Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx=-\frac {\sqrt {c+a^2 c x^2}}{a c x \arctan (a x)}-\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)},x\right )}{a} \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx=\int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+a^2 c x^2}}{a c x \arctan (a x)}-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)} \, dx}{a} \\ \end{align*}
Not integrable
Time = 1.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx=\int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx \]
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Not integrable
Time = 7.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {1}{x \arctan \left (a x \right )^{2} \sqrt {a^{2} c \,x^{2}+c}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx=\int { \frac {1}{\sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 1.91 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx=\int \frac {1}{x \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]
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Not integrable
Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx=\int { \frac {1}{\sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 65.52 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx=\int { \frac {1}{\sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
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