3.7.41 \(\int \frac {1}{x^3 (c+a^2 c x^2)^3 \arctan (a x)^3} \, dx\) [641]

3.7.41.1 Optimal result

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

output

-1/2/a/c^3/x^3/arctan(a*x)^2+a/c^3/x/arctan(a*x)^2-1/2*a^3*x/c^3/(a^2*x^2+ 
1)^2/arctan(a*x)^2-a^3*x/c^3/(a^2*x^2+1)/arctan(a*x)^2-2*a^2/c^3/(a^2*x^2+ 
1)^2/arctan(a*x)+3/2*a^2/c^3/(a^2*x^2+1)/arctan(a*x)-a^2*(-a^2*x^2+1)/c^3/ 
(a^2*x^2+1)/arctan(a*x)-5/2*a^2*Si(2*arctan(a*x))/c^3-a^2*Si(4*arctan(a*x) 
)/c^3-3/2*Unintegrable(1/x^4/arctan(a*x)^2,x)/a/c^3+a*Unintegrable(1/x^2/a 
rctan(a*x)^2,x)/c^3
 

3.7.41.2 Mathematica [N/A]

Not integrable

Time = 3.92 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx \]

input

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

output

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

3.7.41.3 Rubi [N/A]

Not integrable

Time = 4.03 (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 = {5501, 27, 5501, 5461, 5377, 5501, 5461, 5377, 5467, 5503, 5437, 5499, 5437, 5505, 4906, 27, 2009, 3042, 3780}

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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 5461

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

\(\Big \downarrow \) 5377

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 5461

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

\(\Big \downarrow \) 5377

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

\(\Big \downarrow \) 5467

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

\(\Big \downarrow \) 5503

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

\(\Big \downarrow \) 5437

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

\(\Big \downarrow \) 5499

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

\(\Big \downarrow \) 5437

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

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

input

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

output

$Aborted
 

3.7.41.3.1 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 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

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 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 5467

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*(x_))/((d_) + (e_.)*(x_)^2)^2, 
 x_Symbol] :> Simp[x*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2 
))), x] + (-Simp[(1 - c^2*x^2)*((a + b*ArcTan[c*x])^(p + 2)/(b^2*e*(p + 1)* 
(p + 2)*(d + e*x^2))), x] - Simp[4/(b^2*(p + 1)*(p + 2))   Int[x*((a + b*Ar 
cTan[c*x])^(p + 2)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && 
EqQ[e, c^2*d] && LtQ[p, -1] && NeQ[p, -2]
 

rule 5499

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/e   Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar 
cTan[c*x])^p, x], x] - Simp[d/e   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] && IGtQ[m, 1] && NeQ[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])
 

3.7.41.4 Maple [N/A] (verified)

Not integrable

Time = 84.94 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

input

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

output

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

3.7.41.5 Fricas [N/A]

Not integrable

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

input

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

3.7.41.6 Sympy [N/A]

Not integrable

Time = 2.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\frac {\int \frac {1}{a^{6} x^{9} \operatorname {atan}^{3}{\left (a x \right )} + 3 a^{4} x^{7} \operatorname {atan}^{3}{\left (a x \right )} + 3 a^{2} x^{5} \operatorname {atan}^{3}{\left (a x \right )} + x^{3} \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{3}} \]

input

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

output

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

3.7.41.7 Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 174, normalized size of antiderivative = 7.91 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3}} \,d x } \]

input

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

output

1/2*(2*(a^6*c^3*x^8 + 2*a^4*c^3*x^6 + a^2*c^3*x^4)*arctan(a*x)^2*integrate 
((21*a^4*x^4 + 19*a^2*x^2 + 6)/((a^8*c^3*x^11 + 3*a^6*c^3*x^9 + 3*a^4*c^3* 
x^7 + a^2*c^3*x^5)*arctan(a*x)), x) - a*x + (7*a^2*x^2 + 3)*arctan(a*x))/( 
(a^6*c^3*x^8 + 2*a^4*c^3*x^6 + a^2*c^3*x^4)*arctan(a*x)^2)
 

3.7.41.8 Giac [N/A]

Not integrable

Time = 184.69 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3}} \,d x } \]

input

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

output

sage0*x
 

3.7.41.9 Mupad [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

input

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

output

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