Integrand size = 24, antiderivative size = 129 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=-\frac {2 x^2}{3 a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {8 x^3}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^3 c^3} \]
-2/3*x^2/a/c^3/(a^2*x^2+1)^2/arctan(a*x)^(3/2)+4/3*FresnelC(2*2^(1/2)/Pi^( 1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3/c^3-8/3*x/a^2/c^3/(a^2*x^2+1) ^2/arctan(a*x)^(1/2)+8/3*x^3/c^3/(a^2*x^2+1)^2/arctan(a*x)^(1/2)
Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.01 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\frac {\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^3}-\frac {16 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a^3}+\frac {-\frac {8 x^2}{a \left (1+a^2 x^2\right )^2}+\frac {32 x^3 \arctan (a x)}{\left (1+a^2 x^2\right )^2}-\frac {32 x \arctan (a x)}{\left (a+a^3 x^2\right )^2}+\frac {4 \sqrt {2} (-i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )}{a^3}+\frac {4 \sqrt {2} (i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )}{a^3}+\frac {7 (-i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )}{a^3}+\frac {7 (i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{a^3}}{\arctan (a x)^{3/2}}}{12 c^3} \]
((2*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/a^3 - (16*Sqrt[Pi ]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/a^3 + ((-8*x^2)/(a*(1 + a^2*x^ 2)^2) + (32*x^3*ArcTan[a*x])/(1 + a^2*x^2)^2 - (32*x*ArcTan[a*x])/(a + a^3 *x^2)^2 + (4*Sqrt[2]*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcTan[a*x ]])/a^3 + (4*Sqrt[2]*(I*ArcTan[a*x])^(3/2)*Gamma[1/2, (2*I)*ArcTan[a*x]])/ a^3 + (7*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcTan[a*x]])/a^3 + (7 *(I*ArcTan[a*x])^(3/2)*Gamma[1/2, (4*I)*ArcTan[a*x]])/a^3)/ArcTan[a*x]^(3/ 2))/(12*c^3)
Leaf count is larger than twice the leaf count of optimal. \(359\) vs. \(2(129)=258\).
Time = 2.11 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.78, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {5503, 27, 5503, 5439, 3042, 3793, 2009, 5505, 3042, 3793, 2009, 4906, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2}{\arctan (a x)^{5/2} \left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5503 |
| \(\displaystyle \frac {4 \int \frac {x}{c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^{3/2}}dx}{3 a}-\frac {4}{3} a \int \frac {x^3}{c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^{3/2}}dx-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)^{3/2}}dx}{3 a c^3}-\frac {4 a \int \frac {x^3}{\left (a^2 x^2+1\right )^3 \arctan (a x)^{3/2}}dx}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 5503 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {1}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{a}-6 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {4 a \left (\frac {6 \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{a}-2 a \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 5439 |
| \(\displaystyle \frac {4 \left (-6 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {2 \int \frac {1}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a^2}-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {4 a \left (\frac {6 \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{a}-2 a \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 \left (-6 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {2 \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^4}{\sqrt {\arctan (a x)}}d\arctan (a x)}{a^2}-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {4 a \left (\frac {6 \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{a}-2 a \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {4 \left (-6 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {2 \int \left (\frac {\cos (2 \arctan (a x))}{2 \sqrt {\arctan (a x)}}+\frac {\cos (4 \arctan (a x))}{8 \sqrt {\arctan (a x)}}+\frac {3}{8 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{a^2}-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {4 a \left (\frac {6 \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{a}-2 a \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (-6 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{a^2}-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {4 a \left (\frac {6 \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{a}-2 a \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 5505 |
| \(\displaystyle \frac {4 \left (-\frac {6 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a^2}+\frac {2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{a^2}-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {4 a \left (\frac {6 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a^4}-\frac {2 \int \frac {a^4 x^4}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a^4}-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 \left (-\frac {6 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a^2}+\frac {2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{a^2}-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {4 a \left (-\frac {2 \int \frac {\sin (\arctan (a x))^4}{\sqrt {\arctan (a x)}}d\arctan (a x)}{a^4}+\frac {6 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a^4}-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {4 \left (-\frac {6 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a^2}+\frac {2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{a^2}-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {4 a \left (-\frac {2 \int \left (-\frac {\cos (2 \arctan (a x))}{2 \sqrt {\arctan (a x)}}+\frac {\cos (4 \arctan (a x))}{8 \sqrt {\arctan (a x)}}+\frac {3}{8 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{a^4}+\frac {6 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a^4}-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (-\frac {6 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a^2}+\frac {2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{a^2}-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {4 a \left (\frac {6 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a^4}-\frac {2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{a^4}-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {4 \left (-\frac {6 \int \left (\frac {1}{8 \sqrt {\arctan (a x)}}-\frac {\cos (4 \arctan (a x))}{8 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{a^2}+\frac {2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{a^2}-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {4 a \left (\frac {6 \int \left (\frac {1}{8 \sqrt {\arctan (a x)}}-\frac {\cos (4 \arctan (a x))}{8 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{a^4}-\frac {2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{a^4}-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (-\frac {6 \left (\frac {1}{4} \sqrt {\arctan (a x)}-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{a^2}+\frac {2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{a^2}-\frac {2 x}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 a c^3}-\frac {2 x^2}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}-\frac {4 a \left (\frac {6 \left (\frac {1}{4} \sqrt {\arctan (a x)}-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{a^4}-\frac {2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{a^4}-\frac {2 x^3}{a \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}\right )}{3 c^3}\) |
(-2*x^2)/(3*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) - (4*a*((-2*x^3)/(a*( 1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (6*(Sqrt[ArcTan[a*x]]/4 - (Sqrt[Pi/2]* FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/8))/a^4 - (2*((3*Sqrt[ArcTan[a*x ]])/4 + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/8 - (Sqrt[Pi ]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/2))/a^4))/(3*c^3) + (4*((-2*x) /(a*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (6*(Sqrt[ArcTan[a*x]]/4 - (Sqrt[P i/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/8))/a^2 + (2*((3*Sqrt[ArcTa n[a*x]])/4 + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/8 + (Sq rt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/2))/a^2))/(3*a*c^3)
3.11.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] + (-Simp[c*((m + 2*q + 2)/(b*(p + 1))) Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 1)) Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x]) /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] & & LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1) Subst[Int[(a + b*x)^p*(Sin[x]^m/ Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p }, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q ] || GtQ[d, 0])
Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.53
| method | result | size |
| default | \(-\frac {-16 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \arctan \left (a x \right )^{\frac {3}{2}}+8 \sin \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\cos \left (4 \arctan \left (a x \right )\right )+1}{12 a^{3} c^{3} \arctan \left (a x \right )^{\frac {3}{2}}}\) | \(68\) |
-1/12/a^3/c^3*(-16*2^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arctan(a*x )^(1/2))*arctan(a*x)^(3/2)+8*sin(4*arctan(a*x))*arctan(a*x)-cos(4*arctan(a *x))+1)/arctan(a*x)^(3/2)
Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {x^{2}}{a^{6} x^{6} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]
Integral(x**2/(a**6*x**6*atan(a*x)**(5/2) + 3*a**4*x**4*atan(a*x)**(5/2) + 3*a**2*x**2*atan(a*x)**(5/2) + atan(a*x)**(5/2)), x)/c**3
Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]