Integrand size = 18, antiderivative size = 38 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^2} \, dx=-\frac {\text {arctanh}(a x)}{x}-a^2 x \text {arctanh}(a x)+a \log (x)-a \log \left (1-a^2 x^2\right ) \] Output:
-arctanh(a*x)/x-a^2*x*arctanh(a*x)+a*ln(x)-a*ln(-a^2*x^2+1)
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^2} \, dx=-\frac {\text {arctanh}(a x)}{x}-a^2 x \text {arctanh}(a x)+a \log (x)-a \log \left (1-a^2 x^2\right ) \] Input:
Integrate[((1 - a^2*x^2)*ArcTanh[a*x])/x^2,x]
Output:
-(ArcTanh[a*x]/x) - a^2*x*ArcTanh[a*x] + a*Log[x] - a*Log[1 - a^2*x^2]
Time = 0.35 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.66, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6576, 6436, 240, 6452, 243, 47, 14, 16}
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.
\(\displaystyle \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^2} \, dx\) |
\(\Big \downarrow \) 6576 |
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x^2}dx-a^2 \int \text {arctanh}(a x)dx\) |
\(\Big \downarrow \) 6436 |
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x^2}dx-a^2 \left (x \text {arctanh}(a x)-a \int \frac {x}{1-a^2 x^2}dx\right )\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x^2}dx-a^2 \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle a \int \frac {1}{x \left (1-a^2 x^2\right )}dx-\left (a^2 \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )\right )-\frac {\text {arctanh}(a x)}{x}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx^2-\left (a^2 \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )\right )-\frac {\text {arctanh}(a x)}{x}\) |
\(\Big \downarrow \) 47 |
\(\displaystyle \frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx^2+\int \frac {1}{x^2}dx^2\right )-\left (a^2 \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )\right )-\frac {\text {arctanh}(a x)}{x}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx^2+\log \left (x^2\right )\right )-\left (a^2 \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )\right )-\frac {\text {arctanh}(a x)}{x}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -\left (a^2 \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )-\frac {\text {arctanh}(a x)}{x}\) |
Input:
Int[((1 - a^2*x^2)*ArcTanh[a*x])/x^2,x]
Output:
-(ArcTanh[a*x]/x) + (a*(Log[x^2] - Log[1 - a^2*x^2]))/2 - a^2*(x*ArcTanh[a *x] + Log[1 - a^2*x^2]/(2*a))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_)^(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]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x ^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ [p, 1] && IntegerQ[q]))
Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08
method | result | size |
parts | \(-a^{2} x \,\operatorname {arctanh}\left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{x}-a \left (-\ln \left (x \right )+\ln \left (a x +1\right )+\ln \left (a x -1\right )\right )\) | \(41\) |
derivativedivides | \(a \left (-a x \,\operatorname {arctanh}\left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{a x}+\ln \left (a x \right )-\ln \left (a x -1\right )-\ln \left (a x +1\right )\right )\) | \(44\) |
default | \(a \left (-a x \,\operatorname {arctanh}\left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{a x}+\ln \left (a x \right )-\ln \left (a x -1\right )-\ln \left (a x +1\right )\right )\) | \(44\) |
parallelrisch | \(\frac {-a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+a \ln \left (x \right ) x -2 \ln \left (a x -1\right ) a x -2 a x \,\operatorname {arctanh}\left (a x \right )-\operatorname {arctanh}\left (a x \right )}{x}\) | \(47\) |
risch | \(-\frac {\left (a^{2} x^{2}+1\right ) \ln \left (a x +1\right )}{2 x}+\frac {x^{2} \ln \left (-a x +1\right ) a^{2}+2 a \ln \left (x \right ) x -2 a \ln \left (a^{2} x^{2}-1\right ) x +\ln \left (-a x +1\right )}{2 x}\) | \(69\) |
meijerg | \(\frac {a \left (4 \ln \left (x \right )+4 \ln \left (i a \right )+\frac {2 \ln \left (1-\sqrt {a^{2} x^{2}}\right )-2 \ln \left (1+\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (-a^{2} x^{2}+1\right )\right )}{4}+\frac {a \left (\frac {2 a^{2} x^{2} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (-a^{2} x^{2}+1\right )\right )}{4}\) | \(133\) |
Input:
int((-a^2*x^2+1)*arctanh(a*x)/x^2,x,method=_RETURNVERBOSE)
Output:
-a^2*x*arctanh(a*x)-arctanh(a*x)/x-a*(-ln(x)+ln(a*x+1)+ln(a*x-1))
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^2} \, dx=-\frac {2 \, a x \log \left (a^{2} x^{2} - 1\right ) - 2 \, a x \log \left (x\right ) + {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, x} \] Input:
integrate((-a^2*x^2+1)*arctanh(a*x)/x^2,x, algorithm="fricas")
Output:
-1/2*(2*a*x*log(a^2*x^2 - 1) - 2*a*x*log(x) + (a^2*x^2 + 1)*log(-(a*x + 1) /(a*x - 1)))/x
Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^2} \, dx=\begin {cases} - a^{2} x \operatorname {atanh}{\left (a x \right )} + a \log {\left (x \right )} - 2 a \log {\left (x - \frac {1}{a} \right )} - 2 a \operatorname {atanh}{\left (a x \right )} - \frac {\operatorname {atanh}{\left (a x \right )}}{x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate((-a**2*x**2+1)*atanh(a*x)/x**2,x)
Output:
Piecewise((-a**2*x*atanh(a*x) + a*log(x) - 2*a*log(x - 1/a) - 2*a*atanh(a* x) - atanh(a*x)/x, Ne(a, 0)), (0, True))
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^2} \, dx=-a {\left (\log \left (a x + 1\right ) + \log \left (a x - 1\right ) - \log \left (x\right )\right )} - {\left (a^{2} x + \frac {1}{x}\right )} \operatorname {artanh}\left (a x\right ) \] Input:
integrate((-a^2*x^2+1)*arctanh(a*x)/x^2,x, algorithm="maxima")
Output:
-a*(log(a*x + 1) + log(a*x - 1) - log(x)) - (a^2*x + 1/x)*arctanh(a*x)
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (38) = 76\).
Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.82 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^2} \, dx=-a {\left (\frac {2 \, \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 1} + \log \left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}}\right ) - \log \left ({\left | \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 1 \right |}\right )\right )} \] Input:
integrate((-a^2*x^2+1)*arctanh(a*x)/x^2,x, algorithm="giac")
Output:
-a*(2*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)^2 /(a*x - 1)^2 - 1) + log((a*x + 1)^2/(a*x - 1)^2) - log(abs((a*x + 1)^2/(a* x - 1)^2 - 1)))
Time = 3.84 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^2} \, dx=a\,\ln \left (x\right )-a\,\ln \left (a^2\,x^2-1\right )-\frac {\mathrm {atanh}\left (a\,x\right )}{x}-a^2\,x\,\mathrm {atanh}\left (a\,x\right ) \] Input:
int(-(atanh(a*x)*(a^2*x^2 - 1))/x^2,x)
Output:
a*log(x) - a*log(a^2*x^2 - 1) - atanh(a*x)/x - a^2*x*atanh(a*x)
Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^2} \, dx=\frac {-\mathit {atanh} \left (a x \right ) a^{2} x^{2}-2 \mathit {atanh} \left (a x \right ) a x -\mathit {atanh} \left (a x \right )-2 \,\mathrm {log}\left (a^{2} x -a \right ) a x +\mathrm {log}\left (x \right ) a x}{x} \] Input:
int((-a^2*x^2+1)*atanh(a*x)/x^2,x)
Output:
( - atanh(a*x)*a**2*x**2 - 2*atanh(a*x)*a*x - atanh(a*x) - 2*log(a**2*x - a)*a*x + log(x)*a*x)/x