Integrand size = 22, antiderivative size = 91 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}+\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}-\frac {2 c^4}{a^2 x}-c^4 x+\frac {2 c^4 \log (x)}{a} \]
-1/7*c^4/a^8/x^7+1/3*c^4/a^7/x^6+2/5*c^4/a^6/x^5-3/2*c^4/a^5/x^4+3*c^4/a^3 /x^2-2*c^4/a^2/x-c^4*x+2*c^4*ln(x)/a
Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}+\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}-\frac {2 c^4}{a^2 x}-c^4 x+\frac {2 c^4 \log (x)}{a} \]
-1/7*c^4/(a^8*x^7) + c^4/(3*a^7*x^6) + (2*c^4)/(5*a^6*x^5) - (3*c^4)/(2*a^ 5*x^4) + (3*c^4)/(a^3*x^2) - (2*c^4)/(a^2*x) - c^4*x + (2*c^4*Log[x])/a
Time = 0.39 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.79, 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.
\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx\) |
\(\Big \downarrow \) 6707 |
\(\displaystyle \frac {c^4 \int \frac {e^{-2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^4}{x^8}dx}{a^8}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {c^4 \int \frac {(1-a x)^5 (a x+1)^3}{x^8}dx}{a^8}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {c^4 \int \left (-a^8+\frac {2 a^7}{x}+\frac {2 a^6}{x^2}-\frac {6 a^5}{x^3}+\frac {6 a^3}{x^5}-\frac {2 a^2}{x^6}-\frac {2 a}{x^7}+\frac {1}{x^8}\right )dx}{a^8}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^4 \left (a^8 (-x)+2 a^7 \log (x)-\frac {2 a^6}{x}+\frac {3 a^5}{x^2}-\frac {3 a^3}{2 x^4}+\frac {2 a^2}{5 x^5}+\frac {a}{3 x^6}-\frac {1}{7 x^7}\right )}{a^8}\) |
(c^4*(-1/7*1/x^7 + a/(3*x^6) + (2*a^2)/(5*x^5) - (3*a^3)/(2*x^4) + (3*a^5) /x^2 - (2*a^6)/x - a^8*x + 2*a^7*Log[x]))/a^8
3.7.70.3.1 Defintions of rubi rules used
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 = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {c^{4} \left (-a^{8} x +2 a^{7} \ln \left (x \right )-\frac {3 a^{3}}{2 x^{4}}+\frac {a}{3 x^{6}}+\frac {3 a^{5}}{x^{2}}-\frac {2 a^{6}}{x}-\frac {1}{7 x^{7}}+\frac {2 a^{2}}{5 x^{5}}\right )}{a^{8}}\) | \(65\) |
risch | \(-c^{4} x +\frac {-2 a^{6} c^{4} x^{6}+3 a^{5} c^{4} x^{5}-\frac {3}{2} a^{3} c^{4} x^{3}+\frac {2}{5} a^{2} c^{4} x^{2}+\frac {1}{3} a \,c^{4} x -\frac {1}{7} c^{4}}{a^{8} x^{7}}+\frac {2 c^{4} \ln \left (x \right )}{a}\) | \(82\) |
parallelrisch | \(\frac {-210 a^{8} c^{4} x^{8}+420 c^{4} \ln \left (x \right ) a^{7} x^{7}-420 a^{6} c^{4} x^{6}+630 a^{5} c^{4} x^{5}-315 a^{3} c^{4} x^{3}+84 a^{2} c^{4} x^{2}+70 a \,c^{4} x -30 c^{4}}{210 a^{8} x^{7}}\) | \(90\) |
norman | \(\frac {a^{7} c^{4} x^{8}+a^{5} c^{4} x^{6}-\frac {c^{4}}{7 a}+\frac {4 c^{4} x}{21}+\frac {11 a \,c^{4} x^{2}}{15}-\frac {11 a^{2} c^{4} x^{3}}{10}-\frac {3 a^{3} c^{4} x^{4}}{2}+3 a^{4} c^{4} x^{5}-a^{8} c^{4} x^{9}}{a^{7} x^{7} \left (a x +1\right )}+\frac {2 c^{4} \ln \left (x \right )}{a}\) | \(114\) |
meijerg | \(-\frac {c^{4} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}+\frac {5 c^{4} x}{a x +1}-\frac {10 c^{4} \left (\frac {3 a x}{3 a x +3}+2 \ln \left (a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (a \right )-\frac {1}{a x}\right )}{a}+\frac {10 c^{4} \left (\frac {5 a x}{5 a x +5}+4 \ln \left (a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (a \right )-\frac {1}{3 x^{3} a^{3}}+\frac {1}{a^{2} x^{2}}-\frac {3}{a x}\right )}{a}-\frac {5 c^{4} \left (\frac {7 a x}{7 a x +7}+6 \ln \left (a x +1\right )-1-6 \ln \left (x \right )-6 \ln \left (a \right )-\frac {1}{5 x^{5} a^{5}}+\frac {1}{2 a^{4} x^{4}}-\frac {1}{x^{3} a^{3}}+\frac {2}{a^{2} x^{2}}-\frac {5}{a x}\right )}{a}+\frac {c^{4} \left (\frac {9 a x}{9 a x +9}+8 \ln \left (a x +1\right )-1-8 \ln \left (x \right )-8 \ln \left (a \right )-\frac {1}{7 x^{7} a^{7}}+\frac {1}{3 a^{6} x^{6}}-\frac {3}{5 x^{5} a^{5}}+\frac {1}{a^{4} x^{4}}-\frac {5}{3 x^{3} a^{3}}+\frac {3}{a^{2} x^{2}}-\frac {7}{a x}\right )}{a}\) | \(326\) |
Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {210 \, a^{8} c^{4} x^{8} - 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) + 420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]
-1/210*(210*a^8*c^4*x^8 - 420*a^7*c^4*x^7*log(x) + 420*a^6*c^4*x^6 - 630*a ^5*c^4*x^5 + 315*a^3*c^4*x^3 - 84*a^2*c^4*x^2 - 70*a*c^4*x + 30*c^4)/(a^8* x^7)
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {- a^{8} c^{4} x + 2 a^{7} c^{4} \log {\left (x \right )} - \frac {420 a^{6} c^{4} x^{6} - 630 a^{5} c^{4} x^{5} + 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} - 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}} \]
(-a**8*c**4*x + 2*a**7*c**4*log(x) - (420*a**6*c**4*x**6 - 630*a**5*c**4*x **5 + 315*a**3*c**4*x**3 - 84*a**2*c**4*x**2 - 70*a*c**4*x + 30*c**4)/(210 *x**7))/a**8
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.90 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-c^{4} x + \frac {2 \, c^{4} \log \left (x\right )}{a} - \frac {420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]
-c^4*x + 2*c^4*log(x)/a - 1/210*(420*a^6*c^4*x^6 - 630*a^5*c^4*x^5 + 315*a ^3*c^4*x^3 - 84*a^2*c^4*x^2 - 70*a*c^4*x + 30*c^4)/(a^8*x^7)
Time = 0.28 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.76 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {2 \, c^{4} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {2 \, c^{4} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} + \frac {{\left (210 \, c^{4} - \frac {719 \, c^{4}}{a x + 1} - \frac {427 \, c^{4}}{{\left (a x + 1\right )}^{2}} + \frac {5271 \, c^{4}}{{\left (a x + 1\right )}^{3}} - \frac {9485 \, c^{4}}{{\left (a x + 1\right )}^{4}} + \frac {7490 \, c^{4}}{{\left (a x + 1\right )}^{5}} - \frac {2730 \, c^{4}}{{\left (a x + 1\right )}^{6}} + \frac {420 \, c^{4}}{{\left (a x + 1\right )}^{7}}\right )} {\left (a x + 1\right )}}{210 \, a {\left (\frac {1}{a x + 1} - 1\right )}^{7}} \]
-2*c^4*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/a + 2*c^4*log(abs(-1/(a*x + 1) + 1))/a + 1/210*(210*c^4 - 719*c^4/(a*x + 1) - 427*c^4/(a*x + 1)^2 + 52 71*c^4/(a*x + 1)^3 - 9485*c^4/(a*x + 1)^4 + 7490*c^4/(a*x + 1)^5 - 2730*c^ 4/(a*x + 1)^6 + 420*c^4/(a*x + 1)^7)*(a*x + 1)/(a*(1/(a*x + 1) - 1)^7)
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4\,\left (\frac {a\,x}{3}+\frac {2\,a^2\,x^2}{5}-\frac {3\,a^3\,x^3}{2}+3\,a^5\,x^5-2\,a^6\,x^6-a^8\,x^8+2\,a^7\,x^7\,\ln \left (x\right )-\frac {1}{7}\right )}{a^8\,x^7} \]