Integrand size = 18, antiderivative size = 55 \[ \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {x}{4 a \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{4 a^2}+\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )} \] Output:
-1/4*x/a/(-a^2*x^2+1)-1/4*arctanh(a*x)/a^2+1/2*arctanh(a*x)/a^2/(-a^2*x^2+ 1)
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.20 \[ \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=\frac {2 a x-4 \text {arctanh}(a x)+\left (-1+a^2 x^2\right ) \log (1-a x)+\log (1+a x)-a^2 x^2 \log (1+a x)}{8 a^2 \left (-1+a^2 x^2\right )} \] Input:
Integrate[(x*ArcTanh[a*x])/(1 - a^2*x^2)^2,x]
Output:
(2*a*x - 4*ArcTanh[a*x] + (-1 + a^2*x^2)*Log[1 - a*x] + Log[1 + a*x] - a^2 *x^2*Log[1 + a*x])/(8*a^2*(-1 + a^2*x^2))
Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {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.
\(\displaystyle \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6556 |
\(\displaystyle \frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\) |
Input:
Int[(x*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)
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])
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])
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 = 0.38 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.67
method | result | size |
parallelrisch | \(-\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-a x +\operatorname {arctanh}\left (a x \right )}{4 \left (a^{2} x^{2}-1\right ) a^{2}}\) | \(37\) |
derivativedivides | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )}{2 \left (a^{2} x^{2}-1\right )}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8}}{a^{2}}\) | \(57\) |
default | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )}{2 \left (a^{2} x^{2}-1\right )}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8}}{a^{2}}\) | \(57\) |
parts | \(-\frac {\operatorname {arctanh}\left (a x \right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}+\frac {\frac {1}{4 \left (a x +1\right ) a}-\frac {\ln \left (a x +1\right )}{4 a}+\frac {1}{4 a \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{4 a}}{2 a}\) | \(74\) |
risch | \(-\frac {\ln \left (a x +1\right )}{4 a^{2} \left (a^{2} x^{2}-1\right )}-\frac {\ln \left (a x +1\right ) a^{2} x^{2}-x^{2} \ln \left (-a x +1\right ) a^{2}-2 a x -\ln \left (a x +1\right )-\ln \left (-a x +1\right )}{8 a^{2} \left (a x -1\right ) \left (a x +1\right )}\) | \(93\) |
orering | \(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (2 a^{2} x^{2}+1\right ) \operatorname {arctanh}\left (a x \right )}{2 a^{2} \left (-a^{2} x^{2}+1\right )^{2}}-\frac {\left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (\frac {\operatorname {arctanh}\left (a x \right )}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {x a}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {4 x^{2} \operatorname {arctanh}\left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{3}}\right )}{4 a^{2}}\) | \(119\) |
Input:
int(x*arctanh(a*x)/(-a^2*x^2+1)^2,x,method=_RETURNVERBOSE)
Output:
-1/4*(a^2*x^2*arctanh(a*x)-a*x+arctanh(a*x))/(a^2*x^2-1)/a^2
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=\frac {2 \, a x - {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{8 \, {\left (a^{4} x^{2} - a^{2}\right )}} \] Input:
integrate(x*arctanh(a*x)/(-a^2*x^2+1)^2,x, algorithm="fricas")
Output:
1/8*(2*a*x - (a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1)))/(a^4*x^2 - a^2)
Time = 0.44 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=\begin {cases} - \frac {a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{4 a^{4} x^{2} - 4 a^{2}} + \frac {a x}{4 a^{4} x^{2} - 4 a^{2}} - \frac {\operatorname {atanh}{\left (a x \right )}}{4 a^{4} x^{2} - 4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x*atanh(a*x)/(-a**2*x**2+1)**2,x)
Output:
Piecewise((-a**2*x**2*atanh(a*x)/(4*a**4*x**2 - 4*a**2) + a*x/(4*a**4*x**2 - 4*a**2) - atanh(a*x)/(4*a**4*x**2 - 4*a**2), Ne(a, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.13 \[ \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=\frac {\frac {2 \, x}{a^{2} x^{2} - 1} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}}{8 \, a} - \frac {\operatorname {artanh}\left (a x\right )}{2 \, {\left (a^{2} x^{2} - 1\right )} a^{2}} \] Input:
integrate(x*arctanh(a*x)/(-a^2*x^2+1)^2,x, algorithm="maxima")
Output:
1/8*(2*x/(a^2*x^2 - 1) - log(a*x + 1)/a + log(a*x - 1)/a)/a - 1/2*arctanh( a*x)/((a^2*x^2 - 1)*a^2)
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (47) = 94\).
Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.80 \[ \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {1}{16} \, {\left ({\left (\frac {a x + 1}{{\left (a x - 1\right )} a^{3}} + \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right ) - \frac {a x + 1}{{\left (a x - 1\right )} a^{3}} + \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} a \] Input:
integrate(x*arctanh(a*x)/(-a^2*x^2+1)^2,x, algorithm="giac")
Output:
-1/16*(((a*x + 1)/((a*x - 1)*a^3) + (a*x - 1)/((a*x + 1)*a^3))*log(-(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)) - (a*x + 1)/((a*x - 1)*a^3) + (a*x - 1)/((a*x + 1)*a^3))*a
Time = 3.53 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.67 \[ \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {\mathrm {atanh}\left (a\,x\right )}{4\,a^2}-\frac {\frac {\mathrm {atanh}\left (a\,x\right )}{2}-\frac {a\,x}{4}}{a^2\,\left (a^2\,x^2-1\right )} \] Input:
int((x*atanh(a*x))/(a^2*x^2 - 1)^2,x)
Output:
- atanh(a*x)/(4*a^2) - (atanh(a*x)/2 - (a*x)/4)/(a^2*(a^2*x^2 - 1))
Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.56 \[ \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2} \, dx=\frac {-4 \mathit {atanh} \left (a x \right ) a^{2} x^{2}-\mathrm {log}\left (a^{2} x -a \right ) a^{2} x^{2}+\mathrm {log}\left (a^{2} x -a \right )+\mathrm {log}\left (a^{2} x +a \right ) a^{2} x^{2}-\mathrm {log}\left (a^{2} x +a \right )+2 a x}{8 a^{2} \left (a^{2} x^{2}-1\right )} \] Input:
int(x*atanh(a*x)/(-a^2*x^2+1)^2,x)
Output:
( - 4*atanh(a*x)*a**2*x**2 - log(a**2*x - a)*a**2*x**2 + log(a**2*x - a) + log(a**2*x + a)*a**2*x**2 - log(a**2*x + a) + 2*a*x)/(8*a**2*(a**2*x**2 - 1))