\(\int \frac {1}{x^3 (c+a^2 c x^2)^3 \arctan (a x)^2} \, dx\) [564]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=-\frac {1}{a c^3 x^3 \arctan (a x)}+\frac {2 a}{c^3 x \arctan (a x)}-\frac {a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}-\frac {2 a^3 x}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {5 a^2 \operatorname {CosIntegral}(2 \arctan (a x))}{2 c^3}+\frac {a^2 \operatorname {CosIntegral}(4 \arctan (a x))}{2 c^3}-\frac {3 \text {Int}\left (\frac {1}{x^4 \arctan (a x)},x\right )}{a c^3}+\frac {2 a \text {Int}\left (\frac {1}{x^2 \arctan (a x)},x\right )}{c^3} \] Output:

-1/a/c^3/x^3/arctan(a*x)+2*a/c^3/x/arctan(a*x)-a^3*x/c^3/(a^2*x^2+1)^2/arc 
tan(a*x)-2*a^3*x/c^3/(a^2*x^2+1)/arctan(a*x)+5/2*a^2*Ci(2*arctan(a*x))/c^3 
+1/2*a^2*Ci(4*arctan(a*x))/c^3-3*Defer(Int)(1/x^4/arctan(a*x),x)/a/c^3+2*a 
*Defer(Int)(1/x^2/arctan(a*x),x)/c^3
 

Mathematica [N/A]

Not integrable

Time = 2.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 4.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 5461

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

\(\Big \downarrow \) 5377

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 5461

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

\(\Big \downarrow \) 5377

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

\(\Big \downarrow \) 5503

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

\(\Big \downarrow \) 5439

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 60.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{2}} \,d x } \] Input:

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

Output:

integral(1/((a^6*c^3*x^9 + 3*a^4*c^3*x^7 + 3*a^2*c^3*x^5 + c^3*x^3)*arctan 
(a*x)^2), x)
 

Sympy [N/A]

Not integrable

Time = 1.56 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {\int \frac {1}{a^{6} x^{9} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{7} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{5} \operatorname {atan}^{2}{\left (a x \right )} + x^{3} \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \] Input:

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

Output:

Integral(1/(a**6*x**9*atan(a*x)**2 + 3*a**4*x**7*atan(a*x)**2 + 3*a**2*x** 
5*atan(a*x)**2 + x**3*atan(a*x)**2), x)/c**3
 

Maxima [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 139, normalized size of antiderivative = 6.32 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{2}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{2}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {\int \frac {1}{\mathit {atan} \left (a x \right )^{2} a^{6} x^{9}+3 \mathit {atan} \left (a x \right )^{2} a^{4} x^{7}+3 \mathit {atan} \left (a x \right )^{2} a^{2} x^{5}+\mathit {atan} \left (a x \right )^{2} x^{3}}d x}{c^{3}} \] Input:

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

Output:

int(1/(atan(a*x)**2*a**6*x**9 + 3*atan(a*x)**2*a**4*x**7 + 3*atan(a*x)**2* 
a**2*x**5 + atan(a*x)**2*x**3),x)/c**3