\(\int \frac {1}{x^2 (c+a^2 c x^2)^2 \arctan (a x)^3} \, dx\) [631]

Optimal result

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

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

Mathematica [N/A]

Not integrable

Time = 1.91 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 15, 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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5437

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

\(\Big \downarrow \) 5461

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

\(\Big \downarrow \) 5377

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

\(\Big \downarrow \) 5503

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

\(\Big \downarrow \) 5439

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 1.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{3}} \,d x } \] Input:

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

Output:

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

Sympy [N/A]

Not integrable

Time = 1.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {\int \frac {1}{a^{4} x^{6} \operatorname {atan}^{3}{\left (a x \right )} + 2 a^{2} x^{4} \operatorname {atan}^{3}{\left (a x \right )} + x^{2} \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \] Input:

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

Output:

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

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 6.45 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{3}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

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

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

Output:

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

Mupad [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {\int \frac {1}{\mathit {atan} \left (a x \right )^{3} a^{4} x^{6}+2 \mathit {atan} \left (a x \right )^{3} a^{2} x^{4}+\mathit {atan} \left (a x \right )^{3} x^{2}}d x}{c^{2}} \] Input:

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

Output:

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