Integrand size = 18, antiderivative size = 95 \[ \int x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=\frac {1-a^2 x^2}{12 a^2}+\frac {x \text {arctanh}(a x)}{3 a}+\frac {x \left (1-a^2 x^2\right ) \text {arctanh}(a x)}{6 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{4 a^2}+\frac {\log \left (1-a^2 x^2\right )}{6 a^2} \] Output:
1/12*(-a^2*x^2+1)/a^2+1/3*x*arctanh(a*x)/a+1/6*x*(-a^2*x^2+1)*arctanh(a*x) /a-1/4*(-a^2*x^2+1)^2*arctanh(a*x)^2/a^2+1/6*ln(-a^2*x^2+1)/a^2
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=\frac {-a^2 x^2+\left (6 a x-2 a^3 x^3\right ) \text {arctanh}(a x)-3 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2+2 \log \left (1-a^2 x^2\right )}{12 a^2} \] Input:
Integrate[x*(1 - a^2*x^2)*ArcTanh[a*x]^2,x]
Output:
(-(a^2*x^2) + (6*a*x - 2*a^3*x^3)*ArcTanh[a*x] - 3*(-1 + a^2*x^2)^2*ArcTan h[a*x]^2 + 2*Log[1 - a^2*x^2])/(12*a^2)
Time = 0.40 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6556, 6504, 6436, 240}
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.
\(\displaystyle \int x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx\) |
\(\Big \downarrow \) 6556 |
\(\displaystyle \frac {\int \left (1-a^2 x^2\right ) \text {arctanh}(a x)dx}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{4 a^2}\) |
\(\Big \downarrow \) 6504 |
\(\displaystyle \frac {\frac {2}{3} \int \text {arctanh}(a x)dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {1-a^2 x^2}{6 a}}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{4 a^2}\) |
\(\Big \downarrow \) 6436 |
\(\displaystyle \frac {\frac {2}{3} \left (x \text {arctanh}(a x)-a \int \frac {x}{1-a^2 x^2}dx\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {1-a^2 x^2}{6 a}}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{4 a^2}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{4 a^2}\) |
Input:
Int[x*(1 - a^2*x^2)*ArcTanh[a*x]^2,x]
Output:
-1/4*((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/a^2 + ((1 - a^2*x^2)/(6*a) + (x*(1 - a^2*x^2)*ArcTanh[a*x])/3 + (2*(x*ArcTanh[a*x] + Log[1 - a^2*x^2]/(2*a)))/ 3)/(2*a)
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
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])
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symb ol] :> Simp[b*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2)^q *((a + b*ArcTanh[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e *x^2)^(q - 1)*(a + b*ArcTanh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]
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]
Time = 0.22 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.88
method | result | size |
parallelrisch | \(-\frac {3 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}+2 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-6 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}+a^{2} x^{2}-6 a x \,\operatorname {arctanh}\left (a x \right )+3 \operatorname {arctanh}\left (a x \right )^{2}-4 \ln \left (a x -1\right )-4 \,\operatorname {arctanh}\left (a x \right )}{12 a^{2}}\) | \(84\) |
derivativedivides | \(\frac {-\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4}-\frac {a^{3} x^{3} \operatorname {arctanh}\left (a x \right )}{6}+\frac {a x \,\operatorname {arctanh}\left (a x \right )}{2}-\frac {a^{2} x^{2}}{12}+\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}}{a^{2}}\) | \(86\) |
default | \(\frac {-\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4}-\frac {a^{3} x^{3} \operatorname {arctanh}\left (a x \right )}{6}+\frac {a x \,\operatorname {arctanh}\left (a x \right )}{2}-\frac {a^{2} x^{2}}{12}+\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}}{a^{2}}\) | \(86\) |
parts | \(-\frac {x^{4} a^{2} \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {\operatorname {arctanh}\left (a x \right )^{2} x^{2}}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a^{2}}+\frac {-\frac {a^{3} x^{3} \operatorname {arctanh}\left (a x \right )}{3}+a x \,\operatorname {arctanh}\left (a x \right )-\frac {a^{2} x^{2}}{6}+\frac {\ln \left (a x -1\right )}{3}+\frac {\ln \left (a x +1\right )}{3}}{2 a^{2}}\) | \(87\) |
risch | \(-\frac {\left (a^{2} x^{2}-1\right )^{2} \ln \left (a x +1\right )^{2}}{16 a^{2}}+\frac {\left (3 x^{4} \ln \left (-a x +1\right ) a^{4}-2 a^{3} x^{3}-6 x^{2} \ln \left (-a x +1\right ) a^{2}+6 a x +3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{24 a^{2}}-\frac {\ln \left (-a x +1\right )^{2} a^{2} x^{4}}{16}+\frac {a \,x^{3} \ln \left (-a x +1\right )}{12}+\frac {x^{2} \ln \left (-a x +1\right )^{2}}{8}-\frac {x^{2}}{12}-\frac {x \ln \left (-a x +1\right )}{4 a}-\frac {\ln \left (-a x +1\right )^{2}}{16 a^{2}}+\frac {\ln \left (a^{2} x^{2}-1\right )}{6 a^{2}}\) | \(180\) |
Input:
int(x*(-a^2*x^2+1)*arctanh(a*x)^2,x,method=_RETURNVERBOSE)
Output:
-1/12*(3*a^4*x^4*arctanh(a*x)^2+2*a^3*x^3*arctanh(a*x)-6*a^2*x^2*arctanh(a *x)^2+a^2*x^2-6*a*x*arctanh(a*x)+3*arctanh(a*x)^2-4*ln(a*x-1)-4*arctanh(a* x))/a^2
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96 \[ \int x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=-\frac {4 \, a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 8 \, \log \left (a^{2} x^{2} - 1\right )}{48 \, a^{2}} \] Input:
integrate(x*(-a^2*x^2+1)*arctanh(a*x)^2,x, algorithm="fricas")
Output:
-1/48*(4*a^2*x^2 + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 + 4*(a^3*x^3 - 3*a*x)*log(-(a*x + 1)/(a*x - 1)) - 8*log(a^2*x^2 - 1))/a^2
Time = 0.37 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=\begin {cases} - \frac {a^{2} x^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{4} - \frac {a x^{3} \operatorname {atanh}{\left (a x \right )}}{6} + \frac {x^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{2} - \frac {x^{2}}{12} + \frac {x \operatorname {atanh}{\left (a x \right )}}{2 a} + \frac {\log {\left (x - \frac {1}{a} \right )}}{3 a^{2}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{4 a^{2}} + \frac {\operatorname {atanh}{\left (a x \right )}}{3 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x*(-a**2*x**2+1)*atanh(a*x)**2,x)
Output:
Piecewise((-a**2*x**4*atanh(a*x)**2/4 - a*x**3*atanh(a*x)/6 + x**2*atanh(a *x)**2/2 - x**2/12 + x*atanh(a*x)/(2*a) + log(x - 1/a)/(3*a**2) - atanh(a* x)**2/(4*a**2) + atanh(a*x)/(3*a**2), Ne(a, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.78 \[ \int x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=-\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{4 \, a^{2}} - \frac {{\left (x^{2} - \frac {2 \, \log \left (a x + 1\right )}{a^{2}} - \frac {2 \, \log \left (a x - 1\right )}{a^{2}}\right )} a + 2 \, {\left (a^{2} x^{3} - 3 \, x\right )} \operatorname {artanh}\left (a x\right )}{12 \, a} \] Input:
integrate(x*(-a^2*x^2+1)*arctanh(a*x)^2,x, algorithm="maxima")
Output:
-1/4*(a^2*x^2 - 1)^2*arctanh(a*x)^2/a^2 - 1/12*((x^2 - 2*log(a*x + 1)/a^2 - 2*log(a*x - 1)/a^2)*a + 2*(a^2*x^3 - 3*x)*arctanh(a*x))/a
Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (82) = 164\).
Time = 0.12 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.21 \[ \int x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=-\frac {1}{3} \, a {\left (\frac {{\left (\frac {3 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{3} a^{3}}{{\left (a x - 1\right )}^{3}} - \frac {3 \, {\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} + \frac {3 \, {\left (a x + 1\right )} a^{3}}{a x - 1} - a^{3}} + \frac {3 \, {\left (a x + 1\right )}^{2} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{{\left (\frac {{\left (a x + 1\right )}^{4} a^{3}}{{\left (a x - 1\right )}^{4}} - \frac {4 \, {\left (a x + 1\right )}^{3} a^{3}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} - \frac {4 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + a^{3}\right )} {\left (a x - 1\right )}^{2}} + \frac {a x + 1}{{\left (\frac {{\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} - \frac {2 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + a^{3}\right )} {\left (a x - 1\right )}} + \frac {\log \left (-\frac {a x + 1}{a x - 1} + 1\right )}{a^{3}} - \frac {\log \left (-\frac {a x + 1}{a x - 1}\right )}{a^{3}}\right )} \] Input:
integrate(x*(-a^2*x^2+1)*arctanh(a*x)^2,x, algorithm="giac")
Output:
-1/3*a*((3*(a*x + 1)/(a*x - 1) - 1)*log(-(a*x + 1)/(a*x - 1))/((a*x + 1)^3 *a^3/(a*x - 1)^3 - 3*(a*x + 1)^2*a^3/(a*x - 1)^2 + 3*(a*x + 1)*a^3/(a*x - 1) - a^3) + 3*(a*x + 1)^2*log(-(a*x + 1)/(a*x - 1))^2/(((a*x + 1)^4*a^3/(a *x - 1)^4 - 4*(a*x + 1)^3*a^3/(a*x - 1)^3 + 6*(a*x + 1)^2*a^3/(a*x - 1)^2 - 4*(a*x + 1)*a^3/(a*x - 1) + a^3)*(a*x - 1)^2) + (a*x + 1)/(((a*x + 1)^2* a^3/(a*x - 1)^2 - 2*(a*x + 1)*a^3/(a*x - 1) + a^3)*(a*x - 1)) + log(-(a*x + 1)/(a*x - 1) + 1)/a^3 - log(-(a*x + 1)/(a*x - 1))/a^3)
Time = 3.89 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=\frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^2}{2}-\frac {{\mathrm {atanh}\left (a\,x\right )}^2}{4\,a^2}-\frac {x^2}{12}+\frac {\ln \left (a^2\,x^2-1\right )}{6\,a^2}+\frac {x\,\mathrm {atanh}\left (a\,x\right )}{2\,a}-\frac {a\,x^3\,\mathrm {atanh}\left (a\,x\right )}{6}-\frac {a^2\,x^4\,{\mathrm {atanh}\left (a\,x\right )}^2}{4} \] Input:
int(-x*atanh(a*x)^2*(a^2*x^2 - 1),x)
Output:
(x^2*atanh(a*x)^2)/2 - atanh(a*x)^2/(4*a^2) - x^2/12 + log(a^2*x^2 - 1)/(6 *a^2) + (x*atanh(a*x))/(2*a) - (a*x^3*atanh(a*x))/6 - (a^2*x^4*atanh(a*x)^ 2)/4
Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=\frac {-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+6 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-3 \mathit {atanh} \left (a x \right )^{2}-2 \mathit {atanh} \left (a x \right ) a^{3} x^{3}+6 \mathit {atanh} \left (a x \right ) a x +4 \mathit {atanh} \left (a x \right )+4 \,\mathrm {log}\left (a^{2} x -a \right )-a^{2} x^{2}}{12 a^{2}} \] Input:
int(x*(-a^2*x^2+1)*atanh(a*x)^2,x)
Output:
( - 3*atanh(a*x)**2*a**4*x**4 + 6*atanh(a*x)**2*a**2*x**2 - 3*atanh(a*x)** 2 - 2*atanh(a*x)*a**3*x**3 + 6*atanh(a*x)*a*x + 4*atanh(a*x) + 4*log(a**2* x - a) - a**2*x**2)/(12*a**2)