Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\frac {1}{a^3 c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^2},x\right )}{a^2 c} \]
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Not integrable
Time = 0.25 (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 = {} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx}{a^2}+\frac {\int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^2 c} \\ & = \frac {1}{a^3 c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx}{a}+\frac {\int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^2 c} \\ & = \frac {1}{a^3 c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^2 c}+\frac {\sqrt {1+a^2 x^2} \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \arctan (a x)} \, dx}{a c \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{a^3 c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^2 c}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arctan (a x)\right )}{a^3 c \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{a^3 c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^2 c} \\ \end{align*}
Not integrable
Time = 6.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx \]
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Not integrable
Time = 2.50 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {x^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{2}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 2.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int \frac {x^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]
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Not integrable
Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 60.96 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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