Integrand size = 24, antiderivative size = 24 \[ \int \frac {\sqrt {\arctan (a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {2 a^3 \arctan (a x)^{3/2}}{3 c}+\frac {\text {Int}\left (\frac {\sqrt {\arctan (a x)}}{x^4},x\right )}{c}-\frac {a^2 \text {Int}\left (\frac {\sqrt {\arctan (a x)}}{x^2},x\right )}{c} \]
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Not integrable
Time = 0.13 (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 = {} \[ \int \frac {\sqrt {\arctan (a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {\sqrt {\arctan (a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\sqrt {\arctan (a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\sqrt {\arctan (a x)}}{x^4} \, dx}{c} \\ & = a^4 \int \frac {\sqrt {\arctan (a x)}}{c+a^2 c x^2} \, dx+\frac {\int \frac {\sqrt {\arctan (a x)}}{x^4} \, dx}{c}-\frac {a^2 \int \frac {\sqrt {\arctan (a x)}}{x^2} \, dx}{c} \\ & = \frac {2 a^3 \arctan (a x)^{3/2}}{3 c}+\frac {\int \frac {\sqrt {\arctan (a x)}}{x^4} \, dx}{c}-\frac {a^2 \int \frac {\sqrt {\arctan (a x)}}{x^2} \, dx}{c} \\ \end{align*}
Not integrable
Time = 5.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\arctan (a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {\sqrt {\arctan (a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx \]
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Not integrable
Time = 4.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {\sqrt {\arctan \left (a x \right )}}{x^{4} \left (a^{2} c \,x^{2}+c \right )}d x\]
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Exception generated. \[ \int \frac {\sqrt {\arctan (a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 2.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {\arctan (a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{2} x^{6} + x^{4}}\, dx}{c} \]
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Exception generated. \[ \int \frac {\sqrt {\arctan (a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 28.70 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt {\arctan (a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\sqrt {\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \]
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Not integrable
Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {\arctan (a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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