Integrand size = 24, antiderivative size = 129 \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {3 x \sqrt {\arctan (a x)}}{2 a c \sqrt {c+a^2 c x^2}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a^2 c \sqrt {c+a^2 c x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5050, 5025, 5024, 3377, 3386, 3432} \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {3 x \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}} \]
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Rule 3377
Rule 3386
Rule 3432
Rule 5024
Rule 5025
Rule 5050
Rubi steps \begin{align*} \text {integral}& = -\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {3 \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{2 a} \\ & = -\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\sqrt {\arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{2 a c \sqrt {c+a^2 c x^2}} \\ & = -\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sqrt {x} \cos (x) \, dx,x,\arctan (a x)\right )}{2 a^2 c \sqrt {c+a^2 c x^2}} \\ & = \frac {3 x \sqrt {\arctan (a x)}}{2 a c \sqrt {c+a^2 c x^2}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{4 a^2 c \sqrt {c+a^2 c x^2}} \\ & = \frac {3 x \sqrt {\arctan (a x)}}{2 a c \sqrt {c+a^2 c x^2}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{2 a^2 c \sqrt {c+a^2 c x^2}} \\ & = \frac {3 x \sqrt {\arctan (a x)}}{2 a c \sqrt {c+a^2 c x^2}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a^2 c \sqrt {c+a^2 c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.99 \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {4 (3 a x-2 \arctan (a x)) \arctan (a x)+3 \sqrt {1+a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )+3 \sqrt {1+a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )}{8 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \]
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\[\int \frac {x \arctan \left (a x \right )^{\frac {3}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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