Integrand size = 26, antiderivative size = 26 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {16}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {4 \sqrt {\frac {2 \pi }{3}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {8 \text {Int}\left (\frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}},x\right )}{3 a^2}+\frac {20}{3} \text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}},x\right ) \]
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Not integrable
Time = 0.51 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (8 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2}{3 a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {16}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\left (16 a^2\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2}{3 a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {16}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {\left (16 a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{3 a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {16}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{3 a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {16}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\sin (x)}{4 \sqrt {x}}+\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{3 a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {16}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{3 a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {16}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{3 a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {16}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {4 \sqrt {\frac {2 \pi }{3}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {20}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{3 a^2} \\ \end{align*}
Not integrable
Time = 9.93 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx \]
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Not integrable
Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
\[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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