Integrand size = 24, antiderivative size = 58 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {\sqrt {\arctan (a x)}}{4 a^3 c^3}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a^3 c^3} \]
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Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5090, 4491, 3385, 3433} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {\sqrt {\arctan (a x)}}{4 a^3 c^3}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a^3 c^3} \]
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Rule 3385
Rule 3433
Rule 4491
Rule 5090
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^3 c^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{8 \sqrt {x}}-\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{a^3 c^3} \\ & = \frac {\sqrt {\arctan (a x)}}{4 a^3 c^3}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{8 a^3 c^3} \\ & = \frac {\sqrt {\arctan (a x)}}{4 a^3 c^3}-\frac {\text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{4 a^3 c^3} \\ & = \frac {\sqrt {\arctan (a x)}}{4 a^3 c^3}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a^3 c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.95 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {-2 \sqrt {2 \pi } \sqrt {\arctan (a x)^2} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+16 \sqrt {\pi } \sqrt {\arctan (a x)^2} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)} \left (64 \sqrt {\arctan (a x)^2}+4 \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+4 \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+7 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+7 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )\right )}{256 a^3 c^3 \sqrt {\arctan (a x)^2}} \]
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Time = 1.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }+\pi \,\operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{16 c^{3} a^{3} \sqrt {\pi }}\) | \(47\) |
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Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {\int \frac {x^{2}}{a^{6} x^{6} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{4} x^{4} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{2} x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}} + \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3} \sqrt {\arctan \left (a x\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\int \frac {x^2}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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