\(\int \frac {1}{x (1-a^2 x^2)^3 \text {arctanh}(a x)^2} \, dx\) [335]

Optimal result

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

-1/a/x/arctanh(a*x)-a*x/(-a^2*x^2+1)^2/arctanh(a*x)-a*x/(-a^2*x^2+1)/arcta 
nh(a*x)+3/2*Chi(2*arctanh(a*x))+1/2*Chi(4*arctanh(a*x))-Defer(Int)(1/x^2/a 
rctanh(a*x),x)/a
 

Mathematica [N/A]

Not integrable

Time = 3.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx \] Input:

Integrate[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]
 

Output:

Integrate[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^2), x]
 

Rubi [N/A]

Not integrable

Time = 2.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 17, 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.

\(\displaystyle \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx\)

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6552

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

\(\Big \downarrow \) 6468

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

\(\Big \downarrow \) 6594

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

\(\Big \downarrow \) 6530

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

\(\displaystyle a^2 \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {1}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )+a^2 \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}+\frac {3}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle a^2 \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\)

\(\Big \downarrow \) 6596

\(\displaystyle a^2 \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (\frac {\int \frac {a^2 x^2}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (\frac {\int -\frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\)

\(\Big \downarrow \) 25

\(\displaystyle a^2 \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (-\frac {\int \frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\)

\(\Big \downarrow \) 3793

\(\displaystyle a^2 \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (-\frac {\int \left (\frac {1}{2 \text {arctanh}(a x)}-\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle a^2 \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}+a^2 \left (\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{a x \text {arctanh}(a x)}\)

\(\Big \downarrow \) 5971

\(\displaystyle a^2 \left (\frac {3 \int \left (\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}-\frac {1}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}+a^2 \left (\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{a x \text {arctanh}(a x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}+a^2 \left (\frac {3 \left (\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))-\frac {1}{8} \log (\text {arctanh}(a x))\right )}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{a x \text {arctanh}(a x)}\)

Input:

Int[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 1.81 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}d x\]

Input:

int(1/x/(-a^2*x^2+1)^3/arctanh(a*x)^2,x)
 

Output:

int(1/x/(-a^2*x^2+1)^3/arctanh(a*x)^2,x)
 

Fricas [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

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

Output:

integral(-1/((a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x)*arctanh(a*x)^2), x)
 

Sympy [N/A]

Not integrable

Time = 2.82 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=- \int \frac {1}{a^{6} x^{7} \operatorname {atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname {atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atanh}^{2}{\left (a x \right )} - x \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \] Input:

integrate(1/x/(-a**2*x**2+1)**3/atanh(a*x)**2,x)
 

Output:

-Integral(1/(a**6*x**7*atanh(a*x)**2 - 3*a**4*x**5*atanh(a*x)**2 + 3*a**2* 
x**3*atanh(a*x)**2 - x*atanh(a*x)**2), x)
 

Maxima [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 6.95 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

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

Output:

-2/((a^5*x^5 - 2*a^3*x^3 + a*x)*log(a*x + 1) - (a^5*x^5 - 2*a^3*x^3 + a*x) 
*log(-a*x + 1)) + integrate(-2*(5*a^2*x^2 - 1)/((a^7*x^8 - 3*a^5*x^6 + 3*a 
^3*x^4 - a*x^2)*log(a*x + 1) - (a^7*x^8 - 3*a^5*x^6 + 3*a^3*x^4 - a*x^2)*l 
og(-a*x + 1)), x)
 

Giac [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

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

Output:

integrate(-1/((a^2*x^2 - 1)^3*x*arctanh(a*x)^2), x)
 

Mupad [N/A]

Not integrable

Time = 3.89 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:

int(-1/(x*atanh(a*x)^2*(a^2*x^2 - 1)^3),x)
 

Output:

-int(1/(x*atanh(a*x)^2*(a^2*x^2 - 1)^3), x)
 

Reduce [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.59 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\left (\int \frac {1}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{7}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{5}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{3}-\mathit {atanh} \left (a x \right )^{2} x}d x \right ) \] Input:

int(1/x/(-a^2*x^2+1)^3/atanh(a*x)^2,x)
 

Output:

 - int(1/(atanh(a*x)**2*a**6*x**7 - 3*atanh(a*x)**2*a**4*x**5 + 3*atanh(a* 
x)**2*a**2*x**3 - atanh(a*x)**2*x),x)