Integrand size = 18, antiderivative size = 62 \[ \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)}{4 a^2 c^2}-\frac {\arctan (a x)}{2 a^2 c^2 \left (1+a^2 x^2\right )} \]
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Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5050, 205, 211} \[ \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {\arctan (a x)}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{4 a^2 c^2}+\frac {x}{4 a c^2 \left (a^2 x^2+1\right )} \]
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Rule 205
Rule 211
Rule 5050
Rubi steps \begin{align*} \text {integral}& = -\frac {\arctan (a x)}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a} \\ & = \frac {x}{4 a c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\int \frac {1}{c+a^2 c x^2} \, dx}{4 a c} \\ & = \frac {x}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)}{4 a^2 c^2}-\frac {\arctan (a x)}{2 a^2 c^2 \left (1+a^2 x^2\right )} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.63 \[ \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {a x+\left (-1+a^2 x^2\right ) \arctan (a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )} \]
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66
method | result | size |
parallelrisch | \(\frac {a^{2} \arctan \left (a x \right ) x^{2}+a x -\arctan \left (a x \right )}{4 c^{2} \left (a^{2} x^{2}+1\right ) a^{2}}\) | \(41\) |
derivativedivides | \(\frac {-\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\frac {a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )}{2}}{2 c^{2}}}{a^{2}}\) | \(53\) |
default | \(\frac {-\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\frac {a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )}{2}}{2 c^{2}}}{a^{2}}\) | \(53\) |
parts | \(-\frac {\arctan \left (a x \right )}{2 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\frac {x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )}{2 a}}{2 a \,c^{2}}\) | \(57\) |
risch | \(\frac {i \ln \left (i a x +1\right )}{4 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}-\frac {i \left (2 \ln \left (-i a x +1\right )+\ln \left (a x -i\right ) a^{2} x^{2}+\ln \left (a x -i\right )-\ln \left (-a x -i\right ) a^{2} x^{2}-\ln \left (-a x -i\right )+2 i a x \right )}{8 \left (a x +i\right ) a^{2} c^{2} \left (a x -i\right )}\) | \(118\) |
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none
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.65 \[ \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {a x + {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{4 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} \]
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Time = 0.35 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.32 \[ \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\begin {cases} \frac {a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} + 4 a^{2} c^{2}} + \frac {a x}{4 a^{4} c^{2} x^{2} + 4 a^{2} c^{2}} - \frac {\operatorname {atan}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} + 4 a^{2} c^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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none
Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.95 \[ \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\frac {x}{a^{2} c x^{2} + c} + \frac {\arctan \left (a x\right )}{a c}}{4 \, a c} - \frac {\arctan \left (a x\right )}{2 \, {\left (a^{2} c x^{2} + c\right )} a^{2} c} \]
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\[ \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
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Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.65 \[ \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {a\,x-\mathrm {atan}\left (a\,x\right )+a^2\,x^2\,\mathrm {atan}\left (a\,x\right )}{4\,a^2\,c^2\,\left (a^2\,x^2+1\right )} \]
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