Integrand size = 24, antiderivative size = 256 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {135 \sqrt {\arctan (a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\arctan (a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \arctan (a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{256 a^4 c^3} \]
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Time = 0.36 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5064, 5060, 5056, 5050, 5024, 3393, 3385, 3433, 5090} \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{256 a^4 c^3}-\frac {3 \arctan (a x)^{5/2}}{32 a^4 c^3}-\frac {135 \sqrt {\arctan (a x)}}{2048 a^4 c^3}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}+\frac {45 \sqrt {\arctan (a x)}}{256 a^4 c^3 \left (a^2 x^2+1\right )}+\frac {15 x \arctan (a x)^{3/2}}{64 a^3 c^3 \left (a^2 x^2+1\right )} \]
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Rule 3385
Rule 3393
Rule 3433
Rule 5024
Rule 5050
Rule 5056
Rule 5060
Rule 5064
Rule 5090
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{8} (5 a) \int \frac {x^4 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx \\ & = -\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {1}{512} (15 a) \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx-\frac {15 \int \frac {x^2 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a c} \\ & = -\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \arctan (a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \text {Subst}\left (\int \frac {\sin ^4(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{512 a^4 c^3}-\frac {45 \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{128 a^2 c} \\ & = -\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\arctan (a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \arctan (a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \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 )}{512 a^4 c^3}-\frac {45 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{512 a^3 c} \\ & = \frac {45 \sqrt {\arctan (a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\arctan (a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \arctan (a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{4096 a^4 c^3}-\frac {15 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{1024 a^4 c^3}-\frac {45 \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{512 a^4 c^3} \\ & = \frac {45 \sqrt {\arctan (a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\arctan (a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \arctan (a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{2048 a^4 c^3}-\frac {15 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{512 a^4 c^3}-\frac {45 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{512 a^4 c^3} \\ & = -\frac {135 \sqrt {\arctan (a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\arctan (a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \arctan (a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{1024 a^4 c^3}-\frac {45 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{1024 a^4 c^3} \\ & = -\frac {135 \sqrt {\arctan (a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\arctan (a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \arctan (a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{1024 a^4 c^3}-\frac {45 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{512 a^4 c^3} \\ & = -\frac {135 \sqrt {\arctan (a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\arctan (a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \arctan (a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{256 a^4 c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.60 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.40 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {510 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {14400 \arctan (a x)+5760 a^2 x^2 \arctan (a x)-16320 a^4 x^4 \arctan (a x)+30720 a x \arctan (a x)^2+51200 a^3 x^3 \arctan (a x)^2-12288 \arctan (a x)^3-24576 a^2 x^2 \arctan (a x)^3+20480 a^4 x^4 \arctan (a x)^3-4080 \sqrt {\pi } \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+900 i \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )-900 i \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+135 i \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )-135 i \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{\left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}}{131072 a^4 c^3} \]
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Time = 2.56 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.61
\[\frac {-1024 \arctan \left (a x \right )^{\frac {5}{2}} \cos \left (2 \arctan \left (a x \right )\right ) \sqrt {\pi }+256 \arctan \left (a x \right )^{\frac {5}{2}} \cos \left (4 \arctan \left (a x \right )\right ) \sqrt {\pi }+1280 \arctan \left (a x \right )^{\frac {3}{2}} \sin \left (2 \arctan \left (a x \right )\right ) \sqrt {\pi }-160 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sin \left (4 \arctan \left (a x \right )\right )+15 \pi \sqrt {2}\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+960 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \cos \left (2 \arctan \left (a x \right )\right )-60 \cos \left (4 \arctan \left (a x \right )\right ) \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }-480 \pi \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )}{8192 c^{3} a^{4} \sqrt {\pi }}\]
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Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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