Integrand size = 21, antiderivative size = 219 \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {45 \sqrt {\arctan (a x)}}{256 a c^3}+\frac {3 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{32 a c^3} \]
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Time = 0.21 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5020, 5012, 5050, 5024, 3393, 3385, 3433} \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 x \arctan (a x)^{3/2}}{8 c^3 \left (a^2 x^2+1\right )}+\frac {x \arctan (a x)^{3/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )}+\frac {3 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{32 a c^3}+\frac {3 \arctan (a x)^{5/2}}{20 a c^3}-\frac {45 \sqrt {\arctan (a x)}}{256 a c^3} \]
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Rule 3385
Rule 3393
Rule 3433
Rule 5012
Rule 5020
Rule 5024
Rule 5050
Rubi steps \begin{align*} \text {integral}& = \frac {3 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3}{64} \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {3 \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c} \\ & = \frac {3 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{5/2}}{20 a c^3}-\frac {3 \text {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{64 a c^3}-\frac {(9 a) \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c} \\ & = \frac {3 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{5/2}}{20 a c^3}-\frac {3 \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 )}{64 a c^3}-\frac {9 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{64 c} \\ & = -\frac {9 \sqrt {\arctan (a x)}}{256 a c^3}+\frac {3 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{5/2}}{20 a c^3}-\frac {3 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{512 a c^3}-\frac {3 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{128 a c^3}-\frac {9 \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{64 a c^3} \\ & = -\frac {9 \sqrt {\arctan (a x)}}{256 a c^3}+\frac {3 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{5/2}}{20 a c^3}-\frac {3 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{256 a c^3}-\frac {3 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{64 a c^3}-\frac {9 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{64 a c^3} \\ & = -\frac {45 \sqrt {\arctan (a x)}}{256 a c^3}+\frac {3 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a c^3}-\frac {9 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{128 a c^3} \\ & = -\frac {45 \sqrt {\arctan (a x)}}{256 a c^3}+\frac {3 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a c^3}-\frac {9 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{64 a c^3} \\ & = -\frac {45 \sqrt {\arctan (a x)}}{256 a c^3}+\frac {3 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{32 a c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.71 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.62 \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\frac {64 \sqrt {\arctan (a x)} \left (-15 \left (-17+6 a^2 x^2+15 a^4 x^4\right )+160 a x \left (5+3 a^2 x^2\right ) \arctan (a x)+192 \left (1+a^2 x^2\right )^2 \arctan (a x)^2\right )}{\left (1+a^2 x^2\right )^2}+450 \left (12 \sqrt {\arctan (a x)}+\sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )\right )+90 \sqrt {\arctan (a x)} \left (8+\frac {\Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )}{\sqrt {-i \arctan (a x)}}+\frac {\Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{\sqrt {i \arctan (a x)}}\right )-\frac {255 \left (24 \arctan (a x)-4 i \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+4 i \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )-i \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+i \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )\right )}{\sqrt {\arctan (a x)}}}{81920 a c^3} \]
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Time = 7.53 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {768 \arctan \left (a x \right )^{3}+1280 \arctan \left (a x \right )^{2} \sin \left (2 \arctan \left (a x \right )\right )+160 \arctan \left (a x \right )^{2} \sin \left (4 \arctan \left (a x \right )\right )-15 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }+960 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-480 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }+60 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{5120 c^{3} a \sqrt {\arctan \left (a x \right )}}\) | \(132\) |
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Exception generated. \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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