\(\int (1-a^2 x^2) \text {arctanh}(a x)^2 \, dx\) [176]

Optimal result

Integrand size = 17, antiderivative size = 115 \[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=-\frac {x}{3}+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}+\frac {2 \text {arctanh}(a x)^2}{3 a}+\frac {2}{3} x \text {arctanh}(a x)^2+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2-\frac {4 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{3 a}-\frac {2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{3 a} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.62 \[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=-\frac {a x+(-1+a x)^2 (2+a x) \text {arctanh}(a x)^2+\text {arctanh}(a x) \left (-1+a^2 x^2+4 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )-2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )}{3 a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6506, 24, 6436, 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.

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

\(\Big \downarrow \) 6506

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 6436

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcTanh[c*x^n]) 
^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] 
 && (EqQ[n, 1] || EqQ[p, 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 6506

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x 
_Symbol] :> Simp[b*p*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^(p - 1)/(2*c*q*(2* 
q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^p/(2*q + 1)), x] 
 + Simp[2*d*(q/(2*q + 1))   Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, 
 x], x] - Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1)))   Int[(d + e*x^2)^(q - 1)* 
(a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c 
^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 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 = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}}{3}+\operatorname {arctanh}\left (a x \right )^{2} a x -\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{3}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{3}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {a x}{3}-\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}+\frac {\ln \left (a x -1\right )^{2}}{6}-\frac {2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (a x +1\right )^{2}}{6}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3}}{a}\)	\(154\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}}{3}+\operatorname {arctanh}\left (a x \right )^{2} a x -\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{3}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{3}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {a x}{3}-\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}+\frac {\ln \left (a x -1\right )^{2}}{6}-\frac {2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (a x +1\right )^{2}}{6}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3}}{a}\)	\(154\)
parts	\(-\frac {x^{3} a^{2} \operatorname {arctanh}\left (a x \right )^{2}}{3}+x \operatorname {arctanh}\left (a x \right )^{2}-\frac {x^{2} \operatorname {arctanh}\left (a x \right ) a}{3}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{3 a}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{3 a}+\frac {-a x -\frac {\ln \left (a x -1\right )}{2}+\frac {\ln \left (a x +1\right )}{2}+\frac {\ln \left (a x -1\right )^{2}}{2}-2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )-\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )-\frac {\ln \left (a x +1\right )^{2}}{2}+\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3 a}\)	\(158\)
risch	\(-\frac {x}{3}+\frac {a \ln \left (-a x +1\right ) x^{2}}{6}+\frac {\ln \left (-a x +1\right ) \ln \left (a x +1\right )}{6 a}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{3 a}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{3 a}-\frac {a \ln \left (a x +1\right ) x^{2}}{6}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{2 a}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{a}-\frac {x \ln \left (-a x +1\right )}{2}-\frac {\ln \left (a x +1\right )}{3 a}-\frac {5 \ln \left (a x -1\right )}{9 a}-\frac {\ln \left (-a x +1\right )}{9 a}-\frac {\ln \left (a x +1\right ) x}{2}-\frac {2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{3 a}+\frac {\ln \left (-a x +1\right )^{2} x}{4}-\frac {\ln \left (-a x +1\right )^{2}}{6 a}+\frac {\ln \left (a x +1\right )^{2} x}{4}+\frac {\ln \left (a x +1\right )^{2}}{6 a}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{3}}{12}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{3}}{12}-\frac {37}{54 a}-\frac {\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right ) \ln \left (-a x +1\right )}{2 a}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{3}}{6}\)	\(327\)

Input:

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

Output:

1/a*(-1/3*arctanh(a*x)^2*a^3*x^3+arctanh(a*x)^2*a*x-1/3*a^2*x^2*arctanh(a* 
x)+2/3*arctanh(a*x)*ln(a*x-1)+2/3*arctanh(a*x)*ln(a*x+1)-1/3*a*x-1/6*ln(a* 
x-1)+1/6*ln(a*x+1)+1/6*ln(a*x-1)^2-2/3*dilog(1/2*a*x+1/2)-1/3*ln(a*x-1)*ln 
(1/2*a*x+1/2)-1/6*ln(a*x+1)^2+1/3*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+ 
1/2))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.25 \[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=-\frac {1}{6} \, a^{2} {\left (\frac {2 \, a x + \log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2} + \log \left (a x - 1\right )}{a^{3}} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{3}} - \frac {\log \left (a x + 1\right )}{a^{3}}\right )} - \frac {1}{3} \, {\left (x^{2} - \frac {2 \, \log \left (a x + 1\right )}{a^{2}} - \frac {2 \, \log \left (a x - 1\right )}{a^{2}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {1}{3} \, {\left (a^{2} x^{3} - 3 \, x\right )} \operatorname {artanh}\left (a x\right )^{2} \] Input:

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

Output:

-1/6*a^2*((2*a*x + log(a*x + 1)^2 - 2*log(a*x + 1)*log(a*x - 1) - log(a*x 
- 1)^2 + log(a*x - 1))/a^3 + 4*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1 
/2*a*x + 1/2))/a^3 - log(a*x + 1)/a^3) - 1/3*(x^2 - 2*log(a*x + 1)/a^2 - 2 
*log(a*x - 1)/a^2)*a*arctanh(a*x) - 1/3*(a^2*x^3 - 3*x)*arctanh(a*x)^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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