\(\int \frac {(1-a^2 x^2) \text {arctanh}(a x)}{x^3} \, dx\) [168]

Optimal result

Integrand size = 18, antiderivative size = 56 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^3} \, dx=-\frac {a}{2 x}+\frac {1}{2} a^2 \text {arctanh}(a x)-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a^2 \operatorname {PolyLog}(2,-a x)-\frac {1}{2} a^2 \operatorname {PolyLog}(2,a x) \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.21 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^3} \, dx=-\frac {a}{2 x}-\frac {\text {arctanh}(a x)}{2 x^2}-\frac {1}{4} a^2 \log (1-a x)+\frac {1}{4} a^2 \log (1+a x)-\frac {1}{2} a^2 (-\operatorname {PolyLog}(2,-a x)+\operatorname {PolyLog}(2,a x)) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6576, 6446, 6452, 264, 219}

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.

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6446

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 6446

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x 
] + (-Simp[(b/2)*PolyLog[2, (-c)*x], x] + Simp[(b/2)*PolyLog[2, c*x], x]) / 
; FreeQ[{a, b, c}, x]
 

rule 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

rule 6576

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(a^{2} \left (-\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{2 a^{2} x^{2}}-\frac {1}{2 a x}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}+\frac {\operatorname {dilog}\left (a x \right )}{2}+\frac {\operatorname {dilog}\left (a x +1\right )}{2}+\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}\right )\)	\(78\)
default	\(a^{2} \left (-\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{2 a^{2} x^{2}}-\frac {1}{2 a x}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}+\frac {\operatorname {dilog}\left (a x \right )}{2}+\frac {\operatorname {dilog}\left (a x +1\right )}{2}+\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}\right )\)	\(78\)
risch	\(-\frac {a^{2} \operatorname {dilog}\left (-a x +1\right )}{2}+\frac {a^{2} \ln \left (-a x \right )}{4}-\frac {a}{2 x}-\frac {a^{2} \ln \left (-a x +1\right )}{4}+\frac {\ln \left (-a x +1\right )}{4 x^{2}}+\frac {a^{2} \operatorname {dilog}\left (a x +1\right )}{2}-\frac {a^{2} \ln \left (a x \right )}{4}+\frac {a^{2} \ln \left (a x +1\right )}{4}-\frac {\ln \left (a x +1\right )}{4 x^{2}}\)	\(96\)
meijerg	\(\frac {i a^{2} \left (\frac {2 i}{x a}+\frac {2 i \left (-a x +1\right ) \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{x^{2} a^{2}}\right )}{4}+\frac {i a^{2} \left (\frac {2 i a x \operatorname {polylog}\left (2, \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-\frac {2 i a x \operatorname {polylog}\left (2, -\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}\right )}{4}\)	\(101\)
parts	\(-\frac {\operatorname {arctanh}\left (a x \right )}{2 x^{2}}-\operatorname {arctanh}\left (a x \right ) a^{2} \ln \left (x \right )-\frac {a \left (-\frac {a \ln \left (a x +1\right )}{2}+\frac {a \ln \left (a x -1\right )}{2}+\frac {1}{x}+2 a^{2} \left (-\frac {\operatorname {dilog}\left (a x +1\right )}{2 a}-\frac {\ln \left (x \right ) \ln \left (a x +1\right )}{2 a}+\frac {\left (\ln \left (x \right )-\ln \left (a x \right )\right ) \ln \left (-a x +1\right )}{2 a}-\frac {\operatorname {dilog}\left (a x \right )}{2 a}\right )\right )}{2}\)	\(107\)

Input:

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

Output:

a^2*(-arctanh(a*x)*ln(a*x)-1/2*arctanh(a*x)/a^2/x^2-1/2/a/x-1/4*ln(a*x-1)+ 
1/4*ln(a*x+1)+1/2*dilog(a*x)+1/2*dilog(a*x+1)+1/2*ln(a*x)*ln(a*x+1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.45 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^3} \, dx=\frac {1}{4} \, {\left (2 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a - 2 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a + a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} a - \frac {1}{2} \, {\left (a^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (44) = 88\).

Time = 1.15 (sec) , antiderivative size = 330, normalized size of antiderivative = 5.89 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^3} \, dx=a^{2} {\left (\frac {\log \left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}}\right )}{a} - \frac {\log \left ({\left | \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 1 \right |}\right )}{a} + \frac {\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 2}{a {\left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 1\right )}} - \frac {2 \, \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{a - \frac {a {\left (\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1\right )}}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}} - 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{a - \frac {a {\left (\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1\right )}}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}} + 1}\right )}{a {\left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 1\right )}^{2}}\right )} \] Input:

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

Output:

a^2*(log((a*x + 1)^2/(a*x - 1)^2)/a - log(abs((a*x + 1)^2/(a*x - 1)^2 - 1) 
)/a + ((a*x + 1)^2/(a*x - 1)^2 - 2)/(a*((a*x + 1)^2/(a*x - 1)^2 - 1)) - 2* 
log(-(a*((a*x + 1)/(a*x - 1) + 1)/(a - a*(a*((a*x + 1)/(a*x - 1) + 1)/((a* 
x + 1)*a/(a*x - 1) - a) + 1)/(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a/(a* 
x - 1) - a) - 1)) - 1)/(a*((a*x + 1)/(a*x - 1) + 1)/(a - a*(a*((a*x + 1)/( 
a*x - 1) + 1)/((a*x + 1)*a/(a*x - 1) - a) + 1)/(a*((a*x + 1)/(a*x - 1) + 1 
)/((a*x + 1)*a/(a*x - 1) - a) - 1)) + 1))/(a*((a*x + 1)^2/(a*x - 1)^2 - 1) 
^2))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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