Integrand size = 24, antiderivative size = 96 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=-\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^3}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a^4 c^3} \]
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Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5088, 5090, 3393, 3385, 3433, 4491} \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^3}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a^4 c^3}-\frac {2 x^3}{a c^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}} \]
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Rule 3385
Rule 3393
Rule 3433
Rule 4491
Rule 5088
Rule 5090
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {6 \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{a}-(2 a) \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {2 \text {Subst}\left (\int \frac {\sin ^4(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^4 c^3}+\frac {6 \text {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^4 c^3} \\ & = -\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {2 \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 )}{a^4 c^3}+\frac {6 \text {Subst}\left (\int \left (\frac {1}{8 \sqrt {x}}-\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{a^4 c^3} \\ & = -\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{4 a^4 c^3}-\frac {3 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{4 a^4 c^3}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^4 c^3} \\ & = -\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {\text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{2 a^4 c^3}-\frac {3 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{2 a^4 c^3}+\frac {2 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a^4 c^3} \\ & = -\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^3}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a^4 c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.54 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\frac {-2 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+16 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {-\frac {32 a^3 x^3}{\left (1+a^2 x^2\right )^2}+3 i \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )-3 i \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{\sqrt {\arctan (a x)}}}{16 a^4 c^3} \]
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Time = 2.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {2 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }-4 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }+2 \sin \left (2 \arctan \left (a x \right )\right )-\sin \left (4 \arctan \left (a x \right )\right )}{4 c^{3} a^{4} \sqrt {\arctan \left (a x \right )}}\) | \(86\) |
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Exception generated. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\frac {\int \frac {x^{3}}{a^{6} x^{6} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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