\(\int \frac {(1-a^2 x^2)^2 \text {arctanh}(a x)}{x} \, dx\) [197]

Optimal result

Integrand size = 20, antiderivative size = 70 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x} \, dx=-\frac {3 a x}{4}+\frac {a^3 x^3}{12}+\frac {3}{4} \text {arctanh}(a x)-a^2 x^2 \text {arctanh}(a x)+\frac {1}{4} a^4 x^4 \text {arctanh}(a x)-\frac {\operatorname {PolyLog}(2,-a x)}{2}+\frac {\operatorname {PolyLog}(2,a x)}{2} \] Output:

-3/4*a*x+1/12*a^3*x^3+3/4*arctanh(a*x)-a^2*x^2*arctanh(a*x)+1/4*a^4*x^4*ar 
ctanh(a*x)-1/2*polylog(2,-a*x)+1/2*polylog(2,a*x)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.17 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x} \, dx=-\frac {3 a x}{4}+\frac {a^3 x^3}{12}-a^2 x^2 \text {arctanh}(a x)+\frac {1}{4} a^4 x^4 \text {arctanh}(a x)-\frac {3}{8} \log (1-a x)+\frac {3}{8} \log (1+a x)+\frac {1}{2} (-\operatorname {PolyLog}(2,-a x)+\operatorname {PolyLog}(2,a x)) \] Input:

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

Output:

(-3*a*x)/4 + (a^3*x^3)/12 - a^2*x^2*ArcTanh[a*x] + (a^4*x^4*ArcTanh[a*x])/ 
4 - (3*Log[1 - a*x])/8 + (3*Log[1 + a*x])/8 + (-PolyLog[2, -(a*x)] + PolyL 
og[2, a*x])/2
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6574, 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.

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

\(\Big \downarrow \) 6574

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} a^4 x^4 \text {arctanh}(a x)+\frac {a^3 x^3}{12}-a^2 x^2 \text {arctanh}(a x)+\frac {3}{4} \text {arctanh}(a x)-\frac {\operatorname {PolyLog}(2,-a x)}{2}+\frac {\operatorname {PolyLog}(2,a x)}{2}-\frac {3 a x}{4}\)

Input:

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

Output:

(-3*a*x)/4 + (a^3*x^3)/12 + (3*ArcTanh[a*x])/4 - a^2*x^2*ArcTanh[a*x] + (a 
^4*x^4*ArcTanh[a*x])/4 - PolyLog[2, -(a*x)]/2 + PolyLog[2, a*x]/2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6574

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(q_), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a 
+ b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d 
 + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{4}-a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {a^{3} x^{3}}{12}-\frac {3 a x}{4}-\frac {3 \ln \left (a x -1\right )}{8}+\frac {3 \ln \left (a x +1\right )}{8}\)	\(89\)
default	\(\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{4}-a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {a^{3} x^{3}}{12}-\frac {3 a x}{4}-\frac {3 \ln \left (a x -1\right )}{8}+\frac {3 \ln \left (a x +1\right )}{8}\)	\(89\)
parts	\(\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{4}-a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+\operatorname {arctanh}\left (a x \right ) \ln \left (x \right )-\frac {a \left (\frac {2 \operatorname {dilog}\left (a x +1\right )}{a}+\frac {2 \ln \left (x \right ) \ln \left (a x +1\right )}{a}-\frac {2 \left (\ln \left (x \right )-\ln \left (a x \right )\right ) \ln \left (-a x +1\right )}{a}+\frac {2 \operatorname {dilog}\left (a x \right )}{a}-\frac {a^{2} x^{3}}{3}+3 x -\frac {3 \ln \left (a x +1\right )}{2 a}+\frac {3 \ln \left (a x -1\right )}{2 a}\right )}{4}\)	\(124\)
risch	\(\frac {\left (a x +1\right )^{4} \ln \left (a x +1\right )}{8}+\frac {a^{3} x^{3}}{12}-\frac {3 a x}{4}-\frac {\left (a x +1\right )^{3} \ln \left (a x +1\right )}{2}+\frac {\left (a x +1\right )^{2} \ln \left (a x +1\right )}{4}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\left (-a x +1\right )^{4} \ln \left (-a x +1\right )}{8}+\frac {\left (-a x +1\right )^{3} \ln \left (-a x +1\right )}{2}-\frac {\left (-a x +1\right )^{2} \ln \left (-a x +1\right )}{4}-\frac {\left (-a x +1\right ) \ln \left (-a x +1\right )}{2}+\frac {\operatorname {dilog}\left (-a x +1\right )}{2}\)	\(155\)
meijerg	\(-\frac {i \left (\frac {2 i a x \operatorname {polylog}\left (2, \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-\frac {2 i a x \operatorname {polylog}\left (2, -\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}\right )}{4}-\frac {i \left (\frac {i x a \left (5 a^{2} x^{2}+15\right )}{15}+\frac {i x a \left (-5 a^{4} x^{4}+5\right ) \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{10 \sqrt {a^{2} x^{2}}}\right )}{4}-\frac {i \left (-2 i x a +2 i \left (-a x +1\right ) \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )\right )}{2}\)	\(157\)

Input:

int((-a^2*x^2+1)^2*arctanh(a*x)/x,x,method=_RETURNVERBOSE)
 

Output:

1/4*a^4*x^4*arctanh(a*x)-a^2*x^2*arctanh(a*x)+arctanh(a*x)*ln(a*x)-1/2*dil 
og(a*x)-1/2*dilog(a*x+1)-1/2*ln(a*x)*ln(a*x+1)+1/12*a^3*x^3-3/4*a*x-3/8*ln 
(a*x-1)+3/8*ln(a*x+1)
 

Fricas [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )}{x} \,d x } \] Input:

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

Output:

integral((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)/x, x)
 

Sympy [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x} \, dx=\int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}{\left (a x \right )}}{x}\, dx \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.51 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x} \, dx=\frac {1}{24} \, {\left (2 \, a^{2} x^{3} - 18 \, x - \frac {12 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {12 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a} + \frac {9 \, \log \left (a x + 1\right )}{a} - \frac {9 \, \log \left (a x - 1\right )}{a}\right )} a + \frac {1}{4} \, {\left (a^{4} x^{4} - 4 \, a^{2} x^{2} + 2 \, \log \left (x^{2}\right )\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

1/24*(2*a^2*x^3 - 18*x - 12*(log(a*x + 1)*log(x) + dilog(-a*x))/a + 12*(lo 
g(-a*x + 1)*log(x) + dilog(a*x))/a + 9*log(a*x + 1)/a - 9*log(a*x - 1)/a)* 
a + 1/4*(a^4*x^4 - 4*a^2*x^2 + 2*log(x^2))*arctanh(a*x)
 

Giac [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )}{x} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^2}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x} \, dx=\frac {\mathit {atanh} \left (a x \right ) a^{4} x^{4}}{4}-\mathit {atanh} \left (a x \right ) a^{2} x^{2}+\frac {3 \mathit {atanh} \left (a x \right )}{4}+\int \frac {\mathit {atanh} \left (a x \right )}{x}d x +\frac {a^{3} x^{3}}{12}-\frac {3 a x}{4} \] Input:

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

Output:

(3*atanh(a*x)*a**4*x**4 - 12*atanh(a*x)*a**2*x**2 + 9*atanh(a*x) + 12*int( 
atanh(a*x)/x,x) + a**3*x**3 - 9*a*x)/12