\(\int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2} \, dx\) [227]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2} \, dx=-\frac {x}{2 a^3}+\frac {\text {arctanh}(a x)}{2 a^4}-\frac {x^2 \text {arctanh}(a x)}{2 a^2}-\frac {\text {arctanh}(a x)^2}{2 a^4}+\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^4} \] Output:

-1/2*x/a^3+1/2*arctanh(a*x)/a^4-1/2*x^2*arctanh(a*x)/a^2-1/2*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 = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.69 \[ \int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2} \, dx=\frac {-a x+\text {arctanh}(a x)^2+\text {arctanh}(a x) \left (1-a^2 x^2+2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )}{2 a^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6542, 6452, 262, 219, 6546, 6470, 2849, 2752}

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.

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

-(((x^2*ArcTanh[a*x])/2 - (a*(-(x/a^2) + ArcTanh[a*x]/a^3))/2)/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 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 262

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

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 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 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 6542

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

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]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.51

method	result	size
derivativedivides	\(\frac {-\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {a x}{2}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}}{a^{4}}\)	\(131\)
default	\(\frac {-\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {a x}{2}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}}{a^{4}}\)	\(131\)
risch	\(\frac {\ln \left (-a x +1\right ) x^{2}}{4 a^{2}}-\frac {\ln \left (-a x +1\right )}{4 a^{4}}-\frac {x}{2 a^{3}}+\frac {\ln \left (-a x +1\right )^{2}}{8 a^{4}}+\frac {\ln \left (\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (-a x +1\right )}{4 a^{4}}-\frac {\operatorname {dilog}\left (-\frac {a x}{2}+\frac {1}{2}\right )}{4 a^{4}}-\frac {\ln \left (a x +1\right ) x^{2}}{4 a^{2}}+\frac {\ln \left (a x +1\right )}{4 a^{4}}-\frac {\ln \left (a x +1\right )^{2}}{8 a^{4}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{4 a^{4}}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{4 a^{4}}\)	\(148\)
parts	\(-\frac {x^{2} \operatorname {arctanh}\left (a x \right )}{2 a^{2}}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a^{2} x^{2}-1\right )}{2 a^{4}}-\frac {a \left (-\frac {\ln \left (a x +1\right ) \ln \left (a^{2} x^{2}-1\right )}{2 a^{5}}+\frac {\ln \left (a x +1\right )^{2}}{4 a^{5}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{2 a^{5}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2 a^{5}}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{a^{5}}+\frac {\ln \left (a x -1\right ) \ln \left (a^{2} x^{2}-1\right )}{2 a^{5}}-\frac {\ln \left (a x -1\right )^{2}}{4 a^{5}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2 a^{5}}+\frac {x}{a^{4}}-\frac {\ln \left (a x +1\right )}{2 a^{5}}+\frac {\ln \left (a x -1\right )}{2 a^{5}}\right )}{2}\)	\(199\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.38 \[ \int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2} \, dx=-\frac {1}{8} \, a {\left (\frac {4 \, a x - \log \left (a x + 1\right )^{2} + 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + \log \left (a x - 1\right )^{2} + 2 \, \log \left (a x - 1\right )}{a^{5}} - \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{5}} - \frac {2 \, \log \left (a x + 1\right )}{a^{5}}\right )} - \frac {1}{2} \, {\left (\frac {x^{2}}{a^{2}} + \frac {\log \left (a^{2} x^{2} - 1\right )}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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