Integrand size = 22, antiderivative size = 128 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {c^5}{9 a^{10} x^9}+\frac {c^5}{4 a^9 x^8}-\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}+\frac {2 c^5}{5 a^6 x^5}+\frac {3 c^5}{a^5 x^4}+\frac {2 c^5}{3 a^4 x^3}-\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}-c^5 x-\frac {2 c^5 \log (x)}{a} \] Output:
1/9*c^5/a^10/x^9+1/4*c^5/a^9/x^8-3/7*c^5/a^8/x^7-4/3*c^5/a^7/x^6+2/5*c^5/a ^6/x^5+3*c^5/a^5/x^4+2/3*c^5/a^4/x^3-4*c^5/a^3/x^2-3*c^5/a^2/x-c^5*x-2*c^5 *ln(x)/a
Time = 0.01 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {c^5}{9 a^{10} x^9}+\frac {c^5}{4 a^9 x^8}-\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}+\frac {2 c^5}{5 a^6 x^5}+\frac {3 c^5}{a^5 x^4}+\frac {2 c^5}{3 a^4 x^3}-\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}-c^5 x-\frac {2 c^5 \log (x)}{a} \] Input:
Integrate[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^5,x]
Output:
c^5/(9*a^10*x^9) + c^5/(4*a^9*x^8) - (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7* x^6) + (2*c^5)/(5*a^6*x^5) + (3*c^5)/(a^5*x^4) + (2*c^5)/(3*a^4*x^3) - (4* c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) - c^5*x - (2*c^5*Log[x])/a
Time = 0.44 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.78, 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.
| \(\displaystyle \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx\) |
| \(\Big \downarrow \) 6707 |
| \(\displaystyle -\frac {c^5 \int \frac {e^{2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^5}{x^{10}}dx}{a^{10}}\) |
| \(\Big \downarrow \) 6700 |
| \(\displaystyle -\frac {c^5 \int \frac {(1-a x)^4 (a x+1)^6}{x^{10}}dx}{a^{10}}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {c^5 \int \left (a^{10}+\frac {2 a^9}{x}-\frac {3 a^8}{x^2}-\frac {8 a^7}{x^3}+\frac {2 a^6}{x^4}+\frac {12 a^5}{x^5}+\frac {2 a^4}{x^6}-\frac {8 a^3}{x^7}-\frac {3 a^2}{x^8}+\frac {2 a}{x^9}+\frac {1}{x^{10}}\right )dx}{a^{10}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^5 \left (a^{10} x+2 a^9 \log (x)+\frac {3 a^8}{x}+\frac {4 a^7}{x^2}-\frac {2 a^6}{3 x^3}-\frac {3 a^5}{x^4}-\frac {2 a^4}{5 x^5}+\frac {4 a^3}{3 x^6}+\frac {3 a^2}{7 x^7}-\frac {a}{4 x^8}-\frac {1}{9 x^9}\right )}{a^{10}}\) |
Input:
Int[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^5,x]
Output:
-((c^5*(-1/9*1/x^9 - a/(4*x^8) + (3*a^2)/(7*x^7) + (4*a^3)/(3*x^6) - (2*a^ 4)/(5*x^5) - (3*a^5)/x^4 - (2*a^6)/(3*x^3) + (4*a^7)/x^2 + (3*a^8)/x + a^1 0*x + 2*a^9*Log[x]))/a^10)
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 = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(\frac {c^{5} \left (-x \,a^{10}-\frac {4 a^{3}}{3 x^{6}}-\frac {4 a^{7}}{x^{2}}-\frac {3 a^{8}}{x}-\frac {3 a^{2}}{7 x^{7}}+\frac {1}{9 x^{9}}+\frac {3 a^{5}}{x^{4}}-2 a^{9} \ln \left (x \right )+\frac {2 a^{6}}{3 x^{3}}+\frac {2 a^{4}}{5 x^{5}}+\frac {a}{4 x^{8}}\right )}{a^{10}}\) | \(89\) |
| risch | \(-c^{5} x +\frac {-3 a^{8} c^{5} x^{8}-4 a^{7} c^{5} x^{7}+\frac {2}{3} a^{6} c^{5} x^{6}+3 a^{5} c^{5} x^{5}+\frac {2}{5} a^{4} c^{5} x^{4}-\frac {4}{3} a^{3} c^{5} x^{3}-\frac {3}{7} a^{2} c^{5} x^{2}+\frac {1}{4} a \,c^{5} x +\frac {1}{9} c^{5}}{a^{10} x^{9}}-\frac {2 c^{5} \ln \left (x \right )}{a}\) | \(115\) |
| norman | \(\frac {\frac {c^{5}}{9 a}+\frac {c^{5} x}{4}+3 a^{4} c^{5} x^{5}+\frac {2 a^{5} c^{5} x^{6}}{3}-4 a^{6} c^{5} x^{7}-3 a^{7} c^{5} x^{8}-a^{9} c^{5} x^{10}-\frac {3 c^{5} a \,x^{2}}{7}-\frac {4 c^{5} a^{2} x^{3}}{3}+\frac {2 c^{5} a^{3} x^{4}}{5}}{a^{9} x^{9}}-\frac {2 c^{5} \ln \left (x \right )}{a}\) | \(120\) |
| parallelrisch | \(-\frac {1260 x^{10} a^{10} c^{5}+2520 c^{5} \ln \left (x \right ) a^{9} x^{9}+3780 a^{8} c^{5} x^{8}+5040 a^{7} c^{5} x^{7}-840 a^{6} c^{5} x^{6}-3780 a^{5} c^{5} x^{5}-504 a^{4} c^{5} x^{4}+1680 a^{3} c^{5} x^{3}+540 a^{2} c^{5} x^{2}-315 a \,c^{5} x -140 c^{5}}{1260 a^{10} x^{9}}\) | \(123\) |
| meijerg | \(-\frac {c^{5} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \operatorname {arctanh}\left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}-\frac {4 c^{5} \operatorname {arctanh}\left (a x \right )}{a}-\frac {5 c^{5} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \,\operatorname {arctanh}\left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}+\frac {5 c^{5} \left (-\frac {2 a^{4}}{x \left (-a^{2}\right )^{\frac {5}{2}}}-\frac {2 a^{2}}{3 x^{3} \left (-a^{2}\right )^{\frac {5}{2}}}-\frac {2}{5 x^{5} \left (-a^{2}\right )^{\frac {5}{2}}}+\frac {2 a^{5} \operatorname {arctanh}\left (a x \right )}{\left (-a^{2}\right )^{\frac {5}{2}}}\right )}{2 \sqrt {-a^{2}}}+\frac {2 c^{5} \left (-\frac {2 a^{6}}{x \left (-a^{2}\right )^{\frac {7}{2}}}-\frac {2 a^{4}}{3 x^{3} \left (-a^{2}\right )^{\frac {7}{2}}}-\frac {2 a^{2}}{5 x^{5} \left (-a^{2}\right )^{\frac {7}{2}}}-\frac {2}{7 x^{7} \left (-a^{2}\right )^{\frac {7}{2}}}+\frac {2 a^{7} \operatorname {arctanh}\left (a x \right )}{\left (-a^{2}\right )^{\frac {7}{2}}}\right )}{\sqrt {-a^{2}}}-\frac {c^{5} \ln \left (-a^{2} x^{2}+1\right )}{a}-\frac {5 c^{5} \left (2 \ln \left (x \right )+\ln \left (-a^{2}\right )-\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {10 c^{5} \left (\frac {1}{a^{2} x^{2}}-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {10 c^{5} \left (-\frac {1}{2 a^{4} x^{4}}-\frac {1}{a^{2} x^{2}}+2 \ln \left (x \right )+\ln \left (-a^{2}\right )-\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {5 c^{5} \left (\frac {1}{3 a^{6} x^{6}}+\frac {1}{2 a^{4} x^{4}}+\frac {1}{a^{2} x^{2}}-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {c^{5} \left (-\frac {1}{4 a^{8} x^{8}}-\frac {1}{3 a^{6} x^{6}}-\frac {1}{2 a^{4} x^{4}}-\frac {1}{a^{2} x^{2}}+2 \ln \left (x \right )+\ln \left (-a^{2}\right )-\ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {c^{5} \left (-\frac {2 a^{8}}{x \left (-a^{2}\right )^{\frac {9}{2}}}-\frac {2 a^{6}}{3 x^{3} \left (-a^{2}\right )^{\frac {9}{2}}}-\frac {2 a^{4}}{5 x^{5} \left (-a^{2}\right )^{\frac {9}{2}}}-\frac {2 a^{2}}{7 x^{7} \left (-a^{2}\right )^{\frac {9}{2}}}-\frac {2}{9 x^{9} \left (-a^{2}\right )^{\frac {9}{2}}}+\frac {2 a^{9} \operatorname {arctanh}\left (a x \right )}{\left (-a^{2}\right )^{\frac {9}{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(610\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^5,x,method=_RETURNVERBOSE)
Output:
c^5/a^10*(-x*a^10-4/3*a^3/x^6-4*a^7/x^2-3*a^8/x-3/7*a^2/x^7+1/9/x^9+3*a^5/ x^4-2*a^9*ln(x)+2/3*a^6/x^3+2/5*a^4/x^5+1/4*a/x^8)
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-\frac {1260 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \left (x\right ) + 3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^5,x, algorithm="fricas")
Output:
-1/1260*(1260*a^10*c^5*x^10 + 2520*a^9*c^5*x^9*log(x) + 3780*a^8*c^5*x^8 + 5040*a^7*c^5*x^7 - 840*a^6*c^5*x^6 - 3780*a^5*c^5*x^5 - 504*a^4*c^5*x^4 + 1680*a^3*c^5*x^3 + 540*a^2*c^5*x^2 - 315*a*c^5*x - 140*c^5)/(a^10*x^9)
Time = 0.37 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {- a^{10} c^{5} x - 2 a^{9} c^{5} \log {\left (x \right )} - \frac {3780 a^{8} c^{5} x^{8} + 5040 a^{7} c^{5} x^{7} - 840 a^{6} c^{5} x^{6} - 3780 a^{5} c^{5} x^{5} - 504 a^{4} c^{5} x^{4} + 1680 a^{3} c^{5} x^{3} + 540 a^{2} c^{5} x^{2} - 315 a c^{5} x - 140 c^{5}}{1260 x^{9}}}{a^{10}} \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**5,x)
Output:
(-a**10*c**5*x - 2*a**9*c**5*log(x) - (3780*a**8*c**5*x**8 + 5040*a**7*c** 5*x**7 - 840*a**6*c**5*x**6 - 3780*a**5*c**5*x**5 - 504*a**4*c**5*x**4 + 1 680*a**3*c**5*x**3 + 540*a**2*c**5*x**2 - 315*a*c**5*x - 140*c**5)/(1260*x **9))/a**10
Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-c^{5} x - \frac {2 \, c^{5} \log \left (x\right )}{a} - \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^5,x, algorithm="maxima")
Output:
-c^5*x - 2*c^5*log(x)/a - 1/1260*(3780*a^8*c^5*x^8 + 5040*a^7*c^5*x^7 - 84 0*a^6*c^5*x^6 - 3780*a^5*c^5*x^5 - 504*a^4*c^5*x^4 + 1680*a^3*c^5*x^3 + 54 0*a^2*c^5*x^2 - 315*a*c^5*x - 140*c^5)/(a^10*x^9)
Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-c^{5} x - \frac {2 \, c^{5} \log \left ({\left | x \right |}\right )}{a} - \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^5,x, algorithm="giac")
Output:
-c^5*x - 2*c^5*log(abs(x))/a - 1/1260*(3780*a^8*c^5*x^8 + 5040*a^7*c^5*x^7 - 840*a^6*c^5*x^6 - 3780*a^5*c^5*x^5 - 504*a^4*c^5*x^4 + 1680*a^3*c^5*x^3 + 540*a^2*c^5*x^2 - 315*a*c^5*x - 140*c^5)/(a^10*x^9)
Time = 14.34 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-\frac {c^5\,\left (\frac {3\,a^2\,x^2}{7}-\frac {a\,x}{4}+\frac {4\,a^3\,x^3}{3}-\frac {2\,a^4\,x^4}{5}-3\,a^5\,x^5-\frac {2\,a^6\,x^6}{3}+4\,a^7\,x^7+3\,a^8\,x^8+a^{10}\,x^{10}+2\,a^9\,x^9\,\ln \left (x\right )-\frac {1}{9}\right )}{a^{10}\,x^9} \] Input:
int(-((c - c/(a^2*x^2))^5*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
-(c^5*((3*a^2*x^2)/7 - (a*x)/4 + (4*a^3*x^3)/3 - (2*a^4*x^4)/5 - 3*a^5*x^5 - (2*a^6*x^6)/3 + 4*a^7*x^7 + 3*a^8*x^8 + a^10*x^10 + 2*a^9*x^9*log(x) - 1/9))/(a^10*x^9)
Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.71 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {c^{5} \left (-2520 \,\mathrm {log}\left (x \right ) a^{9} x^{9}-1260 a^{10} x^{10}-3780 a^{8} x^{8}-5040 a^{7} x^{7}+840 a^{6} x^{6}+3780 a^{5} x^{5}+504 a^{4} x^{4}-1680 a^{3} x^{3}-540 a^{2} x^{2}+315 a x +140\right )}{1260 a^{10} x^{9}} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^5,x)
Output:
(c**5*( - 2520*log(x)*a**9*x**9 - 1260*a**10*x**10 - 3780*a**8*x**8 - 5040 *a**7*x**7 + 840*a**6*x**6 + 3780*a**5*x**5 + 504*a**4*x**4 - 1680*a**3*x* *3 - 540*a**2*x**2 + 315*a*x + 140))/(1260*a**10*x**9)