Integrand size = 26, antiderivative size = 160 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=-\frac {2 x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {\frac {3 \pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.28 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5062, 5091, 5090, 4491, 3385, 3433} \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=-\frac {2 x^3}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\frac {3 \pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^2 \sqrt {a^2 c x^2+c}} \]
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Rule 3385
Rule 3433
Rule 4491
Rule 5062
Rule 5090
Rule 5091
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {6 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a} \\ & = -\frac {2 x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \int \frac {x^2}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {x}}-\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{2 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{2 a^4 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {\frac {3 \pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.14 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=\frac {-\frac {8 a^3 c x^3}{1+a^2 x^2}-i c \sqrt {1+a^2 x^2} \left (3 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )-3 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )+\sqrt {3} \left (-\sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )+\sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )\right )\right )}{4 a^4 c^3 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \]
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\[\int \frac {x^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=\int \frac {x^{3}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=\int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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