Integrand size = 20, antiderivative size = 200 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^3 \left (1+a^2 x^2\right )}{35 a^2}+\frac {3 c^3 \left (1+a^2 x^2\right )^2}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3}{168 a^2}-\frac {4 c^3 x \arctan (a x)}{35 a}-\frac {2 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)}{70 a}-\frac {c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)}{28 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^2}{8 a^2}+\frac {2 c^3 \log \left (1+a^2 x^2\right )}{35 a^2} \]
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Time = 0.09 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5050, 4998, 4930, 266} \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^2}{8 a^2}-\frac {c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)}{28 a}-\frac {3 c^3 x \left (a^2 x^2+1\right )^2 \arctan (a x)}{70 a}-\frac {2 c^3 x \left (a^2 x^2+1\right ) \arctan (a x)}{35 a}+\frac {c^3 \left (a^2 x^2+1\right )^3}{168 a^2}+\frac {3 c^3 \left (a^2 x^2+1\right )^2}{280 a^2}+\frac {c^3 \left (a^2 x^2+1\right )}{35 a^2}+\frac {2 c^3 \log \left (a^2 x^2+1\right )}{35 a^2}-\frac {4 c^3 x \arctan (a x)}{35 a} \]
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Rule 266
Rule 4930
Rule 4998
Rule 5050
Rubi steps \begin{align*} \text {integral}& = \frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^2}{8 a^2}-\frac {\int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx}{4 a} \\ & = \frac {c^3 \left (1+a^2 x^2\right )^3}{168 a^2}-\frac {c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)}{28 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^2}{8 a^2}-\frac {(3 c) \int \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx}{14 a} \\ & = \frac {3 c^3 \left (1+a^2 x^2\right )^2}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3}{168 a^2}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)}{70 a}-\frac {c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)}{28 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^2}{8 a^2}-\frac {\left (6 c^2\right ) \int \left (c+a^2 c x^2\right ) \arctan (a x) \, dx}{35 a} \\ & = \frac {c^3 \left (1+a^2 x^2\right )}{35 a^2}+\frac {3 c^3 \left (1+a^2 x^2\right )^2}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3}{168 a^2}-\frac {2 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)}{70 a}-\frac {c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)}{28 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^2}{8 a^2}-\frac {\left (4 c^3\right ) \int \arctan (a x) \, dx}{35 a} \\ & = \frac {c^3 \left (1+a^2 x^2\right )}{35 a^2}+\frac {3 c^3 \left (1+a^2 x^2\right )^2}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3}{168 a^2}-\frac {4 c^3 x \arctan (a x)}{35 a}-\frac {2 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)}{70 a}-\frac {c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)}{28 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^2}{8 a^2}+\frac {1}{35} \left (4 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx \\ & = \frac {c^3 \left (1+a^2 x^2\right )}{35 a^2}+\frac {3 c^3 \left (1+a^2 x^2\right )^2}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3}{168 a^2}-\frac {4 c^3 x \arctan (a x)}{35 a}-\frac {2 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)}{70 a}-\frac {c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)}{28 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^2}{8 a^2}+\frac {2 c^3 \log \left (1+a^2 x^2\right )}{35 a^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.50 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^3 \left (57 a^2 x^2+24 a^4 x^4+5 a^6 x^6-6 a x \left (35+35 a^2 x^2+21 a^4 x^4+5 a^6 x^6\right ) \arctan (a x)+105 \left (1+a^2 x^2\right )^4 \arctan (a x)^2+48 \log \left (1+a^2 x^2\right )\right )}{840 a^2} \]
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Time = 1.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.84
method | result | size |
parts | \(\frac {c^{3} \arctan \left (a x \right )^{2} a^{6} x^{8}}{8}+\frac {c^{3} \arctan \left (a x \right )^{2} a^{4} x^{6}}{2}+\frac {3 c^{3} \arctan \left (a x \right )^{2} a^{2} x^{4}}{4}+\frac {c^{3} \arctan \left (a x \right )^{2} x^{2}}{2}+\frac {c^{3} \arctan \left (a x \right )^{2}}{8 a^{2}}-\frac {c^{3} \left (\frac {\arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 \arctan \left (a x \right ) a^{5} x^{5}}{5}+\arctan \left (a x \right ) x^{3} a^{3}+x \arctan \left (a x \right ) a -\frac {a^{6} x^{6}}{42}-\frac {4 a^{4} x^{4}}{35}-\frac {19 a^{2} x^{2}}{70}-\frac {8 \ln \left (a^{2} x^{2}+1\right )}{35}\right )}{4 a^{2}}\) | \(168\) |
derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}}{2}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}}{4}+\frac {a^{2} c^{3} x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c^{3} \arctan \left (a x \right )^{2}}{8}-\frac {c^{3} \left (\frac {\arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 \arctan \left (a x \right ) a^{5} x^{5}}{5}+\arctan \left (a x \right ) x^{3} a^{3}+x \arctan \left (a x \right ) a -\frac {a^{6} x^{6}}{42}-\frac {4 a^{4} x^{4}}{35}-\frac {19 a^{2} x^{2}}{70}-\frac {8 \ln \left (a^{2} x^{2}+1\right )}{35}\right )}{4}}{a^{2}}\) | \(169\) |
default | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}}{2}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}}{4}+\frac {a^{2} c^{3} x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c^{3} \arctan \left (a x \right )^{2}}{8}-\frac {c^{3} \left (\frac {\arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 \arctan \left (a x \right ) a^{5} x^{5}}{5}+\arctan \left (a x \right ) x^{3} a^{3}+x \arctan \left (a x \right ) a -\frac {a^{6} x^{6}}{42}-\frac {4 a^{4} x^{4}}{35}-\frac {19 a^{2} x^{2}}{70}-\frac {8 \ln \left (a^{2} x^{2}+1\right )}{35}\right )}{4}}{a^{2}}\) | \(169\) |
parallelrisch | \(\frac {105 c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}-30 c^{3} \arctan \left (a x \right ) a^{7} x^{7}+420 a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}+5 a^{6} c^{3} x^{6}-126 a^{5} c^{3} x^{5} \arctan \left (a x \right )+630 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}+24 a^{4} c^{3} x^{4}-210 a^{3} c^{3} x^{3} \arctan \left (a x \right )+420 a^{2} c^{3} x^{2} \arctan \left (a x \right )^{2}+57 a^{2} c^{3} x^{2}-210 a \,c^{3} x \arctan \left (a x \right )+105 c^{3} \arctan \left (a x \right )^{2}+48 c^{3} \ln \left (a^{2} x^{2}+1\right )}{840 a^{2}}\) | \(190\) |
risch | \(-\frac {c^{3} \left (a^{2} x^{2}+1\right )^{4} \ln \left (i a x +1\right )^{2}}{32 a^{2}}+\frac {c^{3} \left (35 a^{8} x^{8} \ln \left (-i a x +1\right )+10 i a^{7} x^{7}+140 a^{6} x^{6} \ln \left (-i a x +1\right )+42 i a^{5} x^{5}+210 x^{4} \ln \left (-i a x +1\right ) a^{4}+70 i a^{3} x^{3}+140 a^{2} x^{2} \ln \left (-i a x +1\right )+70 i a x +35 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{560 a^{2}}-\frac {c^{3} a^{6} x^{8} \ln \left (-i a x +1\right )^{2}}{32}-\frac {i c^{3} a^{5} x^{7} \ln \left (-i a x +1\right )}{56}-\frac {c^{3} a^{4} x^{6} \ln \left (-i a x +1\right )^{2}}{8}-\frac {3 i c^{3} a^{3} x^{5} \ln \left (-i a x +1\right )}{40}+\frac {c^{3} a^{4} x^{6}}{168}-\frac {3 c^{3} a^{2} x^{4} \ln \left (-i a x +1\right )^{2}}{16}-\frac {i c^{3} a \,x^{3} \ln \left (-i a x +1\right )}{8}+\frac {c^{3} a^{2} x^{4}}{35}-\frac {c^{3} x^{2} \ln \left (-i a x +1\right )^{2}}{8}-\frac {i c^{3} x \ln \left (-i a x +1\right )}{8 a}+\frac {19 c^{3} x^{2}}{280}-\frac {c^{3} \ln \left (-i a x +1\right )^{2}}{32 a^{2}}+\frac {2 c^{3} \ln \left (-a^{2} x^{2}-1\right )}{35 a^{2}}\) | \(378\) |
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Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.78 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {5 \, a^{6} c^{3} x^{6} + 24 \, a^{4} c^{3} x^{4} + 57 \, a^{2} c^{3} x^{2} + 48 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) + 105 \, {\left (a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{2} - 6 \, {\left (5 \, a^{7} c^{3} x^{7} + 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} + 35 \, a c^{3} x\right )} \arctan \left (a x\right )}{840 \, a^{2}} \]
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Time = 0.58 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.04 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\begin {cases} \frac {a^{6} c^{3} x^{8} \operatorname {atan}^{2}{\left (a x \right )}}{8} - \frac {a^{5} c^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{28} + \frac {a^{4} c^{3} x^{6} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {a^{4} c^{3} x^{6}}{168} - \frac {3 a^{3} c^{3} x^{5} \operatorname {atan}{\left (a x \right )}}{20} + \frac {3 a^{2} c^{3} x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{4} + \frac {a^{2} c^{3} x^{4}}{35} - \frac {a c^{3} x^{3} \operatorname {atan}{\left (a x \right )}}{4} + \frac {c^{3} x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {19 c^{3} x^{2}}{280} - \frac {c^{3} x \operatorname {atan}{\left (a x \right )}}{4 a} + \frac {2 c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{35 a^{2}} + \frac {c^{3} \operatorname {atan}^{2}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.66 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {{\left (a^{2} c x^{2} + c\right )}^{4} \arctan \left (a x\right )^{2}}{8 \, a^{2} c} + \frac {{\left (5 \, a^{4} c^{4} x^{6} + 24 \, a^{2} c^{4} x^{4} + 57 \, c^{4} x^{2} + \frac {48 \, c^{4} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a - 6 \, {\left (5 \, a^{6} c^{4} x^{7} + 21 \, a^{4} c^{4} x^{5} + 35 \, a^{2} c^{4} x^{3} + 35 \, c^{4} x\right )} \arctan \left (a x\right )}{840 \, a c} \]
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\[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )^{2} \,d x } \]
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Time = 0.52 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.78 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx={\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {c^3}{8\,a^2}+\frac {c^3\,x^2}{2}+\frac {3\,a^2\,c^3\,x^4}{4}+\frac {a^4\,c^3\,x^6}{2}+\frac {a^6\,c^3\,x^8}{8}\right )+\frac {19\,c^3\,x^2}{280}-a^2\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {c^3\,x}{4\,a^3}+\frac {3\,a\,c^3\,x^5}{20}+\frac {c^3\,x^3}{4\,a}+\frac {a^3\,c^3\,x^7}{28}\right )+\frac {2\,c^3\,\ln \left (a^2\,x^2+1\right )}{35\,a^2}+\frac {a^2\,c^3\,x^4}{35}+\frac {a^4\,c^3\,x^6}{168} \]
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