Integrand size = 20, antiderivative size = 81 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=-\frac {x}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\text {Si}(2 \arctan (a x))}{a^2 c^2} \]
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Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5052, 5090, 4491, 12, 3380} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=-\frac {\text {Si}(2 \arctan (a x))}{a^2 c^2}-\frac {x}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 c^2 \left (a^2 x^2+1\right ) \arctan (a x)} \]
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Rule 12
Rule 3380
Rule 4491
Rule 5052
Rule 5090
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-2 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx \\ & = -\frac {x}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {2 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\arctan (a x)\right )}{a^2 c^2} \\ & = -\frac {x}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {2 \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\arctan (a x)\right )}{a^2 c^2} \\ & = -\frac {x}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\arctan (a x)\right )}{a^2 c^2} \\ & = -\frac {x}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\text {Si}(2 \arctan (a x))}{a^2 c^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {-a x+\left (-1+a^2 x^2\right ) \arctan (a x)-2 \left (1+a^2 x^2\right ) \arctan (a x)^2 \text {Si}(2 \arctan (a x))}{2 a^2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2} \]
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Time = 8.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(-\frac {4 \,\operatorname {Si}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+2 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\sin \left (2 \arctan \left (a x \right )\right )}{4 a^{2} c^{2} \arctan \left (a x \right )^{2}}\) | \(51\) |
default | \(-\frac {4 \,\operatorname {Si}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+2 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\sin \left (2 \arctan \left (a x \right )\right )}{4 a^{2} c^{2} \arctan \left (a x \right )^{2}}\) | \(51\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.67 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {{\left (-i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) + {\left (i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - a x + {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{2 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )} \arctan \left (a x\right )^{2}} \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {\int \frac {x}{a^{4} x^{4} \operatorname {atan}^{3}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}} \,d x } \]
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Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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