Integrand size = 22, antiderivative size = 74 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {x}{c^2}-\frac {1}{4 a c^2 (1+a x)^2}+\frac {7}{4 a c^2 (1+a x)}-\frac {\log (1-a x)}{8 a c^2}+\frac {17 \log (1+a x)}{8 a c^2} \] Output:
-x/c^2-1/4/a/c^2/(a*x+1)^2+7/4/a/c^2/(a*x+1)-1/8*ln(-a*x+1)/a/c^2+17/8*ln( a*x+1)/a/c^2
Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {12+6 a x-16 a^2 x^2-8 a^3 x^3-(1+a x)^2 \log (1-a x)+17 (1+a x)^2 \log (1+a x)}{8 a (c+a c x)^2} \] Input:
Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^2),x]
Output:
(12 + 6*a*x - 16*a^2*x^2 - 8*a^3*x^3 - (1 + a*x)^2*Log[1 - a*x] + 17*(1 + a*x)^2*Log[1 + a*x])/(8*a*(c + a*c*x)^2)
Time = 0.64 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6707, 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.
\(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx\) |
\(\Big \downarrow \) 6707 |
\(\displaystyle \frac {a^4 \int \frac {e^{-2 \text {arctanh}(a x)} x^4}{\left (1-a^2 x^2\right )^2}dx}{c^2}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {a^4 \int \frac {x^4}{(1-a x) (a x+1)^3}dx}{c^2}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {a^4 \int \left (\frac {17}{8 a^4 (a x+1)}-\frac {7}{4 a^4 (a x+1)^2}+\frac {1}{2 a^4 (a x+1)^3}-\frac {1}{a^4}-\frac {1}{8 a^4 (a x-1)}\right )dx}{c^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^4 \left (\frac {7}{4 a^5 (a x+1)}-\frac {1}{4 a^5 (a x+1)^2}-\frac {\log (1-a x)}{8 a^5}+\frac {17 \log (a x+1)}{8 a^5}-\frac {x}{a^4}\right )}{c^2}\) |
Input:
Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^2),x]
Output:
(a^4*(-(x/a^4) - 1/(4*a^5*(1 + a*x)^2) + 7/(4*a^5*(1 + a*x)) - Log[1 - a*x ]/(8*a^5) + (17*Log[1 + a*x])/(8*a^5)))/c^2
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])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[d^p Int[(u/x^(2*p))*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x ] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {a^{4} \left (-\frac {x}{a^{4}}+\frac {17 \ln \left (a x +1\right )}{8 a^{5}}-\frac {1}{4 a^{5} \left (a x +1\right )^{2}}+\frac {7}{4 a^{5} \left (a x +1\right )}-\frac {\ln \left (a x -1\right )}{8 a^{5}}\right )}{c^{2}}\) | \(61\) |
risch | \(-\frac {x}{c^{2}}+\frac {\frac {7 c^{2} x}{4}+\frac {3 c^{2}}{2 a}}{c^{4} \left (a x +1\right )^{2}}+\frac {17 \ln \left (-a x -1\right )}{8 a \,c^{2}}-\frac {\ln \left (a x -1\right )}{8 a \,c^{2}}\) | \(63\) |
norman | \(\frac {\frac {7 a \,x^{2}}{2 c}-\frac {a^{4} x^{5}}{c}+\frac {9 x}{4 c}-\frac {5 a^{2} x^{3}}{4 c}-\frac {7 a^{3} x^{4}}{2 c}}{\left (a x +1\right )^{3} c \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{8 a \,c^{2}}+\frac {17 \ln \left (a x +1\right )}{8 a \,c^{2}}\) | \(97\) |
parallelrisch | \(\frac {-8 a^{3} x^{3}-a^{2} \ln \left (a x -1\right ) x^{2}+17 \ln \left (a x +1\right ) x^{2} a^{2}-28 a^{2} x^{2}-2 a \ln \left (a x -1\right ) x +34 \ln \left (a x +1\right ) x a -18 a x -\ln \left (a x -1\right )+17 \ln \left (a x +1\right )}{8 c^{2} \left (a x +1\right )^{2} a}\) | \(101\) |
Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^2,x,method=_RETURNVERBOSE)
Output:
a^4/c^2*(-x/a^4+17/8*ln(a*x+1)/a^5-1/4/a^5/(a*x+1)^2+7/4/a^5/(a*x+1)-1/8/a ^5*ln(a*x-1))
Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {8 \, a^{3} x^{3} + 16 \, a^{2} x^{2} - 6 \, a x - 17 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 12}{8 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^2,x, algorithm="fricas")
Output:
-1/8*(8*a^3*x^3 + 16*a^2*x^2 - 6*a*x - 17*(a^2*x^2 + 2*a*x + 1)*log(a*x + 1) + (a^2*x^2 + 2*a*x + 1)*log(a*x - 1) - 12)/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2)
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=- a^{4} \left (\frac {- 7 a x - 6}{4 a^{7} c^{2} x^{2} + 8 a^{6} c^{2} x + 4 a^{5} c^{2}} + \frac {x}{a^{4} c^{2}} + \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{8} - \frac {17 \log {\left (x + \frac {1}{a} \right )}}{8}}{a^{5} c^{2}}\right ) \] Input:
integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2)**2,x)
Output:
-a**4*((-7*a*x - 6)/(4*a**7*c**2*x**2 + 8*a**6*c**2*x + 4*a**5*c**2) + x/( a**4*c**2) + (log(x - 1/a)/8 - 17*log(x + 1/a)/8)/(a**5*c**2))
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {7 \, a x + 6}{4 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} - \frac {x}{c^{2}} + \frac {17 \, \log \left (a x + 1\right )}{8 \, a c^{2}} - \frac {\log \left (a x - 1\right )}{8 \, a c^{2}} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^2,x, algorithm="maxima")
Output:
1/4*(7*a*x + 6)/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2) - x/c^2 + 17/8*log(a*x + 1)/(a*c^2) - 1/8*log(a*x - 1)/(a*c^2)
Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {a x + 1}{a c^{2}} - \frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a c^{2}} - \frac {\log \left ({\left | -\frac {2}{a x + 1} + 1 \right |}\right )}{8 \, a c^{2}} + \frac {\frac {7 \, a^{5} c^{2}}{a x + 1} - \frac {a^{5} c^{2}}{{\left (a x + 1\right )}^{2}}}{4 \, a^{6} c^{4}} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^2,x, algorithm="giac")
Output:
-(a*x + 1)/(a*c^2) - 2*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/(a*c^2) - 1/ 8*log(abs(-2/(a*x + 1) + 1))/(a*c^2) + 1/4*(7*a^5*c^2/(a*x + 1) - a^5*c^2/ (a*x + 1)^2)/(a^6*c^4)
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {\frac {7\,x}{4}+\frac {3}{2\,a}}{a^2\,c^2\,x^2+2\,a\,c^2\,x+c^2}-\frac {x}{c^2}-\frac {\ln \left (a\,x-1\right )}{8\,a\,c^2}+\frac {17\,\ln \left (a\,x+1\right )}{8\,a\,c^2} \] Input:
int(-(a^2*x^2 - 1)/((c - c/(a^2*x^2))^2*(a*x + 1)^2),x)
Output:
((7*x)/4 + 3/(2*a))/(c^2 + a^2*c^2*x^2 + 2*a*c^2*x) - x/c^2 - log(a*x - 1) /(8*a*c^2) + (17*log(a*x + 1))/(8*a*c^2)
Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.42 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {-\mathrm {log}\left (a x -1\right ) a^{2} x^{2}-2 \,\mathrm {log}\left (a x -1\right ) a x -\mathrm {log}\left (a x -1\right )+17 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}+34 \,\mathrm {log}\left (a x +1\right ) a x +17 \,\mathrm {log}\left (a x +1\right )-8 a^{3} x^{3}-19 a^{2} x^{2}+9}{8 a \,c^{2} \left (a^{2} x^{2}+2 a x +1\right )} \] Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^2,x)
Output:
( - log(a*x - 1)*a**2*x**2 - 2*log(a*x - 1)*a*x - log(a*x - 1) + 17*log(a* x + 1)*a**2*x**2 + 34*log(a*x + 1)*a*x + 17*log(a*x + 1) - 8*a**3*x**3 - 1 9*a**2*x**2 + 9)/(8*a*c**2*(a**2*x**2 + 2*a*x + 1))