\(\int \frac {x^3 \text {arctanh}(a x)}{(1-a^2 x^2)^2} \, dx\) [259]

Optimal result

Integrand size = 20, antiderivative size = 109 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {x}{4 a^3 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{4 a^4}+\frac {\text {arctanh}(a x)}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{2 a^4}-\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.59 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {4 \text {arctanh}(a x)^2-2 \text {arctanh}(a x) \left (\cosh (2 \text {arctanh}(a x))-4 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+\sinh (2 \text {arctanh}(a x))}{8 a^4} \] Input:

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

Output:

-1/8*(4*ArcTanh[a*x]^2 - 2*ArcTanh[a*x]*(Cosh[2*ArcTanh[a*x]] - 4*Log[1 + 
E^(-2*ArcTanh[a*x])]) - 4*PolyLog[2, -E^(-2*ArcTanh[a*x])] + Sinh[2*ArcTan 
h[a*x]])/a^4
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6590, 6546, 6470, 2849, 2752, 6556, 215, 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.

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

\(\Big \downarrow \) 6590

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 
, 0]
 

rule 6546

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

rule 6556

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q 
_.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q 
+ 1))), x] + Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTanh[c* 
x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[p, 0] && NeQ[q, -1]
 

rule 6590

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[1/e   Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*A 
rcTanh[c*x])^p, x], x] - Simp[d/e   Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcT 
anh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && In 
tegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )}{4 a x +4}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8}}{a^{4}}\)	\(159\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )}{4 a x +4}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8}}{a^{4}}\)	\(159\)
parts	\(\frac {\operatorname {arctanh}\left (a x \right )}{4 a^{4} \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2 a^{4}}-\frac {\operatorname {arctanh}\left (a x \right )}{4 a^{4} \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2 a^{4}}-\frac {a \left (-\frac {\ln \left (a x -1\right )^{2}}{2 a^{5}}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{a^{5}}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{a^{5}}-\frac {2 \left (-\frac {\ln \left (a x +1\right )^{2}}{4}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}\right )}{a^{5}}-\frac {1}{2 a^{5} \left (a x +1\right )}+\frac {\ln \left (a x +1\right )}{2 a^{5}}-\frac {1}{2 a^{5} \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{2 a^{5}}\right )}{4}\)	\(205\)
risch	\(\frac {\ln \left (a x +1\right )^{2}}{8 a^{4}}+\frac {\ln \left (a x -1\right )}{16 a^{4}}-\frac {\ln \left (a x +1\right ) x}{16 a^{3} \left (a x -1\right )}-\frac {\ln \left (a x +1\right )}{16 a^{4} \left (a x -1\right )}+\frac {\ln \left (a x +1\right )}{8 a^{4} \left (a x +1\right )}+\frac {1}{8 a^{4} \left (a x +1\right )}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{4 a^{4}}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{4 a^{4}}-\frac {\ln \left (-a x +1\right )^{2}}{8 a^{4}}-\frac {\ln \left (-a x -1\right )}{16 a^{4}}-\frac {\ln \left (-a x +1\right ) x}{16 a^{3} \left (-a x -1\right )}+\frac {\ln \left (-a x +1\right )}{16 a^{4} \left (-a x -1\right )}-\frac {\ln \left (-a x +1\right )}{8 a^{4} \left (-a x +1\right )}-\frac {1}{8 a^{4} \left (-a x +1\right )}-\frac {\ln \left (\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (-a x +1\right )}{4 a^{4}}+\frac {\operatorname {dilog}\left (-\frac {a x}{2}+\frac {1}{2}\right )}{4 a^{4}}\)	\(254\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.62 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {1}{8} \, a {\left (\frac {{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2 \, a x - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )}{a^{7} x^{2} - a^{5}} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{5}} + \frac {\log \left (a x + 1\right )}{a^{5}}\right )} - \frac {1}{2} \, {\left (\frac {1}{a^{6} x^{2} - a^{4}} - \frac {\log \left (a^{2} x^{2} - 1\right )}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

-1/8*a*(((a^2*x^2 - 1)*log(a*x + 1)^2 - 2*(a^2*x^2 - 1)*log(a*x + 1)*log(a 
*x - 1) - (a^2*x^2 - 1)*log(a*x - 1)^2 - 2*a*x - (a^2*x^2 - 1)*log(a*x - 1 
))/(a^7*x^2 - a^5) + 4*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 
 1/2))/a^5 + log(a*x + 1)/a^5) - 1/2*(1/(a^6*x^2 - a^4) - log(a^2*x^2 - 1) 
/a^4)*arctanh(a*x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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