Integrand size = 22, antiderivative size = 135 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {3}{8 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^2}{8 a^3 c^2}-\frac {3 \arctan (a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a^3 c^2} \]
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Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5056, 5050, 5012, 267} \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\arctan (a x)^4}{8 a^3 c^2}+\frac {3 \arctan (a x)^2}{8 a^3 c^2}-\frac {x \arctan (a x)^3}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {3 x \arctan (a x)}{4 a^2 c^2 \left (a^2 x^2+1\right )}-\frac {3 \arctan (a x)^2}{4 a^3 c^2 \left (a^2 x^2+1\right )}+\frac {3}{8 a^3 c^2 \left (a^2 x^2+1\right )} \]
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Rule 267
Rule 5012
Rule 5050
Rule 5056
Rubi steps \begin{align*} \text {integral}& = -\frac {x \arctan (a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a^3 c^2}+\frac {3 \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a} \\ & = -\frac {3 \arctan (a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a^3 c^2}+\frac {3 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2} \\ & = \frac {3 x \arctan (a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^2}{8 a^3 c^2}-\frac {3 \arctan (a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a^3 c^2}-\frac {3 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a} \\ & = \frac {3}{8 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^2}{8 a^3 c^2}-\frac {3 \arctan (a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a^3 c^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.55 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {3+6 a x \arctan (a x)+3 \left (-1+a^2 x^2\right ) \arctan (a x)^2-4 a x \arctan (a x)^3+\left (1+a^2 x^2\right ) \arctan (a x)^4}{8 a^3 c^2 \left (1+a^2 x^2\right )} \]
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Time = 1.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.65
method | result | size |
parallelrisch | \(\frac {\arctan \left (a x \right )^{4} x^{2} a^{2}+3 x^{2} \arctan \left (a x \right )^{2} a^{2}-4 \arctan \left (a x \right )^{3} a x -3 a^{2} x^{2}+\arctan \left (a x \right )^{4}+6 x \arctan \left (a x \right ) a -3 \arctan \left (a x \right )^{2}}{8 c^{2} \left (a^{2} x^{2}+1\right ) a^{3}}\) | \(88\) |
derivativedivides | \(\frac {-\frac {\arctan \left (a x \right )^{3} a x}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{4}}{2 c^{2}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4}+\frac {\arctan \left (a x \right )^{2}}{2 a^{2} x^{2}+2}-\frac {\arctan \left (a x \right ) a x}{2 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )^{2}}{4}-\frac {1}{4 \left (a^{2} x^{2}+1\right )}\right )}{2 c^{2}}}{a^{3}}\) | \(114\) |
default | \(\frac {-\frac {\arctan \left (a x \right )^{3} a x}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{4}}{2 c^{2}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4}+\frac {\arctan \left (a x \right )^{2}}{2 a^{2} x^{2}+2}-\frac {\arctan \left (a x \right ) a x}{2 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )^{2}}{4}-\frac {1}{4 \left (a^{2} x^{2}+1\right )}\right )}{2 c^{2}}}{a^{3}}\) | \(114\) |
parts | \(-\frac {x \arctan \left (a x \right )^{3}}{2 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{4}}{2 a^{3} c^{2}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4 a^{3}}-\frac {-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {x \arctan \left (a x \right ) a}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )^{2}}{4}+\frac {1}{4 a^{2} x^{2}+4}}{a^{3}}\right )}{2 c^{2}}\) | \(124\) |
risch | \(\frac {\ln \left (i a x +1\right )^{4}}{128 c^{2} a^{3}}-\frac {\left (a^{2} x^{2} \ln \left (-i a x +1\right )+\ln \left (-i a x +1\right )+2 i a x \right ) \ln \left (i a x +1\right )^{3}}{32 a^{3} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {3 \left (a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+4 i a x \ln \left (-i a x +1\right )-2 a^{2} x^{2}+\ln \left (-i a x +1\right )^{2}+2\right ) \ln \left (i a x +1\right )^{2}}{64 a^{3} c^{2} \left (a x +i\right ) \left (a x -i\right )}-\frac {\left (a^{2} x^{2} \ln \left (-i a x +1\right )^{3}-6 a^{2} x^{2} \ln \left (-i a x +1\right )+\ln \left (-i a x +1\right )^{3}+6 i a x \ln \left (-i a x +1\right )^{2}+6 \ln \left (-i a x +1\right )+12 i a x \right ) \ln \left (i a x +1\right )}{32 a^{3} c^{2} \left (a x +i\right ) \left (a x -i\right )}+\frac {a^{2} x^{2} \ln \left (-i a x +1\right )^{4}-12 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+\ln \left (-i a x +1\right )^{4}+8 i a x \ln \left (-i a x +1\right )^{3}+12 \ln \left (-i a x +1\right )^{2}+48 i a x \ln \left (-i a x +1\right )+48}{128 a^{3} c^{2} \left (a x +i\right ) \left (a x -i\right )}\) | \(379\) |
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Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.56 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {4 \, a x \arctan \left (a x\right )^{3} - {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 6 \, a x \arctan \left (a x\right ) - 3 \, {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )^{2} - 3}{8 \, {\left (a^{5} c^{2} x^{2} + a^{3} c^{2}\right )}} \]
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\[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
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Time = 0.34 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.61 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {1}{2} \, {\left (\frac {x}{a^{4} c^{2} x^{2} + a^{2} c^{2}} - \frac {\arctan \left (a x\right )}{a^{3} c^{2}}\right )} \arctan \left (a x\right )^{3} - \frac {3 \, {\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 1\right )} a \arctan \left (a x\right )^{2}}{4 \, {\left (a^{6} c^{2} x^{2} + a^{4} c^{2}\right )}} - \frac {1}{8} \, {\left (\frac {{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} + 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 3\right )} a^{2}}{a^{8} c^{2} x^{2} + a^{6} c^{2}} - \frac {2 \, {\left (2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 3 \, a x + 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a \arctan \left (a x\right )}{a^{7} c^{2} x^{2} + a^{5} c^{2}}\right )} a \]
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\[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
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Time = 0.51 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.88 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {3}{2\,a^2\,\left (4\,a^3\,c^2\,x^2+4\,a\,c^2\right )}+{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {3}{8\,a^3\,c^2}-\frac {3}{4\,a^5\,c^2\,\left (\frac {1}{a^2}+x^2\right )}\right )+\frac {{\mathrm {atan}\left (a\,x\right )}^4}{8\,a^3\,c^2}+\frac {3\,x\,\mathrm {atan}\left (a\,x\right )}{4\,a^4\,c^2\,\left (\frac {1}{a^2}+x^2\right )}-\frac {x\,{\mathrm {atan}\left (a\,x\right )}^3}{2\,a^4\,c^2\,\left (\frac {1}{a^2}+x^2\right )} \]
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