\(\int \frac {(1-a^2 x^2)^2 \text {arctanh}(a x)}{x^3} \, dx\) [199]

Optimal result

Integrand size = 20, antiderivative size = 62 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^3} \, dx=-\frac {a}{2 x}+\frac {a^3 x}{2}-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a^4 x^2 \text {arctanh}(a x)+a^2 \operatorname {PolyLog}(2,-a x)-a^2 \operatorname {PolyLog}(2,a x) \] Output:

-1/2*a/x+1/2*a^3*x-1/2*arctanh(a*x)/x^2+1/2*a^4*x^2*arctanh(a*x)+a^2*polyl 
og(2,-a*x)-a^2*polylog(2,a*x)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^3} \, dx=-\frac {a}{2 x}+\frac {a^3 x}{2}-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a^4 x^2 \text {arctanh}(a x)-a^2 (-\operatorname {PolyLog}(2,-a x)+\operatorname {PolyLog}(2,a x)) \] Input:

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

Output:

-1/2*a/x + (a^3*x)/2 - ArcTanh[a*x]/(2*x^2) + (a^4*x^2*ArcTanh[a*x])/2 - a 
^2*(-PolyLog[2, -(a*x)] + PolyLog[2, a*x])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 62, 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.

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

\(\Big \downarrow \) 6574

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*a/x + (a^3*x)/2 - ArcTanh[a*x]/(2*x^2) + (a^4*x^2*ArcTanh[a*x])/2 + a 
^2*PolyLog[2, -(a*x)] - a^2*PolyLog[2, a*x]
 

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 = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(a^{2} \left (\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{2}-2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{2 a^{2} x^{2}}+\operatorname {dilog}\left (a x \right )+\operatorname {dilog}\left (a x +1\right )+\ln \left (a x \right ) \ln \left (a x +1\right )+\frac {a x}{2}-\frac {1}{2 a x}\right )\)	\(73\)
default	\(a^{2} \left (\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{2}-2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{2 a^{2} x^{2}}+\operatorname {dilog}\left (a x \right )+\operatorname {dilog}\left (a x +1\right )+\ln \left (a x \right ) \ln \left (a x +1\right )+\frac {a x}{2}-\frac {1}{2 a x}\right )\)	\(73\)
risch	\(\frac {a^{4} \ln \left (a x +1\right ) x^{2}}{4}+\frac {a^{3} x}{2}+a^{2} \operatorname {dilog}\left (a x +1\right )-\frac {a^{2} \ln \left (a x \right )}{4}-\frac {a}{2 x}-\frac {\ln \left (a x +1\right )}{4 x^{2}}-\frac {a^{4} \ln \left (-a x +1\right ) x^{2}}{4}-a^{2} \operatorname {dilog}\left (-a x +1\right )+\frac {a^{2} \ln \left (-a x \right )}{4}+\frac {\ln \left (-a x +1\right )}{4 x^{2}}\)	\(107\)
parts	\(\frac {a^{4} x^{2} \operatorname {arctanh}\left (a x \right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )}{2 x^{2}}-2 \,\operatorname {arctanh}\left (a x \right ) a^{2} \ln \left (x \right )-\frac {a \left (-a^{2} x +\frac {1}{x}+4 a^{2} \left (-\frac {\operatorname {dilog}\left (a x +1\right )}{2 a}-\frac {\ln \left (x \right ) \ln \left (a x +1\right )}{2 a}+\frac {\left (\ln \left (x \right )-\ln \left (a x \right )\right ) \ln \left (-a x +1\right )}{2 a}-\frac {\operatorname {dilog}\left (a x \right )}{2 a}\right )\right )}{2}\)	\(107\)
meijerg	\(\frac {i a^{2} \left (\frac {2 i}{x a}+\frac {2 i \left (-a x +1\right ) \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{x^{2} a^{2}}\right )}{4}+\frac {i a^{2} \left (-2 i x a +2 i \left (-a x +1\right ) \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )\right )}{4}+\frac {i a^{2} \left (\frac {2 i a x \operatorname {polylog}\left (2, \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-\frac {2 i a x \operatorname {polylog}\left (2, -\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}\right )}{2}\)	\(131\)

Input:

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

Output:

a^2*(1/2*a^2*x^2*arctanh(a*x)-2*arctanh(a*x)*ln(a*x)-1/2*arctanh(a*x)/a^2/ 
x^2+dilog(a*x)+dilog(a*x+1)+ln(a*x)*ln(a*x+1)+1/2*a*x-1/2/a/x)
 

Fricas [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )}{x^{3}} \,d x } \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.32 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^3} \, dx=\frac {1}{2} \, {\left (2 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a - 2 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a + \frac {a^{2} x^{2} - 1}{x}\right )} a + \frac {1}{2} \, {\left (a^{4} x^{2} - 2 \, a^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

1/2*(2*(log(a*x + 1)*log(x) + dilog(-a*x))*a - 2*(log(-a*x + 1)*log(x) + d 
ilog(a*x))*a + (a^2*x^2 - 1)/x)*a + 1/2*(a^4*x^2 - 2*a^2*log(x^2) - 1/x^2) 
*arctanh(a*x)
 

Giac [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )}{x^{3}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^3} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^2}{x^3} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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