Integrand size = 24, antiderivative size = 71 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a^4 c^3}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{4 a^4 c^3} \]
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Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5090, 4491, 3386, 3432} \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{4 a^4 c^3}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a^4 c^3} \]
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Rule 3386
Rule 3432
Rule 4491
Rule 5090
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos (x) \sin ^3(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^4 c^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}-\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{a^4 c^3} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{8 a^4 c^3}+\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{4 a^4 c^3} \\ & = -\frac {\text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{4 a^4 c^3}+\frac {\text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{2 a^4 c^3} \\ & = -\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a^4 c^3}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{4 a^4 c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.85 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {-2 \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )-2 \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+\sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+\sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{32 a^4 c^3 \sqrt {\arctan (a x)}} \]
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Time = 1.40 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {\sqrt {\pi }\, \left (-\sqrt {2}\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+4 \,\operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{16 c^{3} a^{4}}\) | \(47\) |
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Exception generated. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {\int \frac {x^{3}}{a^{6} x^{6} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{4} x^{4} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{2} x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}} + \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\int { \frac {x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} \sqrt {\arctan \left (a x\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\int \frac {x^3}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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