Integrand size = 24, antiderivative size = 108 \[ \int \frac {x^2 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\arctan (a x)^{5/2}}{20 a^3 c^3}-\frac {3 \sqrt {\arctan (a x)} \cos (4 \arctan (a x))}{256 a^3 c^3}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a^3 c^3}-\frac {\arctan (a x)^{3/2} \sin (4 \arctan (a x))}{32 a^3 c^3} \]
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Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5090, 4491, 3377, 3385, 3433} \[ \int \frac {x^2 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a^3 c^3}+\frac {\arctan (a x)^{5/2}}{20 a^3 c^3}-\frac {\arctan (a x)^{3/2} \sin (4 \arctan (a x))}{32 a^3 c^3}-\frac {3 \sqrt {\arctan (a x)} \cos (4 \arctan (a x))}{256 a^3 c^3} \]
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Rule 3377
Rule 3385
Rule 3433
Rule 4491
Rule 5090
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^{3/2} \cos ^2(x) \sin ^2(x) \, dx,x,\arctan (a x)\right )}{a^3 c^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {x^{3/2}}{8}-\frac {1}{8} x^{3/2} \cos (4 x)\right ) \, dx,x,\arctan (a x)\right )}{a^3 c^3} \\ & = \frac {\arctan (a x)^{5/2}}{20 a^3 c^3}-\frac {\text {Subst}\left (\int x^{3/2} \cos (4 x) \, dx,x,\arctan (a x)\right )}{8 a^3 c^3} \\ & = \frac {\arctan (a x)^{5/2}}{20 a^3 c^3}-\frac {\arctan (a x)^{3/2} \sin (4 \arctan (a x))}{32 a^3 c^3}+\frac {3 \text {Subst}\left (\int \sqrt {x} \sin (4 x) \, dx,x,\arctan (a x)\right )}{64 a^3 c^3} \\ & = \frac {\arctan (a x)^{5/2}}{20 a^3 c^3}-\frac {3 \sqrt {\arctan (a x)} \cos (4 \arctan (a x))}{256 a^3 c^3}-\frac {\arctan (a x)^{3/2} \sin (4 \arctan (a x))}{32 a^3 c^3}+\frac {3 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{512 a^3 c^3} \\ & = \frac {\arctan (a x)^{5/2}}{20 a^3 c^3}-\frac {3 \sqrt {\arctan (a x)} \cos (4 \arctan (a x))}{256 a^3 c^3}-\frac {\arctan (a x)^{3/2} \sin (4 \arctan (a x))}{32 a^3 c^3}+\frac {3 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{256 a^3 c^3} \\ & = \frac {\arctan (a x)^{5/2}}{20 a^3 c^3}-\frac {3 \sqrt {\arctan (a x)} \cos (4 \arctan (a x))}{256 a^3 c^3}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a^3 c^3}-\frac {\arctan (a x)^{3/2} \sin (4 \arctan (a x))}{32 a^3 c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 353, normalized size of antiderivative = 3.27 \[ \int \frac {x^2 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\frac {64 \sqrt {\arctan (a x)} \left (-15 \left (1-6 a^2 x^2+a^4 x^4\right )+160 a x \left (-1+a^2 x^2\right ) \arctan (a x)+64 \left (1+a^2 x^2\right )^2 \arctan (a x)^2\right )}{\left (1+a^2 x^2\right )^2}+30 \left (12 \sqrt {\arctan (a x)}+\sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )\right )-90 \sqrt {\arctan (a x)} \left (8+\frac {\Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )}{\sqrt {-i \arctan (a x)}}+\frac {\Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{\sqrt {i \arctan (a x)}}\right )+\frac {15 \left (24 \arctan (a x)-4 i \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+4 i \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )-i \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+i \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )\right )}{\sqrt {\arctan (a x)}}}{81920 a^3 c^3} \]
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Time = 6.71 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {15 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }-160 \arctan \left (a x \right )^{2} \sin \left (4 \arctan \left (a x \right )\right )+256 \arctan \left (a x \right )^{3}-60 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{5120 c^{3} a^{3} \sqrt {\arctan \left (a x \right )}}\) | \(81\) |
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Exception generated. \[ \int \frac {x^2 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {x^2 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^2 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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