Integrand size = 22, antiderivative size = 101 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=-\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a^2 c^2} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5052, 5090, 4491, 12, 3386, 3432} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=-\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a^2 c^2}-\frac {2 x}{3 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}} \]
[In]
[Out]
Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 5052
Rule 5090
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {16}{3} \int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {16 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a^2 c^2} \\ & = -\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {16 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a^2 c^2} \\ & = -\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {8 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a^2 c^2} \\ & = -\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {16 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 a^2 c^2} \\ & = -\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a^2 c^2} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.87 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=-\frac {2 \left (a x+\left (2-2 a^2 x^2\right ) \arctan (a x)+4 \sqrt {\pi } \left (1+a^2 x^2\right ) \arctan (a x)^{3/2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}} \]
[In]
[Out]
Time = 1.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.58
method | result | size |
default | \(-\frac {8 \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \arctan \left (a x \right )^{\frac {3}{2}}+4 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\sin \left (2 \arctan \left (a x \right )\right )}{3 c^{2} a^{2} \arctan \left (a x \right )^{\frac {3}{2}}}\) | \(59\) |
[In]
[Out]
Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {x}{a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]
[In]
[Out]
Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
[In]
[Out]