Integrand size = 24, antiderivative size = 129 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=-\frac {2 x^2}{3 a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {8 x^3}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^3 c^3} \]
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Time = 0.47 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5088, 5090, 3393, 3385, 3433, 4491, 5024} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\frac {4 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}+\frac {8 x^3}{3 c^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}} \]
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Rule 3385
Rule 3393
Rule 3433
Rule 4491
Rule 5024
Rule 5088
Rule 5090
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2}{3 a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {4 \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (4 a) \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2 x^2}{3 a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {8 x^3}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-2 \left (8 \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx\right )+\frac {8 \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {1}{3} \left (8 a^2\right ) \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2 x^2}{3 a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {8 x^3}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {8 \text {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a^3 c^3}+\frac {8 \text {Subst}\left (\int \frac {\sin ^4(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a^3 c^3}-2 \frac {8 \text {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^3 c^3} \\ & = -\frac {2 x^2}{3 a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {8 x^3}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {8 \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 )}{3 a^3 c^3}+\frac {8 \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 )}{3 a^3 c^3}-2 \frac {8 \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^3 c^3} \\ & = -\frac {2 x^2}{3 a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {8 x^3}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4 \sqrt {\arctan (a x)}}{a^3 c^3}+2 \frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a^3 c^3}-2 \left (\frac {2 \sqrt {\arctan (a x)}}{a^3 c^3}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^3 c^3}\right ) \\ & = -\frac {2 x^2}{3 a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {8 x^3}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4 \sqrt {\arctan (a x)}}{a^3 c^3}+2 \frac {2 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 a^3 c^3}-2 \left (\frac {2 \sqrt {\arctan (a x)}}{a^3 c^3}-\frac {2 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a^3 c^3}\right ) \\ & = -\frac {2 x^2}{3 a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {8 x^3}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4 \sqrt {\arctan (a x)}}{a^3 c^3}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^3 c^3}-2 \left (\frac {2 \sqrt {\arctan (a x)}}{a^3 c^3}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^3 c^3}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.01 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\frac {\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^3}-\frac {16 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a^3}+\frac {-\frac {8 x^2}{a \left (1+a^2 x^2\right )^2}+\frac {32 x^3 \arctan (a x)}{\left (1+a^2 x^2\right )^2}-\frac {32 x \arctan (a x)}{\left (a+a^3 x^2\right )^2}+\frac {4 \sqrt {2} (-i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )}{a^3}+\frac {4 \sqrt {2} (i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )}{a^3}+\frac {7 (-i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )}{a^3}+\frac {7 (i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{a^3}}{\arctan (a x)^{3/2}}}{12 c^3} \]
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Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.53
method | result | size |
default | \(-\frac {-16 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \arctan \left (a x \right )^{\frac {3}{2}}+8 \sin \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\cos \left (4 \arctan \left (a x \right )\right )+1}{12 a^{3} c^{3} \arctan \left (a x \right )^{\frac {3}{2}}}\) | \(68\) |
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Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {x^{2}}{a^{6} x^{6} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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