Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=-\frac {a^3 x}{c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {a \sqrt {c+a^2 c x^2}}{c^2 x \arctan (a x)}+\frac {a^2 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{c \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {1}{x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2},x\right )}{c}+\frac {a \text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)},x\right )}{c} \]
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Not integrable
Time = 0.51 (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 = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx\right )+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{c} \\ & = a^4 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{c}-\frac {a^2 \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{c} \\ & = -\frac {a^3 x}{c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {a \sqrt {c+a^2 c x^2}}{c^2 x \arctan (a x)}+a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{c}+\frac {a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)} \, dx}{c} \\ & = -\frac {a^3 x}{c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {a \sqrt {c+a^2 c x^2}}{c^2 x \arctan (a x)}+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{c}+\frac {a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)} \, dx}{c}+\frac {\left (a^3 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \arctan (a x)} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = -\frac {a^3 x}{c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {a \sqrt {c+a^2 c x^2}}{c^2 x \arctan (a x)}+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{c}+\frac {a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)} \, dx}{c}+\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}} \\ & = -\frac {a^3 x}{c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {a \sqrt {c+a^2 c x^2}}{c^2 x \arctan (a x)}+\frac {a^2 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{c \sqrt {c+a^2 c x^2}}+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{c}+\frac {a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)} \, dx}{c} \\ \end{align*}
Not integrable
Time = 4.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx \]
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Not integrable
Time = 20.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{2}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 8.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]
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Not integrable
Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]
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Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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