Integrand size = 22, antiderivative size = 91 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {c^4}{7 a^8 x^7}-\frac {c^4}{3 a^7 x^6}+\frac {2 c^4}{5 a^6 x^5}+\frac {3 c^4}{2 a^5 x^4}-\frac {3 c^4}{a^3 x^2}-\frac {2 c^4}{a^2 x}-c^4 x-\frac {2 c^4 \log (x)}{a} \] Output:
-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 = 0.01 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {c^4}{7 a^8 x^7}-\frac {c^4}{3 a^7 x^6}+\frac {2 c^4}{5 a^6 x^5}+\frac {3 c^4}{2 a^5 x^4}-\frac {3 c^4}{a^3 x^2}-\frac {2 c^4}{a^2 x}-c^4 x-\frac {2 c^4 \log (x)}{a} \] Input:
Integrate[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^4,x]
Output:
-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 = 0.40 (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 |
\(\displaystyle \frac {c^4 \int \frac {(1-a x)^3 (a x+1)^5}{x^8}dx}{a^8}\) |
\(\Big \downarrow \) 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 |
\(\displaystyle \frac {c^4 \left (a^8 (-x)-2 a^7 \log (x)-\frac {2 a^6}{x}-\frac {3 a^5}{x^2}+\frac {3 a^3}{2 x^4}+\frac {2 a^2}{5 x^5}-\frac {a}{3 x^6}-\frac {1}{7 x^7}\right )}{a^8}\) |
Input:
Int[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^4,x]
Output:
(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
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 = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {c^{4} \left (-x \,a^{8}-\frac {a}{3 x^{6}}-\frac {3 a^{5}}{x^{2}}-\frac {2 a^{6}}{x}-\frac {1}{7 x^{7}}+\frac {3 a^{3}}{2 x^{4}}-2 a^{7} \ln \left (x \right )+\frac {2 a^{2}}{5 x^{5}}\right )}{a^{8}}\) | \(65\) |
risch | \(-c^{4} x +\frac {-2 a^{6} c^{4} x^{6}-3 a^{5} c^{4} x^{5}+\frac {3}{2} a^{3} c^{4} x^{3}+\frac {2}{5} a^{2} c^{4} x^{2}-\frac {1}{3} a \,c^{4} x -\frac {1}{7} c^{4}}{a^{8} x^{7}}-\frac {2 c^{4} \ln \left (x \right )}{a}\) | \(82\) |
norman | \(\frac {-\frac {c^{4}}{7 a}-\frac {c^{4} x}{3}+\frac {2 a \,c^{4} x^{2}}{5}+\frac {3 a^{2} c^{4} x^{3}}{2}-3 a^{4} c^{4} x^{5}-2 a^{5} c^{4} x^{6}-a^{7} c^{4} x^{8}}{a^{7} x^{7}}-\frac {2 c^{4} \ln \left (x \right )}{a}\) | \(87\) |
parallelrisch | \(-\frac {210 x^{8} a^{8} c^{4}+420 c^{4} \ln \left (x \right ) a^{7} x^{7}+420 a^{6} c^{4} x^{6}+630 a^{5} c^{4} x^{5}-315 a^{3} c^{4} x^{3}-84 a^{2} c^{4} x^{2}+70 a \,c^{4} x +30 c^{4}}{210 a^{8} x^{7}}\) | \(90\) |
meijerg | \(-\frac {c^{4} \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 {3 c^{4} \operatorname {arctanh}\left (a x \right )}{a}-\frac {c^{4} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \,\operatorname {arctanh}\left (a x \right )}{\sqrt {-a^{2}}}\right )}{\sqrt {-a^{2}}}+\frac {c^{4} \left (-\frac {2 a^{2}}{x \left (-a^{2}\right )^{\frac {3}{2}}}-\frac {2}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}}}+\frac {2 a^{3} \operatorname {arctanh}\left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{\sqrt {-a^{2}}}+\frac {3 c^{4} \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 {c^{4} \ln \left (-a^{2} x^{2}+1\right )}{a}-\frac {4 c^{4} \left (2 \ln \left (x \right )+\ln \left (-a^{2}\right )-\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {6 c^{4} \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 {4 c^{4} \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 {c^{4} \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^{4} \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 )}{2 \sqrt {-a^{2}}}\) | \(500\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^4,x,method=_RETURNVERBOSE)
Output:
c^4/a^8*(-x*a^8-1/3*a/x^6-3*a^5/x^2-2*a^6/x-1/7/x^7+3/2*a^3/x^4-2*a^7*ln(x )+2/5*a^2/x^5)
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {210 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) + 420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^4,x, algorithm="fricas")
Output:
-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 = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {- a^{8} c^{4} x - 2 a^{7} c^{4} \log {\left (x \right )} - \frac {420 a^{6} c^{4} x^{6} + 630 a^{5} c^{4} x^{5} - 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} + 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}} \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**4,x)
Output:
(-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 = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.90 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-c^{4} x - \frac {2 \, c^{4} \log \left (x\right )}{a} - \frac {420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^4,x, algorithm="maxima")
Output:
-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 = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-c^{4} x - \frac {2 \, c^{4} \log \left ({\left | x \right |}\right )}{a} - \frac {420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^4,x, algorithm="giac")
Output:
-c^4*x - 2*c^4*log(abs(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 = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {c^4\,\left (\frac {a\,x}{3}-\frac {2\,a^2\,x^2}{5}-\frac {3\,a^3\,x^3}{2}+3\,a^5\,x^5+2\,a^6\,x^6+a^8\,x^8+2\,a^7\,x^7\,\ln \left (x\right )+\frac {1}{7}\right )}{a^8\,x^7} \] Input:
int(-((c - c/(a^2*x^2))^4*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
-(c^4*((a*x)/3 - (2*a^2*x^2)/5 - (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*log(x) + 1/7))/(a^8*x^7)
Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^{4} \left (-420 \,\mathrm {log}\left (x \right ) a^{7} x^{7}-210 a^{8} x^{8}-420 a^{6} x^{6}-630 a^{5} x^{5}+315 a^{3} x^{3}+84 a^{2} x^{2}-70 a x -30\right )}{210 a^{8} x^{7}} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^4,x)
Output:
(c**4*( - 420*log(x)*a**7*x**7 - 210*a**8*x**8 - 420*a**6*x**6 - 630*a**5* x**5 + 315*a**3*x**3 + 84*a**2*x**2 - 70*a*x - 30))/(210*a**8*x**7)