Integrand size = 20, antiderivative size = 68 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^4} \, dx=-\frac {a}{6 x^2}-\frac {\text {arctanh}(a x)}{3 x^3}+\frac {2 a^2 \text {arctanh}(a x)}{x}+a^4 x \text {arctanh}(a x)-\frac {5}{3} a^3 \log (x)+\frac {4}{3} a^3 \log \left (1-a^2 x^2\right ) \] Output:
-1/6*a/x^2-1/3*arctanh(a*x)/x^3+2*a^2*arctanh(a*x)/x+a^4*x*arctanh(a*x)-5/ 3*a^3*ln(x)+4/3*a^3*ln(-a^2*x^2+1)
Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^4} \, dx=-\frac {a}{6 x^2}-\frac {\text {arctanh}(a x)}{3 x^3}+\frac {2 a^2 \text {arctanh}(a x)}{x}+a^4 x \text {arctanh}(a x)-\frac {5}{3} a^3 \log (x)+\frac {4}{3} a^3 \log \left (1-a^2 x^2\right ) \] Input:
Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x])/x^4,x]
Output:
-1/6*a/x^2 - ArcTanh[a*x]/(3*x^3) + (2*a^2*ArcTanh[a*x])/x + a^4*x*ArcTanh [a*x] - (5*a^3*Log[x])/3 + (4*a^3*Log[1 - a^2*x^2])/3
Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6574, 2009}
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.
\(\displaystyle \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^4} \, dx\) |
\(\Big \downarrow \) 6574 |
\(\displaystyle \int \left (a^4 \text {arctanh}(a x)-\frac {2 a^2 \text {arctanh}(a x)}{x^2}+\frac {\text {arctanh}(a x)}{x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^4 x \text {arctanh}(a x)-\frac {5}{3} a^3 \log (x)+\frac {2 a^2 \text {arctanh}(a x)}{x}+\frac {4}{3} a^3 \log \left (1-a^2 x^2\right )-\frac {\text {arctanh}(a x)}{3 x^3}-\frac {a}{6 x^2}\) |
Input:
Int[((1 - a^2*x^2)^2*ArcTanh[a*x])/x^4,x]
Output:
-1/6*a/x^2 - ArcTanh[a*x]/(3*x^3) + (2*a^2*ArcTanh[a*x])/x + a^4*x*ArcTanh [a*x] - (5*a^3*Log[x])/3 + (4*a^3*Log[1 - a^2*x^2])/3
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(a^{3} \left (a x \,\operatorname {arctanh}\left (a x \right )+\frac {2 \,\operatorname {arctanh}\left (a x \right )}{a x}-\frac {\operatorname {arctanh}\left (a x \right )}{3 a^{3} x^{3}}-\frac {1}{6 a^{2} x^{2}}-\frac {5 \ln \left (a x \right )}{3}+\frac {4 \ln \left (a x -1\right )}{3}+\frac {4 \ln \left (a x +1\right )}{3}\right )\) | \(67\) |
default | \(a^{3} \left (a x \,\operatorname {arctanh}\left (a x \right )+\frac {2 \,\operatorname {arctanh}\left (a x \right )}{a x}-\frac {\operatorname {arctanh}\left (a x \right )}{3 a^{3} x^{3}}-\frac {1}{6 a^{2} x^{2}}-\frac {5 \ln \left (a x \right )}{3}+\frac {4 \ln \left (a x -1\right )}{3}+\frac {4 \ln \left (a x +1\right )}{3}\right )\) | \(67\) |
parts | \(a^{4} x \,\operatorname {arctanh}\left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{3 x^{3}}+\frac {2 a^{2} \operatorname {arctanh}\left (a x \right )}{x}-\frac {a \left (\frac {1}{2 x^{2}}+5 a^{2} \ln \left (x \right )-4 a^{2} \ln \left (a x +1\right )-4 a^{2} \ln \left (a x -1\right )\right )}{3}\) | \(70\) |
parallelrisch | \(-\frac {-6 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )+10 \ln \left (x \right ) a^{3} x^{3}-16 \ln \left (a x -1\right ) x^{3} a^{3}-16 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+a^{3} x^{3}-12 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+a x +2 \,\operatorname {arctanh}\left (a x \right )}{6 x^{3}}\) | \(83\) |
risch | \(\frac {\left (3 a^{4} x^{4}+6 a^{2} x^{2}-1\right ) \ln \left (a x +1\right )}{6 x^{3}}-\frac {3 x^{4} \ln \left (-a x +1\right ) a^{4}+10 \ln \left (x \right ) a^{3} x^{3}-8 \ln \left (-a^{2} x^{2}+1\right ) a^{3} x^{3}+6 x^{2} \ln \left (-a x +1\right ) a^{2}+a x -\ln \left (-a x +1\right )}{6 x^{3}}\) | \(108\) |
meijerg | \(-\frac {a^{3} \left (\frac {2}{a^{2} x^{2}}+\frac {4}{9}-\frac {4 \ln \left (x \right )}{3}-\frac {4 \ln \left (i a \right )}{3}-\frac {2 \left (10 a^{2} x^{2}+30\right )}{45 a^{2} x^{2}}-\frac {2 \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{3 a^{2} x^{2} \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{3}\right )}{4}-\frac {a^{3} \left (\frac {2 a^{2} x^{2} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (-a^{2} x^{2}+1\right )\right )}{4}-\frac {a^{3} \left (4 \ln \left (x \right )+4 \ln \left (i a \right )+\frac {2 \ln \left (1-\sqrt {a^{2} x^{2}}\right )-2 \ln \left (1+\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (-a^{2} x^{2}+1\right )\right )}{2}\) | \(240\) |
Input:
int((-a^2*x^2+1)^2*arctanh(a*x)/x^4,x,method=_RETURNVERBOSE)
Output:
a^3*(a*x*arctanh(a*x)+2*arctanh(a*x)/a/x-1/3*arctanh(a*x)/a^3/x^3-1/6/a^2/ x^2-5/3*ln(a*x)+4/3*ln(a*x-1)+4/3*ln(a*x+1))
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.06 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^4} \, dx=\frac {8 \, a^{3} x^{3} \log \left (a^{2} x^{2} - 1\right ) - 10 \, a^{3} x^{3} \log \left (x\right ) - a x + {\left (3 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{6 \, x^{3}} \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)/x^4,x, algorithm="fricas")
Output:
1/6*(8*a^3*x^3*log(a^2*x^2 - 1) - 10*a^3*x^3*log(x) - a*x + (3*a^4*x^4 + 6 *a^2*x^2 - 1)*log(-(a*x + 1)/(a*x - 1)))/x^3
Time = 0.42 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.10 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^4} \, dx=\begin {cases} a^{4} x \operatorname {atanh}{\left (a x \right )} - \frac {5 a^{3} \log {\left (x \right )}}{3} + \frac {8 a^{3} \log {\left (x - \frac {1}{a} \right )}}{3} + \frac {8 a^{3} \operatorname {atanh}{\left (a x \right )}}{3} + \frac {2 a^{2} \operatorname {atanh}{\left (a x \right )}}{x} - \frac {a}{6 x^{2}} - \frac {\operatorname {atanh}{\left (a x \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate((-a**2*x**2+1)**2*atanh(a*x)/x**4,x)
Output:
Piecewise((a**4*x*atanh(a*x) - 5*a**3*log(x)/3 + 8*a**3*log(x - 1/a)/3 + 8 *a**3*atanh(a*x)/3 + 2*a**2*atanh(a*x)/x - a/(6*x**2) - atanh(a*x)/(3*x**3 ), Ne(a, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^4} \, dx=\frac {1}{6} \, {\left (8 \, a^{2} \log \left (a x + 1\right ) + 8 \, a^{2} \log \left (a x - 1\right ) - 10 \, a^{2} \log \left (x\right ) - \frac {1}{x^{2}}\right )} a + \frac {1}{3} \, {\left (3 \, a^{4} x + \frac {6 \, a^{2} x^{2} - 1}{x^{3}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)/x^4,x, algorithm="maxima")
Output:
1/6*(8*a^2*log(a*x + 1) + 8*a^2*log(a*x - 1) - 10*a^2*log(x) - 1/x^2)*a + 1/3*(3*a^4*x + (6*a^2*x^2 - 1)/x^3)*arctanh(a*x)
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (60) = 120\).
Time = 0.13 (sec) , antiderivative size = 274, normalized size of antiderivative = 4.03 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^4} \, dx=\frac {1}{3} \, {\left (8 \, a^{2} \log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right ) - 3 \, a^{2} \log \left ({\left | -\frac {a x + 1}{a x - 1} + 1 \right |}\right ) - 5 \, a^{2} \log \left ({\left | -\frac {a x + 1}{a x - 1} - 1 \right |}\right ) + {\left (\frac {3 \, a^{2}}{\frac {a x + 1}{a x - 1} - 1} - \frac {\frac {3 \, {\left (a x + 1\right )}^{2} a^{2}}{{\left (a x - 1\right )}^{2}} + \frac {12 \, {\left (a x + 1\right )} a^{2}}{a x - 1} + 5 \, a^{2}}{{\left (\frac {a x + 1}{a x - 1} + 1\right )}^{3}}\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right ) + \frac {2 \, {\left (a x + 1\right )} a^{2}}{{\left (a x - 1\right )} {\left (\frac {a x + 1}{a x - 1} + 1\right )}^{2}}\right )} a \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)/x^4,x, algorithm="giac")
Output:
1/3*(8*a^2*log(abs(-a*x - 1)/abs(a*x - 1)) - 3*a^2*log(abs(-(a*x + 1)/(a*x - 1) + 1)) - 5*a^2*log(abs(-(a*x + 1)/(a*x - 1) - 1)) + (3*a^2/((a*x + 1) /(a*x - 1) - 1) - (3*(a*x + 1)^2*a^2/(a*x - 1)^2 + 12*(a*x + 1)*a^2/(a*x - 1) + 5*a^2)/((a*x + 1)/(a*x - 1) + 1)^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)) + 2*(a*x + 1)*a^2/((a*x - 1)*((a*x + 1)/(a*x - 1) + 1)^2))*a
Time = 3.48 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^4} \, dx=\frac {4\,a^3\,\ln \left (a^2\,x^2-1\right )}{3}-\frac {a}{6\,x^2}-\frac {\mathrm {atanh}\left (a\,x\right )}{3\,x^3}-\frac {5\,a^3\,\ln \left (x\right )}{3}+a^4\,x\,\mathrm {atanh}\left (a\,x\right )+\frac {2\,a^2\,\mathrm {atanh}\left (a\,x\right )}{x} \] Input:
int((atanh(a*x)*(a^2*x^2 - 1)^2)/x^4,x)
Output:
(4*a^3*log(a^2*x^2 - 1))/3 - a/(6*x^2) - atanh(a*x)/(3*x^3) - (5*a^3*log(x ))/3 + a^4*x*atanh(a*x) + (2*a^2*atanh(a*x))/x
Time = 0.16 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^4} \, dx=\frac {6 \mathit {atanh} \left (a x \right ) a^{4} x^{4}+16 \mathit {atanh} \left (a x \right ) a^{3} x^{3}+12 \mathit {atanh} \left (a x \right ) a^{2} x^{2}-2 \mathit {atanh} \left (a x \right )+16 \,\mathrm {log}\left (a^{2} x -a \right ) a^{3} x^{3}-10 \,\mathrm {log}\left (x \right ) a^{3} x^{3}-a x}{6 x^{3}} \] Input:
int((-a^2*x^2+1)^2*atanh(a*x)/x^4,x)
Output:
(6*atanh(a*x)*a**4*x**4 + 16*atanh(a*x)*a**3*x**3 + 12*atanh(a*x)*a**2*x** 2 - 2*atanh(a*x) + 16*log(a**2*x - a)*a**3*x**3 - 10*log(x)*a**3*x**3 - a* x)/(6*x**3)