Integrand size = 24, antiderivative size = 180 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=-\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16 \sqrt {\arctan (a x)}}{3 a^3 c^2}-\frac {32 \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a^3 c^2} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5062, 5052, 5050, 5024, 3393, 3385, 3433} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a^3 c^2}+\frac {16 \sqrt {\arctan (a x)}}{3 a^3 c^2}-\frac {2 x^2}{3 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (a^2 x^2+1\right )}-\frac {32 \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (a^2 x^2+1\right )} \]
[In]
[Out]
Rule 3385
Rule 3393
Rule 3433
Rule 5024
Rule 5050
Rule 5052
Rule 5062
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {4 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx}{3 a} \\ & = -\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {64 \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{3 a} \\ & = -\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {32 \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{3 a^2} \\ & = -\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {32 \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a^3 c^2} \\ & = -\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {32 \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{3 a^3 c^2} \\ & = -\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16 \sqrt {\arctan (a x)}}{3 a^3 c^2}-\frac {32 \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {8 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a^3 c^2} \\ & = -\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16 \sqrt {\arctan (a x)}}{3 a^3 c^2}-\frac {32 \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 a^3 c^2} \\ & = -\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16 \sqrt {\arctan (a x)}}{3 a^3 c^2}-\frac {32 \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a^3 c^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\frac {-2 a x (a x+4 \arctan (a x))+4 \sqrt {\pi } \left (1+a^2 x^2\right ) \arctan (a x)^{3/2} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {2} \left (1+a^2 x^2\right ) (-i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+\sqrt {2} \left (1+a^2 x^2\right ) (i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )}{3 a^3 c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}} \]
[In]
[Out]
Time = 1.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.34
method | result | size |
default | \(-\frac {-8 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\pi }\, \arctan \left (a x \right )^{\frac {3}{2}}+4 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\cos \left (2 \arctan \left (a x \right )\right )+1}{3 c^{2} a^{3} \arctan \left (a x \right )^{\frac {3}{2}}}\) | \(62\) |
[In]
[Out]
Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {x^{2}}{a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]
[In]
[Out]
Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
[In]
[Out]