Integrand size = 20, antiderivative size = 20 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=-\frac {x}{2 a c \arctan (a x)^2}+\frac {\text {Int}\left (\frac {1}{\arctan (a x)^2},x\right )}{2 a c} \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {x}{2 a c \arctan (a x)^2}+\frac {\int \frac {1}{\arctan (a x)^2} \, dx}{2 a c} \\ \end{align*}
Not integrable
Time = 0.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx \]
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Not integrable
Time = 8.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \frac {x}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{3}}d x\]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}} \,d x } \]
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Not integrable
Time = 0.72 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\frac {\int \frac {x}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c} \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.75 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}} \,d x } \]
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Not integrable
Time = 48.25 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}} \,d x } \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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