Integrand size = 24, antiderivative size = 118 \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 \sqrt {\arctan (a x)}}{32 a^4 c^3}+\frac {x^4 \sqrt {\arctan (a x)}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{64 a^4 c^3}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{16 a^4 c^3} \]
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Time = 0.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5064, 5090, 3393, 3385, 3433} \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{64 a^4 c^3}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{16 a^4 c^3}-\frac {3 \sqrt {\arctan (a x)}}{32 a^4 c^3}+\frac {x^4 \sqrt {\arctan (a x)}}{4 c^3 \left (a^2 x^2+1\right )^2} \]
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Rule 3385
Rule 3393
Rule 3433
Rule 5064
Rule 5090
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 \sqrt {\arctan (a x)}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{8} a \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx \\ & = \frac {x^4 \sqrt {\arctan (a x)}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {\sin ^4(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{8 a^4 c^3} \\ & = \frac {x^4 \sqrt {\arctan (a x)}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {\text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}-\frac {\cos (2 x)}{2 \sqrt {x}}+\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{8 a^4 c^3} \\ & = -\frac {3 \sqrt {\arctan (a x)}}{32 a^4 c^3}+\frac {x^4 \sqrt {\arctan (a x)}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{64 a^4 c^3}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{16 a^4 c^3} \\ & = -\frac {3 \sqrt {\arctan (a x)}}{32 a^4 c^3}+\frac {x^4 \sqrt {\arctan (a x)}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {\text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{32 a^4 c^3}+\frac {\text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{8 a^4 c^3} \\ & = -\frac {3 \sqrt {\arctan (a x)}}{32 a^4 c^3}+\frac {x^4 \sqrt {\arctan (a x)}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{64 a^4 c^3}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{16 a^4 c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.95 \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-10 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+80 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {\frac {64 \left (-3-6 a^2 x^2+5 a^4 x^4\right ) \arctan (a x)}{\left (1+a^2 x^2\right )^2}-12 i \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+12 i \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+3 i \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )-3 i \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{\sqrt {\arctan (a x)}}}{2048 a^4 c^3} \]
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Time = 6.40 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {\operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }+16 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-8 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }-4 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{128 c^{3} a^{4} \sqrt {\arctan \left (a x \right )}}\) | \(93\) |
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Exception generated. \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{3} \sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{3} \sqrt {\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^3\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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