3.7.40 \(\int \frac {1}{x^2 (c+a^2 c x^2)^3 \arctan (a x)^3} \, dx\) [640]

3.7.40.1 Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=-\frac {1}{2 a c^3 x^2 \arctan (a x)^2}+\frac {a}{2 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}+\frac {a}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}-\frac {2 a^2 x}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}-\frac {a^2 x}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {2 a \operatorname {CosIntegral}(2 \arctan (a x))}{c^3}+\frac {a \operatorname {CosIntegral}(4 \arctan (a x))}{c^3}-\frac {\text {Int}\left (\frac {1}{x^3 \arctan (a x)^2},x\right )}{a c^3} \]

output

-1/2/a/c^3/x^2/arctan(a*x)^2+1/2*a/c^3/(a^2*x^2+1)^2/arctan(a*x)^2+1/2*a/c 
^3/(a^2*x^2+1)/arctan(a*x)^2-2*a^2*x/c^3/(a^2*x^2+1)^2/arctan(a*x)-a^2*x/c 
^3/(a^2*x^2+1)/arctan(a*x)+2*a*Ci(2*arctan(a*x))/c^3+a*Ci(4*arctan(a*x))/c 
^3-Unintegrable(1/x^3/arctan(a*x)^2,x)/a/c^3
 

3.7.40.2 Mathematica [N/A]

Not integrable

Time = 3.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx \]

input

Integrate[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]
 

output

Integrate[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]
 

3.7.40.3 Rubi [N/A]

Not integrable

Time = 2.60 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5501, 27, 5437, 5501, 5437, 5461, 5377, 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.

\(\displaystyle \int \frac {1}{x^2 \arctan (a x)^3 \left (a^2 c x^2+c\right )^3} \, dx\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\int \frac {1}{c^2 x^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^3}dx}{c}-a^2 \int \frac {1}{c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^3}dx}{c^3}-\frac {a^2 \int \frac {1}{\left (a^2 x^2+1\right )^3 \arctan (a x)^3}dx}{c^3}\)

\(\Big \downarrow \) 5437

\(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^3}dx}{c^3}-\frac {a^2 \left (-2 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 x^2+1\right ) \arctan (a x)^3}dx-a^2 \int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)^3}dx}{c^3}-\frac {a^2 \left (-2 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 5437

\(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 x^2+1\right ) \arctan (a x)^3}dx-a^2 \left (-a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^3}-\frac {a^2 \left (-2 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 5461

\(\displaystyle \frac {-\left (a^2 \left (-a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 5377

\(\displaystyle \frac {-\left (a^2 \left (-a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 5503

\(\displaystyle \frac {-\left (a^2 \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{a}-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx}{a}-3 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 5439

\(\displaystyle \frac {-\left (a^2 \left (-a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \frac {1}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (-3 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx+\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\left (a^2 \left (-a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^2}{\arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (-3 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx+\frac {\int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^4}{\arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {-\left (a^2 \left (-a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \left (\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}+\frac {1}{2 \arctan (a x)}\right )d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (-3 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx+\frac {\int \left (\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}+\frac {\cos (4 \arctan (a x))}{8 \arctan (a x)}+\frac {3}{8 \arctan (a x)}\right )d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\left (a^2 \left (-a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (-3 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 5505

\(\displaystyle \frac {-\left (a^2 \left (-a \left (-\frac {\int \frac {a^2 x^2}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (-\frac {3 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\left (a^2 \left (-a \left (-\frac {\int \frac {\sin (\arctan (a x))^2}{\arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (-\frac {3 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {-\left (a^2 \left (-a \left (-\frac {\int \left (\frac {1}{2 \arctan (a x)}-\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}\right )d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (-\frac {3 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\left (a^2 \left (-a \left (-\frac {\frac {1}{2} \log (\arctan (a x))-\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (-\frac {3 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\left (a^2 \left (-a \left (-\frac {\frac {1}{2} \log (\arctan (a x))-\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (-\frac {3 \int \left (\frac {1}{8 \arctan (a x)}-\frac {\cos (4 \arctan (a x))}{8 \arctan (a x)}\right )d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\left (a^2 \left (-a \left (-\frac {\frac {1}{2} \log (\arctan (a x))-\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )\right )-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^3}-\frac {a^2 \left (-2 a \left (-\frac {3 \left (\frac {1}{8} \log (\arctan (a x))-\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))\right )}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right )^2 \arctan (a x)^2}\right )}{c^3}\)

input

Int[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]
 

output

$Aborted
 

3.7.40.3.1 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 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 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 5377

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Sy 
mbol] :> Unintegrable[(d*x)^m*(a + b*ArcTan[c*x^n])^p, x] /; FreeQ[{a, b, c 
, d, m, n, p}, x]
 

rule 5437

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S 
ymbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 
 1))), x] - Simp[2*c*((q + 1)/(b*(p + 1)))   Int[x*(d + e*x^2)^q*(a + b*Arc 
Tan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 
LtQ[q, -1] && LtQ[p, -1]
 

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 5461

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_))/((d_) + (e_ 
.)*(x_)^2), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*( 
p + 1))), x] - Simp[f*(m/(b*c*d*(p + 1)))   Int[(f*x)^(m - 1)*(a + b*ArcTan 
[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] 
&& LtQ[p, -1]
 

rule 5501

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c 
*x])^p, x], x] - Simp[e/d   Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) 
^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* 
q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
 

rule 5503

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]
 

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])
 

3.7.40.4 Maple [N/A] (verified)

Not integrable

Time = 7.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{3}}d x\]

input

int(1/x^2/(a^2*c*x^2+c)^3/arctan(a*x)^3,x)
 

output

int(1/x^2/(a^2*c*x^2+c)^3/arctan(a*x)^3,x)
 

3.7.40.5 Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{3}} \,d x } \]

input

integrate(1/x^2/(a^2*c*x^2+c)^3/arctan(a*x)^3,x, algorithm="fricas")
 

output

integral(1/((a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2)*arctan 
(a*x)^3), x)
 

3.7.40.6 Sympy [N/A]

Not integrable

Time = 2.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\frac {\int \frac {1}{a^{6} x^{8} \operatorname {atan}^{3}{\left (a x \right )} + 3 a^{4} x^{6} \operatorname {atan}^{3}{\left (a x \right )} + 3 a^{2} x^{4} \operatorname {atan}^{3}{\left (a x \right )} + x^{2} \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{3}} \]

input

integrate(1/x**2/(a**2*c*x**2+c)**3/atan(a*x)**3,x)
 

output

Integral(1/(a**6*x**8*atan(a*x)**3 + 3*a**4*x**6*atan(a*x)**3 + 3*a**2*x** 
4*atan(a*x)**3 + x**2*atan(a*x)**3), x)/c**3
 

3.7.40.7 Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 175, normalized size of antiderivative = 7.95 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{3}} \,d x } \]

input

integrate(1/x^2/(a^2*c*x^2+c)^3/arctan(a*x)^3,x, algorithm="maxima")
 

output

1/2*(2*(a^6*c^3*x^7 + 2*a^4*c^3*x^5 + a^2*c^3*x^3)*arctan(a*x)^2*integrate 
((15*a^4*x^4 + 10*a^2*x^2 + 3)/((a^8*c^3*x^10 + 3*a^6*c^3*x^8 + 3*a^4*c^3* 
x^6 + a^2*c^3*x^4)*arctan(a*x)), x) - a*x + 2*(3*a^2*x^2 + 1)*arctan(a*x)) 
/((a^6*c^3*x^7 + 2*a^4*c^3*x^5 + a^2*c^3*x^3)*arctan(a*x)^2)
 

3.7.40.8 Giac [N/A]

Not integrable

Time = 187.20 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{3}} \,d x } \]

input

integrate(1/x^2/(a^2*c*x^2+c)^3/arctan(a*x)^3,x, algorithm="giac")
 

output

sage0*x
 

3.7.40.9 Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

input

int(1/(x^2*atan(a*x)^3*(c + a^2*c*x^2)^3),x)
 

output

int(1/(x^2*atan(a*x)^3*(c + a^2*c*x^2)^3), x)