Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {5 \sqrt {2 \pi } \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 c^3}+\frac {20 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 c^3}+\frac {8 \text {Int}\left (\frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}},x\right )}{3 a^2}+8 \text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}},x\right ) \]
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Not integrable
Time = 0.28 (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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (10 a) \int \frac {1}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {1}{3} \left (80 a^2\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {80 \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 c^3} \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {80 \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{3 c^3} \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {10 \text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 c^3}+\frac {20 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 c^3} \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {20 \text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 c^3}+\frac {40 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 c^3} \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {5 \sqrt {2 \pi } \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 c^3}+\frac {20 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 c^3}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2} \\ \end{align*}
Not integrable
Time = 4.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx \]
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Not integrable
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 28.97 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {1}{a^{6} x^{7} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{4} x^{5} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.67 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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