Integrand size = 25, antiderivative size = 109 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=-\frac {1}{c^3 x}+\frac {a}{12 c^3 (1-a x)^3}+\frac {3 a}{8 c^3 (1-a x)^2}+\frac {23 a}{16 c^3 (1-a x)}-\frac {a}{16 c^3 (1+a x)}+\frac {2 a \log (x)}{c^3}-\frac {9 a \log (1-a x)}{4 c^3}+\frac {a \log (1+a x)}{4 c^3} \] Output:
-1/c^3/x+1/12*a/c^3/(-a*x+1)^3+3/8*a/c^3/(-a*x+1)^2+23/16*a/c^3/(-a*x+1)-1 /16*a/c^3/(a*x+1)+2*a*ln(x)/c^3-9/4*a*ln(-a*x+1)/c^3+1/4*a*ln(a*x+1)/c^3
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {-\frac {48}{x}+\frac {69 a}{1-a x}-\frac {4 a}{(-1+a x)^3}+\frac {18 a}{(-1+a x)^2}-\frac {3 a}{1+a x}+96 a \log (x)-108 a \log (1-a x)+12 a \log (1+a x)}{48 c^3} \] Input:
Integrate[E^(2*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^3),x]
Output:
(-48/x + (69*a)/(1 - a*x) - (4*a)/(-1 + a*x)^3 + (18*a)/(-1 + a*x)^2 - (3* a)/(1 + a*x) + 96*a*Log[x] - 108*a*Log[1 - a*x] + 12*a*Log[1 + a*x])/(48*c ^3)
Time = 0.58 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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.
\(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\int \frac {1}{x^2 (1-a x)^4 (a x+1)^2}dx}{c^3}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\int \left (-\frac {9 a^2}{4 (a x-1)}+\frac {a^2}{4 (a x+1)}+\frac {23 a^2}{16 (a x-1)^2}+\frac {a^2}{16 (a x+1)^2}-\frac {3 a^2}{4 (a x-1)^3}+\frac {a^2}{4 (a x-1)^4}+\frac {2 a}{x}+\frac {1}{x^2}\right )dx}{c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {23 a}{16 (1-a x)}-\frac {a}{16 (a x+1)}+\frac {3 a}{8 (1-a x)^2}+\frac {a}{12 (1-a x)^3}+2 a \log (x)-\frac {9}{4} a \log (1-a x)+\frac {1}{4} a \log (a x+1)-\frac {1}{x}}{c^3}\) |
Input:
Int[E^(2*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^3),x]
Output:
(-x^(-1) + a/(12*(1 - a*x)^3) + (3*a)/(8*(1 - a*x)^2) + (23*a)/(16*(1 - a* x)) - a/(16*(1 + a*x)) + 2*a*Log[x] - (9*a*Log[1 - a*x])/4 + (a*Log[1 + a* x])/4)/c^3
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 = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {-\frac {a}{16 \left (a x +1\right )}+\frac {a \ln \left (a x +1\right )}{4}-\frac {a}{12 \left (a x -1\right )^{3}}+\frac {3 a}{8 \left (a x -1\right )^{2}}-\frac {23 a}{16 \left (a x -1\right )}-\frac {9 a \ln \left (a x -1\right )}{4}-\frac {1}{x}+2 \ln \left (x \right ) a}{c^{3}}\) | \(74\) |
risch | \(\frac {-\frac {5}{2} a^{4} x^{4}+4 a^{3} x^{3}+\frac {7}{6} a^{2} x^{2}-\frac {23}{6} a x +1}{c^{3} x \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )}+\frac {2 a \ln \left (-x \right )}{c^{3}}+\frac {a \ln \left (a x +1\right )}{4 c^{3}}-\frac {9 a \ln \left (-a x +1\right )}{4 c^{3}}\) | \(92\) |
norman | \(\frac {\frac {1}{c}-\frac {11 a^{2} x^{2}}{2 c}+\frac {20 a^{4} x^{4}}{3 c}-\frac {5 a^{6} x^{6}}{2 c}-\frac {3 a^{3} x^{3}}{c}+\frac {9 a^{5} x^{5}}{2 c}-\frac {11 a^{7} x^{7}}{6 c}}{\left (a^{2} x^{2}-1\right )^{3} x \,c^{2}}+\frac {2 a \ln \left (x \right )}{c^{3}}-\frac {9 a \ln \left (a x -1\right )}{4 c^{3}}+\frac {a \ln \left (a x +1\right )}{4 c^{3}}\) | \(122\) |
parallelrisch | \(\frac {12+48 a^{3} x^{3}+62 a^{4} x^{4}-46 a^{5} x^{5}-3 \ln \left (a x +1\right ) x a -48 \ln \left (x \right ) x^{4} a^{4}-24 a \ln \left (x \right ) x +48 a^{2} \ln \left (x \right ) x^{2}-78 a^{2} x^{2}+6 \ln \left (a x +1\right ) x^{2} a^{2}+54 \ln \left (a x -1\right ) x^{4} a^{4}-6 \ln \left (a x +1\right ) x^{4} a^{4}+27 a \ln \left (a x -1\right ) x -27 \ln \left (a x -1\right ) x^{5} a^{5}+3 \ln \left (a x +1\right ) x^{5} a^{5}-54 a^{2} \ln \left (a x -1\right ) x^{2}+24 a^{5} \ln \left (x \right ) x^{5}}{12 c^{3} x \left (a^{2} x^{2}-1\right ) \left (a x -1\right )^{2}}\) | \(201\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)/x^2/(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
1/c^3*(-1/16*a/(a*x+1)+1/4*a*ln(a*x+1)-1/12*a/(a*x-1)^3+3/8*a/(a*x-1)^2-23 /16*a/(a*x-1)-9/4*a*ln(a*x-1)-1/x+2*ln(x)*a)
Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.61 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=-\frac {30 \, a^{4} x^{4} - 48 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 46 \, a x - 3 \, {\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right )} \log \left (a x + 1\right ) + 27 \, {\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 24 \, {\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right )} \log \left (x\right ) - 12}{12 \, {\left (a^{4} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{4} + 2 \, a c^{3} x^{2} - c^{3} x\right )}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)/x^2/(-a^2*c*x^2+c)^3,x, algorithm="fricas ")
Output:
-1/12*(30*a^4*x^4 - 48*a^3*x^3 - 14*a^2*x^2 + 46*a*x - 3*(a^5*x^5 - 2*a^4* x^4 + 2*a^2*x^2 - a*x)*log(a*x + 1) + 27*(a^5*x^5 - 2*a^4*x^4 + 2*a^2*x^2 - a*x)*log(a*x - 1) - 24*(a^5*x^5 - 2*a^4*x^4 + 2*a^2*x^2 - a*x)*log(x) - 12)/(a^4*c^3*x^5 - 2*a^3*c^3*x^4 + 2*a*c^3*x^2 - c^3*x)
Time = 0.47 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {- 15 a^{4} x^{4} + 24 a^{3} x^{3} + 7 a^{2} x^{2} - 23 a x + 6}{6 a^{4} c^{3} x^{5} - 12 a^{3} c^{3} x^{4} + 12 a c^{3} x^{2} - 6 c^{3} x} + \frac {2 a \log {\left (x \right )} - \frac {9 a \log {\left (x - \frac {1}{a} \right )}}{4} + \frac {a \log {\left (x + \frac {1}{a} \right )}}{4}}{c^{3}} \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)/x**2/(-a**2*c*x**2+c)**3,x)
Output:
(-15*a**4*x**4 + 24*a**3*x**3 + 7*a**2*x**2 - 23*a*x + 6)/(6*a**4*c**3*x** 5 - 12*a**3*c**3*x**4 + 12*a*c**3*x**2 - 6*c**3*x) + (2*a*log(x) - 9*a*log (x - 1/a)/4 + a*log(x + 1/a)/4)/c**3
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=-\frac {15 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 23 \, a x - 6}{6 \, {\left (a^{4} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{4} + 2 \, a c^{3} x^{2} - c^{3} x\right )}} + \frac {a \log \left (a x + 1\right )}{4 \, c^{3}} - \frac {9 \, a \log \left (a x - 1\right )}{4 \, c^{3}} + \frac {2 \, a \log \left (x\right )}{c^{3}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)/x^2/(-a^2*c*x^2+c)^3,x, algorithm="maxima ")
Output:
-1/6*(15*a^4*x^4 - 24*a^3*x^3 - 7*a^2*x^2 + 23*a*x - 6)/(a^4*c^3*x^5 - 2*a ^3*c^3*x^4 + 2*a*c^3*x^2 - c^3*x) + 1/4*a*log(a*x + 1)/c^3 - 9/4*a*log(a*x - 1)/c^3 + 2*a*log(x)/c^3
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {a \log \left ({\left | a x + 1 \right |}\right )}{4 \, c^{3}} - \frac {9 \, a \log \left ({\left | a x - 1 \right |}\right )}{4 \, c^{3}} + \frac {2 \, a \log \left ({\left | x \right |}\right )}{c^{3}} - \frac {15 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 23 \, a x - 6}{6 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{3} c^{3} x} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)/x^2/(-a^2*c*x^2+c)^3,x, algorithm="giac")
Output:
1/4*a*log(abs(a*x + 1))/c^3 - 9/4*a*log(abs(a*x - 1))/c^3 + 2*a*log(abs(x) )/c^3 - 1/6*(15*a^4*x^4 - 24*a^3*x^3 - 7*a^2*x^2 + 23*a*x - 6)/((a*x + 1)* (a*x - 1)^3*c^3*x)
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {2\,a\,\ln \left (x\right )}{c^3}-\frac {-\frac {5\,a^4\,x^4}{2}+4\,a^3\,x^3+\frac {7\,a^2\,x^2}{6}-\frac {23\,a\,x}{6}+1}{-a^4\,c^3\,x^5+2\,a^3\,c^3\,x^4-2\,a\,c^3\,x^2+c^3\,x}-\frac {9\,a\,\ln \left (a\,x-1\right )}{4\,c^3}+\frac {a\,\ln \left (a\,x+1\right )}{4\,c^3} \] Input:
int(-(a*x + 1)^2/(x^2*(c - a^2*c*x^2)^3*(a^2*x^2 - 1)),x)
Output:
(2*a*log(x))/c^3 - ((7*a^2*x^2)/6 - (23*a*x)/6 + 4*a^3*x^3 - (5*a^4*x^4)/2 + 1)/(c^3*x - 2*a*c^3*x^2 + 2*a^3*c^3*x^4 - a^4*c^3*x^5) - (9*a*log(a*x - 1))/(4*c^3) + (a*log(a*x + 1))/(4*c^3)
Time = 0.16 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.84 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {-27 \,\mathrm {log}\left (a x -1\right ) a^{5} x^{5}+54 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}-54 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}+27 \,\mathrm {log}\left (a x -1\right ) a x +3 \,\mathrm {log}\left (a x +1\right ) a^{5} x^{5}-6 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}+6 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}-3 \,\mathrm {log}\left (a x +1\right ) a x +24 \,\mathrm {log}\left (x \right ) a^{5} x^{5}-48 \,\mathrm {log}\left (x \right ) a^{4} x^{4}+48 \,\mathrm {log}\left (x \right ) a^{2} x^{2}-24 \,\mathrm {log}\left (x \right ) a x -15 a^{5} x^{5}+48 a^{3} x^{3}-16 a^{2} x^{2}-31 a x +12}{12 c^{3} x \left (a^{4} x^{4}-2 a^{3} x^{3}+2 a x -1\right )} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)/x^2/(-a^2*c*x^2+c)^3,x)
Output:
( - 27*log(a*x - 1)*a**5*x**5 + 54*log(a*x - 1)*a**4*x**4 - 54*log(a*x - 1 )*a**2*x**2 + 27*log(a*x - 1)*a*x + 3*log(a*x + 1)*a**5*x**5 - 6*log(a*x + 1)*a**4*x**4 + 6*log(a*x + 1)*a**2*x**2 - 3*log(a*x + 1)*a*x + 24*log(x)* a**5*x**5 - 48*log(x)*a**4*x**4 + 48*log(x)*a**2*x**2 - 24*log(x)*a*x - 15 *a**5*x**5 + 48*a**3*x**3 - 16*a**2*x**2 - 31*a*x + 12)/(12*c**3*x*(a**4*x **4 - 2*a**3*x**3 + 2*a*x - 1))