 a^2 c^3 x^6}{7560}+\frac {1}{360} a^4 c^3 x^8+\frac {c^3 x \arctan (a x)}{20 a^3}-\frac {c^3 x^3 \arctan (a x)}{60 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {157 c^3 \log \left (1+a^2 x^2\right )}{3150 a^4}-\frac {1}{30} c^3 \text {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 {107 c^3 x^2}{12600 a^2}+\frac {53 c^3 x^4}{6300}+\frac {71 a^2 c^3 x^6}{7560}+\frac {1}{360} a^4 c^3 x^8+\frac {c^3 x \arctan (a x)}{20 a^3}-\frac {c^3 x^3 \arctan (a x)}{60 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {26 c^3 \log \left (1+a^2 x^2\right )}{1575 a^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.52 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^3 \left (-321 a^2 x^2+318 a^4 x^4+355 a^6 x^6+105 a^8 x^8-6 a x \left (-315+105 a^2 x^2+567 a^4 x^4+495 a^6 x^6+140 a^8 x^8\right ) \arctan (a x)+945 \left (1+a^2 x^2\right )^4 \left (-1+4 a^2 x^2\right ) \arctan (a x)^2-624 \log \left (1+a^2 x^2\right )\right )}{37800 a^4} \]
[In]
[Out]
Time = 1.22 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{10} x^{10}}{10}+\frac {3 c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}}{2}+\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}}{4}-\frac {c^{3} \left (\frac {4 \arctan \left (a x \right ) a^{9} x^{9}}{9}+\frac {11 \arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {9 \arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}-x \arctan \left (a x \right ) a +\frac {\arctan \left (a x \right )^{2}}{2}-\frac {a^{8} x^{8}}{18}-\frac {71 a^{6} x^{6}}{378}-\frac {53 a^{4} x^{4}}{315}+\frac {107 a^{2} x^{2}}{630}+\frac {104 \ln \left (a^{2} x^{2}+1\right )}{315}\right )}{20}}{a^{4}}\) | \(188\) |
default | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{10} x^{10}}{10}+\frac {3 c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}}{2}+\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}}{4}-\frac {c^{3} \left (\frac {4 \arctan \left (a x \right ) a^{9} x^{9}}{9}+\frac {11 \arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {9 \arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}-x \arctan \left (a x \right ) a +\frac {\arctan \left (a x \right )^{2}}{2}-\frac {a^{8} x^{8}}{18}-\frac {71 a^{6} x^{6}}{378}-\frac {53 a^{4} x^{4}}{315}+\frac {107 a^{2} x^{2}}{630}+\frac {104 \ln \left (a^{2} x^{2}+1\right )}{315}\right )}{20}}{a^{4}}\) | \(188\) |
parts | \(\frac {a^{6} c^{3} x^{10} \arctan \left (a x \right )^{2}}{10}+\frac {3 a^{4} c^{3} x^{8} \arctan \left (a x \right )^{2}}{8}+\frac {a^{2} c^{3} x^{6} \arctan \left (a x \right )^{2}}{2}+\frac {c^{3} x^{4} \arctan \left (a x \right )^{2}}{4}-\frac {c^{3} \left (\frac {4 a^{5} \arctan \left (a x \right ) x^{9}}{9}+\frac {11 a^{3} \arctan \left (a x \right ) x^{7}}{7}+\frac {9 a \arctan \left (a x \right ) x^{5}}{5}+\frac {\arctan \left (a x \right ) x^{3}}{3 a}-\frac {\arctan \left (a x \right ) x}{a^{3}}+\frac {\arctan \left (a x \right )^{2}}{a^{4}}-\frac {\frac {35 a^{8} x^{8}}{2}+\frac {355 a^{6} x^{6}}{6}+53 a^{4} x^{4}-\frac {107 a^{2} x^{2}}{2}-104 \ln \left (a^{2} x^{2}+1\right )+\frac {315 \arctan \left (a x \right )^{2}}{2}}{315 a^{4}}\right )}{20}\) | \(197\) |
parallelrisch | \(-\frac {-3780 c^{3} \arctan \left (a x \right )^{2} a^{10} x^{10}+840 c^{3} \arctan \left (a x \right ) a^{9} x^{9}-14175 c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}-105 a^{8} c^{3} x^{8}+2970 c^{3} \arctan \left (a x \right ) a^{7} x^{7}-18900 a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}-355 a^{6} c^{3} x^{6}+3402 a^{5} c^{3} x^{5} \arctan \left (a x \right )-9450 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}-318 a^{4} c^{3} x^{4}+630 a^{3} c^{3} x^{3} \arctan \left (a x \right )+321 a^{2} c^{3} x^{2}-1890 a \,c^{3} x \arctan \left (a x \right )+945 c^{3} \arctan \left (a x \right )^{2}+624 c^{3} \ln \left (a^{2} x^{2}+1\right )-321 c^{3}}{37800 a^{4}}\) | \(221\) |
risch | \(-\frac {c^{3} \left (4 a^{10} x^{10}+15 a^{8} x^{8}+20 a^{6} x^{6}+10 a^{4} x^{4}-1\right ) \ln \left (i a x +1\right )^{2}}{160 a^{4}}+\frac {c^{3} \left (1260 a^{10} x^{10} \ln \left (-i a x +1\right )+280 i a^{9} x^{9}+4725 a^{8} x^{8} \ln \left (-i a x +1\right )+990 i a^{7} x^{7}+6300 a^{6} x^{6} \ln \left (-i a x +1\right )+1134 i a^{5} x^{5}+3150 x^{4} \ln \left (-i a x +1\right ) a^{4}+210 i a^{3} x^{3}-630 i a x -315 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{25200 a^{4}}-\frac {c^{3} a^{6} x^{10} \ln \left (-i a x +1\right )^{2}}{40}-\frac {11 i c^{3} a^{3} x^{7} \ln \left (-i a x +1\right )}{280}-\frac {3 c^{3} a^{4} x^{8} \ln \left (-i a x +1\right )^{2}}{32}-\frac {i c^{3} a^{5} x^{9} \ln \left (-i a x +1\right )}{90}+\frac {a^{4} c^{3} x^{8}}{360}-\frac {c^{3} a^{2} x^{6} \ln \left (-i a x +1\right )^{2}}{8}-\frac {9 i c^{3} a \,x^{5} \ln \left (-i a x +1\right )}{200}+\frac {71 a^{2} c^{3} x^{6}}{7560}-\frac {c^{3} x^{4} \ln \left (-i a x +1\right )^{2}}{16}-\frac {i c^{3} x^{3} \ln \left (-i a x +1\right )}{120 a}+\frac {53 c^{3} x^{4}}{6300}+\frac {i c^{3} x \ln \left (-i a x +1\right )}{40 a^{3}}-\frac {107 c^{3} x^{2}}{12600 a^{2}}+\frac {c^{3} \ln \left (-i a x +1\right )^{2}}{160 a^{4}}-\frac {26 c^{3} \ln \left (-a^{2} x^{2}-1\right )}{1575 a^{4}}\) | \(441\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.75 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {105 \, a^{8} c^{3} x^{8} + 355 \, a^{6} c^{3} x^{6} + 318 \, a^{4} c^{3} x^{4} - 321 \, a^{2} c^{3} x^{2} - 624 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) + 945 \, {\left (4 \, a^{10} c^{3} x^{10} + 15 \, a^{8} c^{3} x^{8} + 20 \, a^{6} c^{3} x^{6} + 10 \, a^{4} c^{3} x^{4} - c^{3}\right )} \arctan \left (a x\right )^{2} - 6 \, {\left (140 \, a^{9} c^{3} x^{9} + 495 \, a^{7} c^{3} x^{7} + 567 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3} - 315 \, a c^{3} x\right )} \arctan \left (a x\right )}{37800 \, a^{4}} \]
[In]
[Out]
Time = 0.82 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\begin {cases} \frac {a^{6} c^{3} x^{10} \operatorname {atan}^{2}{\left (a x \right )}}{10} - \frac {a^{5} c^{3} x^{9} \operatorname {atan}{\left (a x \right )}}{45} + \frac {3 a^{4} c^{3} x^{8} \operatorname {atan}^{2}{\left (a x \right )}}{8} + \frac {a^{4} c^{3} x^{8}}{360} - \frac {11 a^{3} c^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{140} + \frac {a^{2} c^{3} x^{6} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {71 a^{2} c^{3} x^{6}}{7560} - \frac {9 a c^{3} x^{5} \operatorname {atan}{\left (a x \right )}}{100} + \frac {c^{3} x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{4} + \frac {53 c^{3} x^{4}}{6300} - \frac {c^{3} x^{3} \operatorname {atan}{\left (a x \right )}}{60 a} - \frac {107 c^{3} x^{2}}{12600 a^{2}} + \frac {c^{3} x \operatorname {atan}{\left (a x \right )}}{20 a^{3}} - \frac {26 c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{1575 a^{4}} - \frac {c^{3} \operatorname {atan}^{2}{\left (a x \right )}}{40 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.84 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=-\frac {1}{6300} \, a {\left (\frac {315 \, c^{3} \arctan \left (a x\right )}{a^{5}} + \frac {140 \, a^{8} c^{3} x^{9} + 495 \, a^{6} c^{3} x^{7} + 567 \, a^{4} c^{3} x^{5} + 105 \, a^{2} c^{3} x^{3} - 315 \, c^{3} x}{a^{4}}\right )} \arctan \left (a x\right ) + \frac {1}{40} \, {\left (4 \, a^{6} c^{3} x^{10} + 15 \, a^{4} c^{3} x^{8} + 20 \, a^{2} c^{3} x^{6} + 10 \, c^{3} x^{4}\right )} \arctan \left (a x\right )^{2} + \frac {105 \, a^{8} c^{3} x^{8} + 355 \, a^{6} c^{3} x^{6} + 318 \, a^{4} c^{3} x^{4} - 321 \, a^{2} c^{3} x^{2} + 945 \, c^{3} \arctan \left (a x\right )^{2} - 624 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{37800 \, a^{4}} \]
[In]
[Out]
\[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{2} \,d x } \]
[In]
[Out]
Time = 0.56 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.74 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx={\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {c^3\,x^4}{4}-\frac {c^3}{40\,a^4}+\frac {a^2\,c^3\,x^6}{2}+\frac {3\,a^4\,c^3\,x^8}{8}+\frac {a^6\,c^3\,x^{10}}{10}\right )+\frac {53\,c^3\,x^4}{6300}-\frac {26\,c^3\,\ln \left (a^2\,x^2+1\right )}{1575\,a^4}-\frac {107\,c^3\,x^2}{12600\,a^2}+\frac {71\,a^2\,c^3\,x^6}{7560}+\frac {a^4\,c^3\,x^8}{360}-a^2\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {11\,a\,c^3\,x^7}{140}-\frac {c^3\,x}{20\,a^5}+\frac {9\,c^3\,x^5}{100\,a}+\frac {c^3\,x^3}{60\,a^3}+\frac {a^3\,c^3\,x^9}{45}\right ) \]
[In]
[Out]