Integrand size = 19, antiderivative size = 129 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {3}{8 a c^2 \left (1+a^2 x^2\right )}-\frac {3 x \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^2}{8 a c^2}+\frac {3 \arctan (a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a c^2} \]
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Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5012, 5050, 267} \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x \arctan (a x)^3}{2 c^2 \left (a^2 x^2+1\right )}+\frac {3 \arctan (a x)^2}{4 a c^2 \left (a^2 x^2+1\right )}-\frac {3 x \arctan (a x)}{4 c^2 \left (a^2 x^2+1\right )}-\frac {3}{8 a c^2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a c^2}-\frac {3 \arctan (a x)^2}{8 a c^2} \]
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Rule 267
Rule 5012
Rule 5050
Rubi steps \begin{align*} \text {integral}& = \frac {x \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a c^2}-\frac {1}{2} (3 a) \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = \frac {3 \arctan (a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a c^2}-\frac {3}{2} \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = -\frac {3 x \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^2}{8 a c^2}+\frac {3 \arctan (a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a c^2}+\frac {1}{4} (3 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = -\frac {3}{8 a c^2 \left (1+a^2 x^2\right )}-\frac {3 x \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^2}{8 a c^2}+\frac {3 \arctan (a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a c^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.55 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {-3-6 a x \arctan (a x)+\left (3-3 a^2 x^2\right ) \arctan (a x)^2+4 a x \arctan (a x)^3+\left (1+a^2 x^2\right ) \arctan (a x)^4}{8 c^2 \left (a+a^3 x^2\right )} \]
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Time = 1.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68
method | result | size |
parallelrisch | \(\frac {\arctan \left (a x \right )^{4} x^{2} a^{2}-3 x^{2} \arctan \left (a x \right )^{2} a^{2}+4 \arctan \left (a x \right )^{3} a x +3 a^{2} x^{2}+\arctan \left (a x \right )^{4}-6 x \arctan \left (a x \right ) a +3 \arctan \left (a x \right )^{2}}{8 c^{2} \left (a^{2} x^{2}+1\right ) a}\) | \(88\) |
derivativedivides | \(\frac {\frac {\arctan \left (a x \right )^{3} a x}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{4}}{2 c^{2}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {x \arctan \left (a x \right ) a}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )^{2}}{4}+\frac {1}{4 a^{2} x^{2}+4}\right )}{2 c^{2}}}{a}\) | \(114\) |
default | \(\frac {\frac {\arctan \left (a x \right )^{3} a x}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{4}}{2 c^{2}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {x \arctan \left (a x \right ) a}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )^{2}}{4}+\frac {1}{4 a^{2} x^{2}+4}\right )}{2 c^{2}}}{a}\) | \(114\) |
parts | \(\frac {x \arctan \left (a x \right )^{3}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{4}}{2 a \,c^{2}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4 a}+\frac {-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {x \arctan \left (a x \right ) a}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )^{2}}{4}+\frac {1}{4 a^{2} x^{2}+4}}{a}\right )}{2 c^{2}}\) | \(120\) |
risch | \(\frac {\ln \left (i a x +1\right )^{4}}{128 c^{2} a}-\frac {\left (a^{2} x^{2} \ln \left (-i a x +1\right )+\ln \left (-i a x +1\right )-2 i a x \right ) \ln \left (i a x +1\right )^{3}}{32 c^{2} \left (a^{2} x^{2}+1\right ) a}+\frac {3 \left (a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+2 a^{2} x^{2}+\ln \left (-i a x +1\right )^{2}-4 i a x \ln \left (-i a x +1\right )-2\right ) \ln \left (i a x +1\right )^{2}}{64 c^{2} \left (a x +i\right ) \left (a x -i\right ) a}-\frac {\left (a^{2} x^{2} \ln \left (-i a x +1\right )^{3}+6 a^{2} x^{2} \ln \left (-i a x +1\right )+\ln \left (-i a x +1\right )^{3}-6 i a x \ln \left (-i a x +1\right )^{2}-6 \ln \left (-i a x +1\right )-12 i a x \right ) \ln \left (i a x +1\right )}{32 c^{2} \left (a x +i\right ) \left (a x -i\right ) a}+\frac {a^{2} x^{2} \ln \left (-i a x +1\right )^{4}+12 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+\ln \left (-i a x +1\right )^{4}-8 i a x \ln \left (-i a x +1\right )^{3}-12 \ln \left (-i a x +1\right )^{2}-48 i a x \ln \left (-i a x +1\right )-48}{128 c^{2} \left (a x +i\right ) \left (a x -i\right ) a}\) | \(379\) |
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Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {4 \, a x \arctan \left (a x\right )^{3} + {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 6 \, a x \arctan \left (a x\right ) - 3 \, {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )^{2} - 3}{8 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \]
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\[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
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Time = 0.33 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.65 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {1}{2} \, {\left (\frac {x}{a^{2} c^{2} x^{2} + c^{2}} + \frac {\arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right )^{3} - \frac {3 \, {\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1\right )} a \arctan \left (a x\right )^{2}}{4 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} - \frac {1}{8} \, {\left (\frac {{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 3\right )} a^{2}}{a^{6} c^{2} x^{2} + a^{4} c^{2}} - \frac {2 \, {\left (2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a \arctan \left (a x\right )}{a^{5} c^{2} x^{2} + a^{3} c^{2}}\right )} a \]
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\[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
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Time = 0.50 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {{\mathrm {atan}\left (a\,x\right )}^4}{8\,a\,c^2}-{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {3}{8\,a\,c^2}-\frac {3}{4\,a^3\,c^2\,\left (\frac {1}{a^2}+x^2\right )}\right )-\frac {3}{2\,a\,\left (4\,a^2\,c^2\,x^2+4\,c^2\right )}-\frac {3\,x\,\mathrm {atan}\left (a\,x\right )}{4\,a^2\,c^2\,\left (\frac {1}{a^2}+x^2\right )}+\frac {x\,{\mathrm {atan}\left (a\,x\right )}^3}{2\,a^2\,c^2\,\left (\frac {1}{a^2}+x^2\right )} \]
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