Integrand size = 20, antiderivative size = 41 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=-\frac {x}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{a^2 c^2} \]
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Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5088, 5090, 3393, 3383, 5024} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)} \]
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Rule 3383
Rule 3393
Rule 5024
Rule 5088
Rule 5090
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx}{a}-a \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx \\ & = -\frac {x}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\text {Subst}\left (\int \frac {\cos ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{a^2 c^2}-\frac {\text {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{a^2 c^2} \\ & = -\frac {x}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\arctan (a x)\right )}{a^2 c^2}+\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\arctan (a x)\right )}{a^2 c^2} \\ & = -\frac {x}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+2 \frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arctan (a x)\right )}{2 a^2 c^2} \\ & = -\frac {x}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{a^2 c^2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {-\frac {a x}{\left (1+a^2 x^2\right ) \arctan (a x)}+\operatorname {CosIntegral}(2 \arctan (a x))}{a^2 c^2} \]
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Time = 8.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {2 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\sin \left (2 \arctan \left (a x \right )\right )}{2 a^{2} c^{2} \arctan \left (a x \right )}\) | \(38\) |
default | \(\frac {2 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\sin \left (2 \arctan \left (a x \right )\right )}{2 a^{2} c^{2} \arctan \left (a x \right )}\) | \(38\) |
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.80 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) + {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2 \, a x}{2 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )} \arctan \left (a x\right )} \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {\int \frac {x}{a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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