Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16 a \sqrt {\arctan (a x)}}{c^2}+\frac {8 a \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{c^2}+\frac {8 \text {Int}\left (\frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}},x\right )}{a^2}+\frac {56}{3} \text {Int}\left (\frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}},x\right ) \]
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Not integrable
Time = 0.33 (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 = {} \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {4 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (8 a) \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{a^2}+\left (16 a^2\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {(16 a) \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^2} \\ & = -\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {(16 a) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{c^2} \\ & = -\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16 a \sqrt {\arctan (a x)}}{c^2}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {(8 a) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^2} \\ & = -\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16 a \sqrt {\arctan (a x)}}{c^2}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {(16 a) \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{c^2} \\ & = -\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16 a \sqrt {\arctan (a x)}}{c^2}+\frac {8 a \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{c^2}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{a^2} \\ \end{align*}
Not integrable
Time = 5.92 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx \]
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Not integrable
Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 16.82 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {1}{a^{4} x^{6} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 2 a^{2} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]
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Exception generated. \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 227.73 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Not integrable
Time = 0.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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