Integrand size = 22, antiderivative size = 181 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{64 a^3 c^3}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3} \]
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Time = 0.20 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5084, 5012, 5050, 205, 211, 5020} \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\arctan (a x)^3}{24 a^3 c^3}-\frac {\arctan (a x)}{64 a^3 c^3}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}-\frac {x}{64 a^2 c^3 \left (a^2 x^2+1\right )}+\frac {x}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )}-\frac {\arctan (a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )^2} \]
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Rule 205
Rule 211
Rule 5012
Rule 5020
Rule 5050
Rule 5084
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx}{a^2}+\frac {\int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2 c} \\ & = -\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{2 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{6 a^3 c^3}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx}{8 a^2}-\frac {3 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{a c} \\ & = \frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{2 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3}+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a^2 c}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2 c}+\frac {3 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a c} \\ & = \frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {13 x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3}+\frac {3 \int \frac {1}{c+a^2 c x^2} \, dx}{64 a^2 c^2}-\frac {\int \frac {1}{c+a^2 c x^2} \, dx}{4 a^2 c^2}+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a^2 c} \\ & = \frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {13 \arctan (a x)}{64 a^3 c^3}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3}+\frac {3 \int \frac {1}{c+a^2 c x^2} \, dx}{16 a^2 c^2} \\ & = \frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{64 a^3 c^3}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.52 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 a x-3 a^3 x^3-3 \left (1-6 a^2 x^2+a^4 x^4\right ) \arctan (a x)+24 a x \left (-1+a^2 x^2\right ) \arctan (a x)^2+8 \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{192 a^3 c^3 \left (1+a^2 x^2\right )^2} \]
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Time = 0.62 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.68
method | result | size |
parallelrisch | \(\frac {8 a^{4} \arctan \left (a x \right )^{3} x^{4}-3 \arctan \left (a x \right ) a^{4} x^{4}+24 a^{3} \arctan \left (a x \right )^{2} x^{3}+16 \arctan \left (a x \right )^{3} x^{2} a^{2}-3 a^{3} x^{3}+18 a^{2} \arctan \left (a x \right ) x^{2}-24 a \arctan \left (a x \right )^{2} x +8 \arctan \left (a x \right )^{3}+3 a x -3 \arctan \left (a x \right )}{192 c^{3} \left (a^{2} x^{2}+1\right )^{2} a^{3}}\) | \(123\) |
derivativedivides | \(\frac {\frac {\arctan \left (a x \right )^{2} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {a x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}+\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {\frac {1}{8} a^{3} x^{3}-\frac {1}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )}{16}}{4 c^{3}}}{a^{3}}\) | \(149\) |
default | \(\frac {\frac {\arctan \left (a x \right )^{2} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {a x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}+\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {\frac {1}{8} a^{3} x^{3}-\frac {1}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )}{16}}{4 c^{3}}}{a^{3}}\) | \(149\) |
parts | \(\frac {\arctan \left (a x \right )^{2} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {x \arctan \left (a x \right )^{2}}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{3}}{8 a^{3} c^{3}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3 a^{3}}+\frac {\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {\frac {1}{8} a^{3} x^{3}-\frac {1}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )}{16}}{a^{3}}}{4 c^{3}}\) | \(155\) |
risch | \(\frac {i \ln \left (i a x +1\right )^{3}}{192 a^{3} c^{3}}-\frac {i \left (x^{4} \ln \left (-i a x +1\right ) a^{4}+2 a^{2} x^{2} \ln \left (-i a x +1\right )-2 i a^{3} x^{3}+\ln \left (-i a x +1\right )+2 i a x \right ) \ln \left (i a x +1\right )^{2}}{64 a^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {i \left (a^{4} x^{4} \ln \left (-i a x +1\right )^{2}+2 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-4 i x^{3} \ln \left (-i a x +1\right ) a^{3}-4 a^{2} x^{2}+\ln \left (-i a x +1\right )^{2}+4 i a x \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{64 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}+\frac {i \left (-2 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}+12 i a^{3} x^{3} \ln \left (-i a x +1\right )^{2}+3 \ln \left (-i a x -1\right ) a^{4} x^{4}-3 \ln \left (i a x -1\right ) a^{4} x^{4}+6 i a^{3} x^{3}-4 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}-12 i a x \ln \left (-i a x +1\right )^{2}+6 \ln \left (-i a x -1\right ) a^{2} x^{2}-6 \ln \left (i a x -1\right ) a^{2} x^{2}+24 a^{2} x^{2} \ln \left (-i a x +1\right )-6 i a x -2 \ln \left (-i a x +1\right )^{3}+3 \ln \left (-i a x -1\right )-3 \ln \left (i a x -1\right )\right )}{384 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}\) | \(444\) |
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Time = 0.24 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 24 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )^{2} - 3 \, a x + 3 \, {\left (a^{4} x^{4} - 6 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{192 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \]
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\[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
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Time = 0.33 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.28 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {1}{8} \, {\left (\frac {a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac {\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right )^{2} - \frac {{\left (3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x + 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{192 \, {\left (a^{9} c^{3} x^{4} + 2 \, a^{7} c^{3} x^{2} + a^{5} c^{3}\right )}} + \frac {{\left (a^{2} x^{2} - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a \arctan \left (a x\right )}{8 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]
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\[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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Time = 0.56 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\frac {x}{8\,a^2}-\frac {x^3}{8}}{8\,a^4\,c^3\,x^4+16\,a^2\,c^3\,x^2+8\,c^3}-\frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {x}{8\,a^4\,c^3}-\frac {x^3}{8\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {\mathrm {atan}\left (a\,x\right )}{64\,a^3\,c^3}+\frac {{\mathrm {atan}\left (a\,x\right )}^3}{24\,a^3\,c^3}+\frac {x^2\,\mathrm {atan}\left (a\,x\right )}{8\,a^3\,c^3\,\left (\frac {1}{a^2}+2\,x^2+a^2\,x^4\right )} \]
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