Integrand size = 19, antiderivative size = 100 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{4 a c^2}+\frac {\arctan (a x)}{2 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{6 a c^2} \]
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Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5012, 5050, 205, 211} \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x \arctan (a x)^2}{2 c^2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a c^2 \left (a^2 x^2+1\right )}-\frac {x}{4 c^2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a c^2}-\frac {\arctan (a x)}{4 a c^2} \]
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Rule 205
Rule 211
Rule 5012
Rule 5050
Rubi steps \begin{align*} \text {integral}& = \frac {x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{6 a c^2}-a \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = \frac {\arctan (a x)}{2 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{6 a c^2}-\frac {1}{2} \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = -\frac {x}{4 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)}{2 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{6 a c^2}-\frac {\int \frac {1}{c+a^2 c x^2} \, dx}{4 c} \\ & = -\frac {x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{4 a c^2}+\frac {\arctan (a x)}{2 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{6 a c^2} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.65 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {-3 a x+\left (3-3 a^2 x^2\right ) \arctan (a x)+6 a x \arctan (a x)^2+2 \left (1+a^2 x^2\right ) \arctan (a x)^3}{12 c^2 \left (a+a^3 x^2\right )} \]
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Time = 0.47 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.75
method | result | size |
parallelrisch | \(\frac {2 \arctan \left (a x \right )^{3} x^{2} a^{2}-3 a^{2} \arctan \left (a x \right ) x^{2}+6 a \arctan \left (a x \right )^{2} x +2 \arctan \left (a x \right )^{3}-3 a x +3 \arctan \left (a x \right )}{12 c^{2} \left (a^{2} x^{2}+1\right ) a}\) | \(75\) |
derivativedivides | \(\frac {\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {a x}{4 a^{2} x^{2}+4}+\frac {\arctan \left (a x \right )}{4}}{c^{2}}}{a}\) | \(93\) |
default | \(\frac {\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {a x}{4 a^{2} x^{2}+4}+\frac {\arctan \left (a x \right )}{4}}{c^{2}}}{a}\) | \(93\) |
parts | \(\frac {x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{2 a \,c^{2}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3 a}+\frac {-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {a x}{4 a^{2} x^{2}+4}+\frac {\arctan \left (a x \right )}{4}}{a}}{c^{2}}\) | \(99\) |
risch | \(\frac {i \ln \left (i a x +1\right )^{3}}{48 c^{2} a}-\frac {i \left (a^{2} x^{2} \ln \left (-i a x +1\right )+\ln \left (-i a x +1\right )-2 i a x \right ) \ln \left (i a x +1\right )^{2}}{16 c^{2} \left (a^{2} x^{2}+1\right ) a}+\frac {i \left (a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+\ln \left (-i a x +1\right )^{2}-4 i a x \ln \left (-i a x +1\right )-4\right ) \ln \left (i a x +1\right )}{16 c^{2} \left (a x +i\right ) \left (a x -i\right ) a}-\frac {i \left (a^{2} x^{2} \ln \left (-i a x +1\right )^{3}+6 a^{2} x^{2} \ln \left (-i a x +1\right )-6 \ln \left (i a x +1\right ) a^{2} x^{2}-6 i a x \ln \left (-i a x +1\right )^{2}-12 i a x +\ln \left (-i a x +1\right )^{3}-6 \ln \left (-i a x +1\right )-6 \ln \left (i a x +1\right )\right )}{48 c^{2} \left (a x +i\right ) \left (a x -i\right ) a}\) | \(280\) |
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Time = 0.24 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.67 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {6 \, a x \arctan \left (a x\right )^{2} + 2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \, {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{12 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \]
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\[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.46 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {1}{2} \, {\left (\frac {x}{a^{2} c^{2} x^{2} + c^{2}} + \frac {\arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right )^{2} + \frac {{\left (2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{12 \, {\left (a^{5} c^{2} x^{2} + a^{3} c^{2}\right )}} - \frac {{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1\right )} a \arctan \left (a x\right )}{2 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} \]
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\[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
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Time = 0.57 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.01 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (a\,x\right )}{2\,\left (a^3\,c^2\,x^2+a\,c^2\right )}-\frac {x}{2\,\left (2\,a^2\,c^2\,x^2+2\,c^2\right )}+\frac {x\,{\mathrm {atan}\left (a\,x\right )}^2}{2\,\left (a^2\,c^2\,x^2+c^2\right )}-\frac {\mathrm {atan}\left (a\,x\right )}{4\,a\,c^2}+\frac {{\mathrm {atan}\left (a\,x\right )}^3}{6\,a\,c^2} \]
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