Integrand size = 24, antiderivative size = 73 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {1}{3 x^3}-\frac {a}{2 x^2}-\frac {2 a^2}{x}+\frac {a^3}{2 (1-a x)}+2 a^3 \log (x)-\frac {9}{4} a^3 \log (1-a x)+\frac {1}{4} a^3 \log (1+a x) \] Output:
-1/3/x^3-1/2*a/x^2-2*a^2/x+a^3/(-2*a*x+2)+2*a^3*ln(x)-9/4*a^3*ln(-a*x+1)+1 /4*a^3*ln(a*x+1)
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {1}{12} \left (-\frac {4}{x^3}-\frac {6 a}{x^2}-\frac {24 a^2}{x}+\frac {6 a^3}{1-a x}+24 a^3 \log (x)-27 a^3 \log (1-a x)+3 a^3 \log (1+a x)\right ) \] Input:
Integrate[E^ArcTanh[a*x]/(x^4*(1 - a^2*x^2)^(3/2)),x]
Output:
(-4/x^3 - (6*a)/x^2 - (24*a^2)/x + (6*a^3)/(1 - a*x) + 24*a^3*Log[x] - 27* a^3*Log[1 - a*x] + 3*a^3*Log[1 + a*x])/12
Time = 0.55 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6700, 99, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \int \frac {1}{x^4 (1-a x)^2 (a x+1)}dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {9 a^4}{4 (a x-1)}+\frac {a^4}{4 (a x+1)}+\frac {a^4}{2 (a x-1)^2}+\frac {2 a^3}{x}+\frac {2 a^2}{x^2}+\frac {a}{x^3}+\frac {1}{x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3}{2 (1-a x)}+2 a^3 \log (x)-\frac {9}{4} a^3 \log (1-a x)+\frac {1}{4} a^3 \log (a x+1)-\frac {2 a^2}{x}-\frac {a}{2 x^2}-\frac {1}{3 x^3}\) |
Input:
Int[E^ArcTanh[a*x]/(x^4*(1 - a^2*x^2)^(3/2)),x]
Output:
-1/3*1/x^3 - a/(2*x^2) - (2*a^2)/x + a^3/(2*(1 - a*x)) + 2*a^3*Log[x] - (9 *a^3*Log[1 - a*x])/4 + (a^3*Log[1 + a*x])/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {a^{3} \ln \left (a x +1\right )}{4}-\frac {a^{3}}{2 \left (a x -1\right )}-\frac {9 a^{3} \ln \left (a x -1\right )}{4}-\frac {1}{3 x^{3}}-\frac {a}{2 x^{2}}-\frac {2 a^{2}}{x}+2 a^{3} \ln \left (x \right )\) | \(62\) |
risch | \(\frac {-\frac {5}{2} a^{3} x^{3}+\frac {3}{2} a^{2} x^{2}+\frac {1}{6} a x +\frac {1}{3}}{\left (a x -1\right ) x^{3}}+2 a^{3} \ln \left (-x \right )-\frac {9 a^{3} \ln \left (-a x +1\right )}{4}+\frac {a^{3} \ln \left (a x +1\right )}{4}\) | \(67\) |
norman | \(\frac {\frac {1}{3}-a^{5} x^{5}+\frac {1}{2} a x +\frac {5}{3} a^{2} x^{2}-\frac {5}{2} a^{4} x^{4}}{\left (a^{2} x^{2}-1\right ) x^{3}}+2 a^{3} \ln \left (x \right )-\frac {9 a^{3} \ln \left (a x -1\right )}{4}+\frac {a^{3} \ln \left (a x +1\right )}{4}\) | \(76\) |
parallelrisch | \(\frac {24 a^{5} \ln \left (x \right ) x^{4}-27 \ln \left (a x -1\right ) x^{4} a^{5}+3 \ln \left (a x +1\right ) x^{4} a^{5}-24 \ln \left (x \right ) x^{3} a^{4}+27 \ln \left (a x -1\right ) x^{3} a^{4}-3 \ln \left (a x +1\right ) x^{3} a^{4}-30 a^{4} x^{3}+18 a^{3} x^{2}+2 a^{2} x +4 a}{12 a \,x^{3} \left (a x -1\right )}\) | \(118\) |
meijerg | \(-\frac {a^{3} \left (\frac {1}{a^{2} x^{2}}-1-4 \ln \left (x \right )-2 \ln \left (-a^{2}\right )-\frac {3 a^{2} x^{2}}{-3 a^{2} x^{2}+3}+2 \ln \left (-a^{2} x^{2}+1\right )\right )}{2}+\frac {a^{4} \left (-\frac {2 \left (-15 a^{4} x^{4}+10 a^{2} x^{2}+2\right )}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}} \left (-2 a^{2} x^{2}+2\right )}+\frac {5 a^{3} \operatorname {arctanh}\left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(132\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^2/x^4,x,method=_RETURNVERBOSE)
Output:
1/4*a^3*ln(a*x+1)-1/2*a^3/(a*x-1)-9/4*a^3*ln(a*x-1)-1/3/x^3-1/2*a/x^2-2*a^ 2/x+2*a^3*ln(x)
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {30 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 2 \, a x - 3 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x + 1\right ) + 27 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 24 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (x\right ) - 4}{12 \, {\left (a x^{4} - x^{3}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^2/x^4,x, algorithm="fricas")
Output:
-1/12*(30*a^3*x^3 - 18*a^2*x^2 - 2*a*x - 3*(a^4*x^4 - a^3*x^3)*log(a*x + 1 ) + 27*(a^4*x^4 - a^3*x^3)*log(a*x - 1) - 24*(a^4*x^4 - a^3*x^3)*log(x) - 4)/(a*x^4 - x^3)
Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx=2 a^{3} \log {\left (x \right )} - \frac {9 a^{3} \log {\left (x - \frac {1}{a} \right )}}{4} + \frac {a^{3} \log {\left (x + \frac {1}{a} \right )}}{4} + \frac {- 15 a^{3} x^{3} + 9 a^{2} x^{2} + a x + 2}{6 a x^{4} - 6 x^{3}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**2/x**4,x)
Output:
2*a**3*log(x) - 9*a**3*log(x - 1/a)/4 + a**3*log(x + 1/a)/4 + (-15*a**3*x* *3 + 9*a**2*x**2 + a*x + 2)/(6*a*x**4 - 6*x**3)
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {1}{4} \, a^{3} \log \left (a x + 1\right ) - \frac {9}{4} \, a^{3} \log \left (a x - 1\right ) + 2 \, a^{3} \log \left (x\right ) - \frac {15 \, a^{3} x^{3} - 9 \, a^{2} x^{2} - a x - 2}{6 \, {\left (a x^{4} - x^{3}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^2/x^4,x, algorithm="maxima")
Output:
1/4*a^3*log(a*x + 1) - 9/4*a^3*log(a*x - 1) + 2*a^3*log(x) - 1/6*(15*a^3*x ^3 - 9*a^2*x^2 - a*x - 2)/(a*x^4 - x^3)
Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {1}{4} \, a^{3} \log \left ({\left | a x + 1 \right |}\right ) - \frac {9}{4} \, a^{3} \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a^{3} \log \left ({\left | x \right |}\right ) - \frac {15 \, a^{3} x^{3} - 9 \, a^{2} x^{2} - a x - 2}{6 \, {\left (a x - 1\right )} x^{3}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^2/x^4,x, algorithm="giac")
Output:
1/4*a^3*log(abs(a*x + 1)) - 9/4*a^3*log(abs(a*x - 1)) + 2*a^3*log(abs(x)) - 1/6*(15*a^3*x^3 - 9*a^2*x^2 - a*x - 2)/((a*x - 1)*x^3)
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx=2\,a^3\,\ln \left (x\right )-\frac {9\,a^3\,\ln \left (a\,x-1\right )}{4}+\frac {a^3\,\ln \left (a\,x+1\right )}{4}+\frac {-\frac {5\,a^3\,x^3}{2}+\frac {3\,a^2\,x^2}{2}+\frac {a\,x}{6}+\frac {1}{3}}{a\,x^4-x^3} \] Input:
int((a*x + 1)/(x^4*(a^2*x^2 - 1)^2),x)
Output:
2*a^3*log(x) - (9*a^3*log(a*x - 1))/4 + (a^3*log(a*x + 1))/4 + ((a*x)/6 + (3*a^2*x^2)/2 - (5*a^3*x^3)/2 + 1/3)/(a*x^4 - x^3)
Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.51 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {-27 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}+27 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}+3 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}-3 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}+24 \,\mathrm {log}\left (x \right ) a^{4} x^{4}-24 \,\mathrm {log}\left (x \right ) a^{3} x^{3}-30 a^{4} x^{4}+18 a^{2} x^{2}+2 a x +4}{12 x^{3} \left (a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^2/x^4,x)
Output:
( - 27*log(a*x - 1)*a**4*x**4 + 27*log(a*x - 1)*a**3*x**3 + 3*log(a*x + 1) *a**4*x**4 - 3*log(a*x + 1)*a**3*x**3 + 24*log(x)*a**4*x**4 - 24*log(x)*a* *3*x**3 - 30*a**4*x**4 + 18*a**2*x**2 + 2*a*x + 4)/(12*x**3*(a*x - 1))