Integrand size = 21, antiderivative size = 296 \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {45 x \sqrt {\arctan (a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac {75 \arctan (a x)^{3/2}}{256 a c^3}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a c^3}+\frac {15 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{256 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{2048 a c^3} \]
-75/256*arctan(a*x)^(3/2)/a/c^3+5/32*arctan(a*x)^(3/2)/a/c^3/(a^2*x^2+1)^2 +15/32*arctan(a*x)^(3/2)/a/c^3/(a^2*x^2+1)+1/4*x*arctan(a*x)^(5/2)/c^3/(a^ 2*x^2+1)^2+3/8*x*arctan(a*x)^(5/2)/c^3/(a^2*x^2+1)+3/28*arctan(a*x)^(7/2)/ a/c^3+15/8192*FresnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1 /2)/a/c^3+15/128*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a/c^3-45/ 128*x*arctan(a*x)^(1/2)/c^3/(a^2*x^2+1)-15/256*sin(2*arctan(a*x))*arctan(a *x)^(1/2)/a/c^3-15/2048*sin(4*arctan(a*x))*arctan(a*x)^(1/2)/a/c^3
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.97 \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {57120 a x \arctan (a x)+50400 a^3 x^3 \arctan (a x)-38080 \arctan (a x)^2+13440 a^2 x^2 \arctan (a x)^2+33600 a^4 x^4 \arctan (a x)^2-71680 a x \arctan (a x)^3-43008 a^3 x^3 \arctan (a x)^3-12288 \left (1+a^2 x^2\right )^2 \arctan (a x)^4+3360 \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+3360 \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+105 \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+105 \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{114688 a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}} \]
-1/114688*(57120*a*x*ArcTan[a*x] + 50400*a^3*x^3*ArcTan[a*x] - 38080*ArcTa n[a*x]^2 + 13440*a^2*x^2*ArcTan[a*x]^2 + 33600*a^4*x^4*ArcTan[a*x]^2 - 716 80*a*x*ArcTan[a*x]^3 - 43008*a^3*x^3*ArcTan[a*x]^3 - 12288*(1 + a^2*x^2)^2 *ArcTan[a*x]^4 + 3360*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma [1/2, (-2*I)*ArcTan[a*x]] + 3360*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x ]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + 105*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a* x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + 105*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x ]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(a*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]] )
Time = 1.61 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.11, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5435, 27, 5427, 5439, 3042, 3793, 2009, 5465, 5427, 5505, 4906, 27, 3042, 3786, 3832}
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.
| \(\displaystyle \int \frac {\arctan (a x)^{5/2}}{\left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5435 |
| \(\displaystyle -\frac {15}{64} \int \frac {\sqrt {\arctan (a x)}}{c^3 \left (a^2 x^2+1\right )^3}dx+\frac {3 \int \frac {\arctan (a x)^{5/2}}{c^2 \left (a^2 x^2+1\right )^2}dx}{4 c}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {15 \int \frac {\sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^3}dx}{64 c^3}+\frac {3 \int \frac {\arctan (a x)^{5/2}}{\left (a^2 x^2+1\right )^2}dx}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle -\frac {15 \int \frac {\sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^3}dx}{64 c^3}+\frac {3 \left (-\frac {5}{4} a \int \frac {x \arctan (a x)^{3/2}}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 5439 |
| \(\displaystyle -\frac {15 \int \frac {\sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^2}d\arctan (a x)}{64 a c^3}+\frac {3 \left (-\frac {5}{4} a \int \frac {x \arctan (a x)^{3/2}}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \int \frac {x \arctan (a x)^{3/2}}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}-\frac {15 \int \sqrt {\arctan (a x)} \sin \left (\arctan (a x)+\frac {\pi }{2}\right )^4d\arctan (a x)}{64 a c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \int \frac {x \arctan (a x)^{3/2}}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}-\frac {15 \int \left (\frac {1}{2} \sqrt {\arctan (a x)} \cos (2 \arctan (a x))+\frac {1}{8} \sqrt {\arctan (a x)} \cos (4 \arctan (a x))+\frac {3}{8} \sqrt {\arctan (a x)}\right )d\arctan (a x)}{64 a c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \int \frac {x \arctan (a x)^{3/2}}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {15 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}+\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{64 a c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \left (\frac {3 \int \frac {\sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx}{4 a}-\frac {\arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {15 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}+\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{64 a c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \left (\frac {3 \left (-\frac {1}{4} a \int \frac {x}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a}-\frac {\arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {15 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}+\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{64 a c^3}\) |
| \(\Big \downarrow \) 5505 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \left (\frac {3 \left (-\frac {\int \frac {a x}{\left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}d\arctan (a x)}{4 a}+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a}-\frac {\arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {15 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}+\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{64 a c^3}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \left (\frac {3 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{2 \sqrt {\arctan (a x)}}d\arctan (a x)}{4 a}+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a}-\frac {\arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {15 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}+\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{64 a c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \left (\frac {3 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{\sqrt {\arctan (a x)}}d\arctan (a x)}{8 a}+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a}-\frac {\arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {15 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}+\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{64 a c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \left (\frac {3 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{\sqrt {\arctan (a x)}}d\arctan (a x)}{8 a}+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a}-\frac {\arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {15 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}+\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{64 a c^3}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \left (\frac {3 \left (-\frac {\int \sin (2 \arctan (a x))d\sqrt {\arctan (a x)}}{4 a}+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a}-\frac {\arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {15 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}+\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{64 a c^3}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {3 \left (-\frac {5}{4} a \left (\frac {3 \left (\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a}-\frac {\arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {15 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}+\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{64 a c^3}\) |
(5*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) + (x*ArcTan[a*x]^(5/2))/( 4*c^3*(1 + a^2*x^2)^2) + (3*((x*ArcTan[a*x]^(5/2))/(2*(1 + a^2*x^2)) + Arc Tan[a*x]^(7/2)/(7*a) - (5*a*(-1/2*ArcTan[a*x]^(3/2)/(a^2*(1 + a^2*x^2)) + (3*((x*Sqrt[ArcTan[a*x]])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^(3/2)/(3*a) - (S qrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a)))/(4*a)))/4))/(4*c ^3) - (15*(ArcTan[a*x]^(3/2)/4 - (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[Ar cTan[a*x]]])/64 - (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/8 + (Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/4 + (Sqrt[ArcTan[a*x]]*Sin[4*ArcTan [a*x]])/32))/(64*a*c^3)
3.9.76.3.1 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[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
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[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
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)^2, x_Sym bol] :> Simp[x*((a + b*ArcTan[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b *ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2) Int[x*((a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S ymbol] :> Simp[b*p*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d* (q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d* (q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int[(d + e *x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] & & EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
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_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
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 = 2.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.57
\[\frac {6144 \arctan \left (a x \right )^{\frac {7}{2}} \sqrt {\pi }+14336 \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \sin \left (2 \arctan \left (a x \right )\right )+1792 \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \sin \left (4 \arctan \left (a x \right )\right )+105 \pi \sqrt {2}\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+17920 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \cos \left (2 \arctan \left (a x \right )\right )+1120 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \cos \left (4 \arctan \left (a x \right )\right )+6720 \pi \,\operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-13440 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \sin \left (2 \arctan \left (a x \right )\right )-420 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \sin \left (4 \arctan \left (a x \right )\right )}{57344 c^{3} a \sqrt {\pi }}\]
1/57344/c^3/a/Pi^(1/2)*(6144*arctan(a*x)^(7/2)*Pi^(1/2)+14336*arctan(a*x)^ (5/2)*Pi^(1/2)*sin(2*arctan(a*x))+1792*arctan(a*x)^(5/2)*Pi^(1/2)*sin(4*ar ctan(a*x))+105*Pi*2^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))+1 7920*arctan(a*x)^(3/2)*Pi^(1/2)*cos(2*arctan(a*x))+1120*arctan(a*x)^(3/2)* Pi^(1/2)*cos(4*arctan(a*x))+6720*Pi*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2)) -13440*arctan(a*x)^(1/2)*Pi^(1/2)*sin(2*arctan(a*x))-420*arctan(a*x)^(1/2) *Pi^(1/2)*sin(4*arctan(a*x)))
Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]