Integrand size = 23, antiderivative size = 92 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {2 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a c \sqrt {c+a^2 c x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5022, 5091, 5090, 3386, 3432} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=-\frac {2 \sqrt {2 \pi } \sqrt {a^2 x^2+1} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a c \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \sqrt {a^2 c x^2+c}} \]
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Rule 3386
Rule 3432
Rule 5022
Rule 5090
Rule 5091
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-(2 a) \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {\left (2 a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {2 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a c \sqrt {c+a^2 c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\frac {-2+\sqrt {1+a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )+\sqrt {1+a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \]
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\[\int \frac {1}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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