\(\int \frac {\arctan (a x)^{5/2}}{(c+a^2 c x^2)^3} \, dx\) [875]

Optimal result

Integrand size = 21, antiderivative size = 296 \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {45 x \sqrt {\arctan (a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac {75 \arctan (a x)^{3/2}}{256 a c^3}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a c^3}+\frac {15 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{256 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{2048 a c^3} \] Output:

-45/128*x*arctan(a*x)^(1/2)/c^3/(a^2*x^2+1)-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*2^(1/2)*Pi^(1/2)* 
FresnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))/a/c^3+15/128*Pi^(1/2)*Fresn 
elS(2*arctan(a*x)^(1/2)/Pi^(1/2))/a/c^3-15/256*arctan(a*x)^(1/2)*sin(2*arc 
tan(a*x))/a/c^3-15/2048*arctan(a*x)^(1/2)*sin(4*arctan(a*x))/a/c^3
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.55 \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\frac {16 \sqrt {\arctan (a x)} \left (-105 a x \left (17+15 a^2 x^2\right )-70 \left (-17+6 a^2 x^2+15 a^4 x^4\right ) \arctan (a x)+448 a x \left (5+3 a^2 x^2\right ) \arctan (a x)^2+384 \left (1+a^2 x^2\right )^2 \arctan (a x)^3\right )}{\left (1+a^2 x^2\right )^2}+105 \sqrt {2 \pi } \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+6720 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{57344 a c^3} \] Input:

Integrate[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^3,x]
 

Output:

((16*Sqrt[ArcTan[a*x]]*(-105*a*x*(17 + 15*a^2*x^2) - 70*(-17 + 6*a^2*x^2 + 
 15*a^4*x^4)*ArcTan[a*x] + 448*a*x*(5 + 3*a^2*x^2)*ArcTan[a*x]^2 + 384*(1 
+ a^2*x^2)^2*ArcTan[a*x]^3))/(1 + a^2*x^2)^2 + 105*Sqrt[2*Pi]*FresnelS[2*S 
qrt[2/Pi]*Sqrt[ArcTan[a*x]]] + 6720*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]] 
)/Sqrt[Pi]])/(57344*a*c^3)
 

Rubi [A] (verified)

Time = 1.73 (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

\(\displaystyle \frac {3 \left (-\frac {5}{4} a \left (\frac {3 \left (\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a}-\frac {\arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^{5/2}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{7/2}}{7 a}\right )}{4 c^3}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {15 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}+\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{64 a c^3}\)

Input:

Int[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^3,x]
 

Output:

(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)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3786

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]
 

rule 3793

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]))
 

rule 3832

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]
 

rule 4906

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]
 

rule 5427

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]
 

rule 5435

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]
 

rule 5439

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])
 

rule 5465

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]
 

rule 5505

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])
 

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.57

\[\frac {6144 \arctan \left (a x \right )^{\frac {7}{2}} \sqrt {\pi }+14336 \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \sin \left (2 \arctan \left (a x \right )\right )+1792 \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \sin \left (4 \arctan \left (a x \right )\right )+17920 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \cos \left (2 \arctan \left (a x \right )\right )+1120 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \cos \left (4 \arctan \left (a x \right )\right )+105 \pi \sqrt {2}\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-13440 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \sin \left (2 \arctan \left (a x \right )\right )-420 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \sin \left (4 \arctan \left (a x \right )\right )+6720 \pi \,\operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )}{57344 a \,c^{3} \sqrt {\pi }}\]

Input:

int(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,x)
 

Output:

1/57344/a/c^3/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))+17920*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))+105*Pi*2^(1/2)*FresnelS(2*2^(1/2)/ 
Pi^(1/2)*arctan(a*x)^(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))+6720*Pi*FresnelS(2 
*arctan(a*x)^(1/2)/Pi^(1/2)))
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \] Input:

integrate(atan(a*x)**(5/2)/(a**2*c*x**2+c)**3,x)
 

Output:

Integral(atan(a*x)**(5/2)/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/ 
c**3
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

\[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \] Input:

integrate(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")
 

Output:

integrate(arctan(a*x)^(5/2)/(a^2*c*x^2 + c)^3, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \] Input:

int(atan(a*x)^(5/2)/(c + a^2*c*x^2)^3,x)
 

Output:

int(atan(a*x)^(5/2)/(c + a^2*c*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {96 \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right )^{3} a^{4} x^{4}+192 \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right )^{3} a^{2} x^{2}+96 \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right )^{3}+336 \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}+560 \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right )^{2} a x +420 \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right ) a^{2} x^{2}+560 \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right )+210 \sqrt {\mathit {atan} \left (a x \right )}\, a x -1050 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{5} x^{4}-2100 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{3} x^{2}-1050 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a -105 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{6} x^{4}-210 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{4} x^{2}-105 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{2}}{896 a \,c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:

int(atan(a*x)^(5/2)/(a^2*c*x^2+c)^3,x)
 

Output:

(96*sqrt(atan(a*x))*atan(a*x)**3*a**4*x**4 + 192*sqrt(atan(a*x))*atan(a*x) 
**3*a**2*x**2 + 96*sqrt(atan(a*x))*atan(a*x)**3 + 336*sqrt(atan(a*x))*atan 
(a*x)**2*a**3*x**3 + 560*sqrt(atan(a*x))*atan(a*x)**2*a*x + 420*sqrt(atan( 
a*x))*atan(a*x)*a**2*x**2 + 560*sqrt(atan(a*x))*atan(a*x) + 210*sqrt(atan( 
a*x))*a*x - 1050*int(sqrt(atan(a*x))/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x** 
2 + 1),x)*a**5*x**4 - 2100*int(sqrt(atan(a*x))/(a**6*x**6 + 3*a**4*x**4 + 
3*a**2*x**2 + 1),x)*a**3*x**2 - 1050*int(sqrt(atan(a*x))/(a**6*x**6 + 3*a* 
*4*x**4 + 3*a**2*x**2 + 1),x)*a - 105*int((sqrt(atan(a*x))*x)/(atan(a*x)*a 
**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a 
**6*x**4 - 210*int((sqrt(atan(a*x))*x)/(atan(a*x)*a**6*x**6 + 3*atan(a*x)* 
a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**4*x**2 - 105*int((sqr 
t(atan(a*x))*x)/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x) 
*a**2*x**2 + atan(a*x)),x)*a**2)/(896*a*c**3*(a**4*x**4 + 2*a**2*x**2 + 1) 
)