Integrand size = 22, antiderivative size = 116 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {c^5}{9 a^{10} x^9}+\frac {c^5}{2 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}-\frac {14 c^5}{5 a^6 x^5}+\frac {14 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}+c^5 x+\frac {4 c^5 \log (x)}{a} \]
1/9*c^5/a^10/x^9+1/2*c^5/a^9/x^8+3/7*c^5/a^8/x^7-4/3*c^5/a^7/x^6-14/5*c^5/ a^6/x^5+14/3*c^5/a^4/x^3+4*c^5/a^3/x^2-3*c^5/a^2/x+c^5*x+4*c^5*ln(x)/a
Time = 0.05 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {c^5}{9 a^{10} x^9}+\frac {c^5}{2 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}-\frac {14 c^5}{5 a^6 x^5}+\frac {14 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}+c^5 x+\frac {4 c^5 \log (x)}{a} \]
c^5/(9*a^10*x^9) + c^5/(2*a^9*x^8) + (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7* x^6) - (14*c^5)/(5*a^6*x^5) + (14*c^5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) - ( 3*c^5)/(a^2*x) + c^5*x + (4*c^5*Log[x])/a
Time = 0.47 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.80, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6717, 27, 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.
| \(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^5 e^{4 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle \int \frac {c^5 \left (a^2-\frac {1}{x^2}\right )^5 e^{4 \text {arctanh}(a x)}}{a^{10}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^5 \int e^{4 \text {arctanh}(a x)} \left (a^2-\frac {1}{x^2}\right )^5dx}{a^{10}}\) |
| \(\Big \downarrow \) 6707 |
| \(\displaystyle -\frac {c^5 \int \frac {e^{4 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^5}{x^{10}}dx}{a^{10}}\) |
| \(\Big \downarrow \) 6700 |
| \(\displaystyle -\frac {c^5 \int \frac {(1-a x)^3 (a x+1)^7}{x^{10}}dx}{a^{10}}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {c^5 \int \left (-a^{10}-\frac {4 a^9}{x}-\frac {3 a^8}{x^2}+\frac {8 a^7}{x^3}+\frac {14 a^6}{x^4}-\frac {14 a^4}{x^6}-\frac {8 a^3}{x^7}+\frac {3 a^2}{x^8}+\frac {4 a}{x^9}+\frac {1}{x^{10}}\right )dx}{a^{10}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^5 \left (a^{10} (-x)-4 a^9 \log (x)+\frac {3 a^8}{x}-\frac {4 a^7}{x^2}-\frac {14 a^6}{3 x^3}+\frac {14 a^4}{5 x^5}+\frac {4 a^3}{3 x^6}-\frac {3 a^2}{7 x^7}-\frac {a}{2 x^8}-\frac {1}{9 x^9}\right )}{a^{10}}\) |
-((c^5*(-1/9*1/x^9 - a/(2*x^8) - (3*a^2)/(7*x^7) + (4*a^3)/(3*x^6) + (14*a ^4)/(5*x^5) - (14*a^6)/(3*x^3) - (4*a^7)/x^2 + (3*a^8)/x - a^10*x - 4*a^9* Log[x]))/a^10)
3.8.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.94 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.69
| method | result | size |
| default | \(\frac {c^{5} \left (a^{10} x +\frac {1}{9 x^{9}}+4 a^{9} \ln \left (x \right )+\frac {14 a^{6}}{3 x^{3}}-\frac {4 a^{3}}{3 x^{6}}+\frac {4 a^{7}}{x^{2}}-\frac {3 a^{8}}{x}+\frac {3 a^{2}}{7 x^{7}}+\frac {a}{2 x^{8}}-\frac {14 a^{4}}{5 x^{5}}\right )}{a^{10}}\) | \(80\) |
| risch | \(c^{5} x +\frac {-3 a^{8} c^{5} x^{8}+4 a^{7} c^{5} x^{7}+\frac {14}{3} a^{6} c^{5} x^{6}-\frac {14}{5} a^{4} c^{5} x^{4}-\frac {4}{3} a^{3} c^{5} x^{3}+\frac {3}{7} a^{2} c^{5} x^{2}+\frac {1}{2} a \,c^{5} x +\frac {1}{9} c^{5}}{a^{10} x^{9}}+\frac {4 c^{5} \ln \left (x \right )}{a}\) | \(103\) |
| parallelrisch | \(\frac {630 a^{10} c^{5} x^{10}+2520 c^{5} \ln \left (x \right ) a^{9} x^{9}-1890 a^{8} c^{5} x^{8}+2520 a^{7} c^{5} x^{7}+2940 a^{6} c^{5} x^{6}-1764 a^{4} c^{5} x^{4}-840 a^{3} c^{5} x^{3}+270 a^{2} c^{5} x^{2}+315 a \,c^{5} x +70 c^{5}}{630 a^{10} x^{9}}\) | \(112\) |
| norman | \(\frac {-4 a^{9} c^{5} x^{10}+a^{10} c^{5} x^{11}-\frac {c^{5}}{9 a}-\frac {7 c^{5} x}{18}+\frac {a \,c^{5} x^{2}}{14}+\frac {22 a^{3} c^{5} x^{4}}{15}-\frac {14 a^{4} c^{5} x^{5}}{5}-\frac {14 a^{5} c^{5} x^{6}}{3}+\frac {2 a^{6} c^{5} x^{7}}{3}+7 a^{7} c^{5} x^{8}+\frac {37 c^{5} a^{2} x^{3}}{21}}{\left (a x -1\right ) a^{9} x^{9}}+\frac {4 c^{5} \ln \left (x \right )}{a}\) | \(137\) |
| meijerg | \(-\frac {c^{5} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {4 c^{5} x}{-a x +1}-\frac {5 c^{5} \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 {5 c^{5} \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 {4 c^{5} \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}+\frac {2 c^{5} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (\frac {2 a x}{-2 a x +2}-\ln \left (-a x +1\right )+1+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}+\frac {20 c^{5} \left (\frac {4 a x}{-4 a x +4}-3 \ln \left (-a x +1\right )+1+3 \ln \left (x \right )+3 \ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {2}{a x}\right )}{a}-\frac {20 c^{5} \left (\frac {6 a x}{-6 a x +6}-5 \ln \left (-a x +1\right )+1+5 \ln \left (x \right )+5 \ln \left (-a \right )-\frac {1}{4 a^{4} x^{4}}-\frac {2}{3 x^{3} a^{3}}-\frac {3}{2 a^{2} x^{2}}-\frac {4}{a x}\right )}{a}+\frac {10 c^{5} \left (\frac {8 a x}{-8 a x +8}-7 \ln \left (-a x +1\right )+1+7 \ln \left (x \right )+7 \ln \left (-a \right )-\frac {1}{6 a^{6} x^{6}}-\frac {2}{5 x^{5} a^{5}}-\frac {3}{4 a^{4} x^{4}}-\frac {4}{3 x^{3} a^{3}}-\frac {5}{2 a^{2} x^{2}}-\frac {6}{a x}\right )}{a}-\frac {2 c^{5} \left (\frac {10 a x}{-10 a x +10}-9 \ln \left (-a x +1\right )+1+9 \ln \left (x \right )+9 \ln \left (-a \right )-\frac {1}{8 a^{8} x^{8}}-\frac {2}{7 x^{7} a^{7}}-\frac {1}{2 a^{6} x^{6}}-\frac {4}{5 x^{5} a^{5}}-\frac {5}{4 a^{4} x^{4}}-\frac {2}{x^{3} a^{3}}-\frac {7}{2 a^{2} x^{2}}-\frac {8}{a x}\right )}{a}+\frac {c^{5} \left (-\frac {11 a x}{-11 a x +11}+10 \ln \left (-a x +1\right )-1-10 \ln \left (x \right )-10 \ln \left (-a \right )+\frac {1}{9 x^{9} a^{9}}+\frac {1}{4 a^{8} x^{8}}+\frac {3}{7 x^{7} a^{7}}+\frac {2}{3 a^{6} x^{6}}+\frac {1}{x^{5} a^{5}}+\frac {3}{2 a^{4} x^{4}}+\frac {7}{3 x^{3} a^{3}}+\frac {4}{a^{2} x^{2}}+\frac {9}{a x}\right )}{a}\) | \(775\) |
c^5/a^10*(a^10*x+1/9/x^9+4*a^9*ln(x)+14/3*a^6/x^3-4/3*a^3/x^6+4*a^7/x^2-3* a^8/x+3/7*a^2/x^7+1/2*a/x^8-14/5*a^4/x^5)
Time = 0.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {630 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \left (x\right ) - 1890 \, a^{8} c^{5} x^{8} + 2520 \, a^{7} c^{5} x^{7} + 2940 \, a^{6} c^{5} x^{6} - 1764 \, a^{4} c^{5} x^{4} - 840 \, a^{3} c^{5} x^{3} + 270 \, a^{2} c^{5} x^{2} + 315 \, a c^{5} x + 70 \, c^{5}}{630 \, a^{10} x^{9}} \]
1/630*(630*a^10*c^5*x^10 + 2520*a^9*c^5*x^9*log(x) - 1890*a^8*c^5*x^8 + 25 20*a^7*c^5*x^7 + 2940*a^6*c^5*x^6 - 1764*a^4*c^5*x^4 - 840*a^3*c^5*x^3 + 2 70*a^2*c^5*x^2 + 315*a*c^5*x + 70*c^5)/(a^10*x^9)
Time = 0.34 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {a^{10} c^{5} x + 4 a^{9} c^{5} \log {\left (x \right )} + \frac {- 1890 a^{8} c^{5} x^{8} + 2520 a^{7} c^{5} x^{7} + 2940 a^{6} c^{5} x^{6} - 1764 a^{4} c^{5} x^{4} - 840 a^{3} c^{5} x^{3} + 270 a^{2} c^{5} x^{2} + 315 a c^{5} x + 70 c^{5}}{630 x^{9}}}{a^{10}} \]
(a**10*c**5*x + 4*a**9*c**5*log(x) + (-1890*a**8*c**5*x**8 + 2520*a**7*c** 5*x**7 + 2940*a**6*c**5*x**6 - 1764*a**4*c**5*x**4 - 840*a**3*c**5*x**3 + 270*a**2*c**5*x**2 + 315*a*c**5*x + 70*c**5)/(630*x**9))/a**10
Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.89 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=c^{5} x + \frac {4 \, c^{5} \log \left (x\right )}{a} - \frac {1890 \, a^{8} c^{5} x^{8} - 2520 \, a^{7} c^{5} x^{7} - 2940 \, a^{6} c^{5} x^{6} + 1764 \, a^{4} c^{5} x^{4} + 840 \, a^{3} c^{5} x^{3} - 270 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 70 \, c^{5}}{630 \, a^{10} x^{9}} \]
c^5*x + 4*c^5*log(x)/a - 1/630*(1890*a^8*c^5*x^8 - 2520*a^7*c^5*x^7 - 2940 *a^6*c^5*x^6 + 1764*a^4*c^5*x^4 + 840*a^3*c^5*x^3 - 270*a^2*c^5*x^2 - 315* a*c^5*x - 70*c^5)/(a^10*x^9)
Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.59 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-\frac {4 \, c^{5} \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {4 \, c^{5} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {{\left (630 \, c^{5} + \frac {4049 \, c^{5}}{a x - 1} + \frac {6201 \, c^{5}}{{\left (a x - 1\right )}^{2}} - \frac {18036 \, c^{5}}{{\left (a x - 1\right )}^{3}} - \frac {89124 \, c^{5}}{{\left (a x - 1\right )}^{4}} - \frac {160146 \, c^{5}}{{\left (a x - 1\right )}^{5}} - \frac {153090 \, c^{5}}{{\left (a x - 1\right )}^{6}} - \frac {80220 \, c^{5}}{{\left (a x - 1\right )}^{7}} - \frac {21420 \, c^{5}}{{\left (a x - 1\right )}^{8}} - \frac {2520 \, c^{5}}{{\left (a x - 1\right )}^{9}}\right )} {\left (a x - 1\right )}}{630 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{9}} \]
-4*c^5*log(abs(a*x - 1)/((a*x - 1)^2*abs(a)))/a + 4*c^5*log(abs(-1/(a*x - 1) - 1))/a + 1/630*(630*c^5 + 4049*c^5/(a*x - 1) + 6201*c^5/(a*x - 1)^2 - 18036*c^5/(a*x - 1)^3 - 89124*c^5/(a*x - 1)^4 - 160146*c^5/(a*x - 1)^5 - 1 53090*c^5/(a*x - 1)^6 - 80220*c^5/(a*x - 1)^7 - 21420*c^5/(a*x - 1)^8 - 25 20*c^5/(a*x - 1)^9)*(a*x - 1)/(a*(1/(a*x - 1) + 1)^9)
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.70 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {c^5\,\left (\frac {a\,x}{2}+\frac {3\,a^2\,x^2}{7}-\frac {4\,a^3\,x^3}{3}-\frac {14\,a^4\,x^4}{5}+\frac {14\,a^6\,x^6}{3}+4\,a^7\,x^7-3\,a^8\,x^8+a^{10}\,x^{10}+4\,a^9\,x^9\,\ln \left (x\right )+\frac {1}{9}\right )}{a^{10}\,x^9} \]