Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=-\frac {1}{a c^2 x \arctan (a x)}+\frac {a x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{c^2}-\frac {\text {Int}\left (\frac {1}{x^2 \arctan (a x)},x\right )}{a c^2} \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx\right )+\frac {\int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx}{c} \\ & = -\frac {1}{a c^2 x \arctan (a x)}+\frac {a x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-a \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx+a^3 \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx-\frac {\int \frac {1}{x^2 \arctan (a x)} \, dx}{a c^2} \\ & = -\frac {1}{a c^2 x \arctan (a x)}+\frac {a x}{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 )}{c^2}+\frac {\text {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{c^2}-\frac {\int \frac {1}{x^2 \arctan (a x)} \, dx}{a c^2} \\ & = -\frac {1}{a c^2 x \arctan (a x)}+\frac {a x}{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 )}{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 )}{c^2}-\frac {\int \frac {1}{x^2 \arctan (a x)} \, dx}{a c^2} \\ & = -\frac {1}{a c^2 x \arctan (a x)}+\frac {a x}{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 c^2}-\frac {\int \frac {1}{x^2 \arctan (a x)} \, dx}{a c^2} \\ & = -\frac {1}{a c^2 x \arctan (a x)}+\frac {a x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{c^2}-\frac {\int \frac {1}{x^2 \arctan (a x)} \, dx}{a c^2} \\ \end{align*}
Not integrable
Time = 1.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx \]
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Not integrable
Time = 12.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{2}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 1.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {\int \frac {1}{a^{4} x^{5} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{3} \operatorname {atan}^{2}{\left (a x \right )} + x \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.64 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 84.22 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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