Integrand size = 22, antiderivative size = 78 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3}{5 a^6 x^5}+\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}-\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x-\frac {2 c^3 \log (x)}{a} \] Output:
1/5*c^3/a^6/x^5+1/2*c^3/a^5/x^4-1/3*c^3/a^4/x^3-2*c^3/a^3/x^2-c^3/a^2/x-c^ 3*x-2*c^3*ln(x)/a
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3}{5 a^6 x^5}+\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}-\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x-\frac {2 c^3 \log (x)}{a} \] Input:
Integrate[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^3,x]
Output:
c^3/(5*a^6*x^5) + c^3/(2*a^5*x^4) - c^3/(3*a^4*x^3) - (2*c^3)/(a^3*x^2) - c^3/(a^2*x) - c^3*x - (2*c^3*Log[x])/a
Time = 0.63 (sec) , antiderivative size = 61, 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.
\(\displaystyle \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx\) |
\(\Big \downarrow \) 6707 |
\(\displaystyle -\frac {c^3 \int \frac {e^{2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^3}{x^6}dx}{a^6}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle -\frac {c^3 \int \frac {(1-a x)^2 (a x+1)^4}{x^6}dx}{a^6}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {c^3 \int \left (a^6+\frac {2 a^5}{x}-\frac {a^4}{x^2}-\frac {4 a^3}{x^3}-\frac {a^2}{x^4}+\frac {2 a}{x^5}+\frac {1}{x^6}\right )dx}{a^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {c^3 \left (a^6 x+2 a^5 \log (x)+\frac {a^4}{x}+\frac {2 a^3}{x^2}+\frac {a^2}{3 x^3}-\frac {a}{2 x^4}-\frac {1}{5 x^5}\right )}{a^6}\) |
Input:
Int[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^3,x]
Output:
-((c^3*(-1/5*1/x^5 - a/(2*x^4) + a^2/(3*x^3) + (2*a^3)/x^2 + a^4/x + a^6*x + 2*a^5*Log[x]))/a^6)
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 = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {c^{3} \left (-x \,a^{6}-\frac {2 a^{3}}{x^{2}}-\frac {a^{4}}{x}+\frac {a}{2 x^{4}}-2 a^{5} \ln \left (x \right )-\frac {a^{2}}{3 x^{3}}+\frac {1}{5 x^{5}}\right )}{a^{6}}\) | \(57\) |
risch | \(-c^{3} x +\frac {-a^{4} c^{3} x^{4}-2 a^{3} c^{3} x^{3}-\frac {1}{3} a^{2} c^{3} x^{2}+\frac {1}{2} a \,c^{3} x +\frac {1}{5} c^{3}}{a^{6} x^{5}}-\frac {2 c^{3} \ln \left (x \right )}{a}\) | \(71\) |
norman | \(\frac {\frac {c^{3}}{5 a}+\frac {c^{3} x}{2}-\frac {a \,c^{3} x^{2}}{3}-2 a^{2} c^{3} x^{3}-a^{3} c^{3} x^{4}-a^{5} c^{3} x^{6}}{a^{5} x^{5}}-\frac {2 c^{3} \ln \left (x \right )}{a}\) | \(76\) |
parallelrisch | \(-\frac {30 a^{6} c^{3} x^{6}+60 c^{3} \ln \left (x \right ) a^{5} x^{5}+30 a^{4} c^{3} x^{4}+60 a^{3} c^{3} x^{3}+10 a^{2} c^{3} x^{2}-15 a \,c^{3} x -6 c^{3}}{30 a^{6} x^{5}}\) | \(79\) |
meijerg | \(-\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \operatorname {arctanh}\left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}-\frac {2 c^{3} \operatorname {arctanh}\left (a x \right )}{a}+\frac {c^{3} \left (-\frac {2 a^{2}}{x \left (-a^{2}\right )^{\frac {3}{2}}}-\frac {2}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}}}+\frac {2 a^{3} \operatorname {arctanh}\left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{\sqrt {-a^{2}}}-\frac {c^{3} \ln \left (-a^{2} x^{2}+1\right )}{a}-\frac {3 c^{3} \left (2 \ln \left (x \right )+\ln \left (-a^{2}\right )-\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {3 c^{3} \left (\frac {1}{a^{2} x^{2}}-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {c^{3} \left (-\frac {1}{2 a^{4} x^{4}}-\frac {1}{a^{2} x^{2}}+2 \ln \left (x \right )+\ln \left (-a^{2}\right )-\ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {c^{3} \left (-\frac {2 a^{4}}{x \left (-a^{2}\right )^{\frac {5}{2}}}-\frac {2 a^{2}}{3 x^{3} \left (-a^{2}\right )^{\frac {5}{2}}}-\frac {2}{5 x^{5} \left (-a^{2}\right )^{\frac {5}{2}}}+\frac {2 a^{5} \operatorname {arctanh}\left (a x \right )}{\left (-a^{2}\right )^{\frac {5}{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(320\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^3,x,method=_RETURNVERBOSE)
Output:
c^3/a^6*(-x*a^6-2*a^3/x^2-a^4/x+1/2*a/x^4-2*a^5*ln(x)-1/3*a^2/x^3+1/5/x^5)
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {30 \, a^{6} c^{3} x^{6} + 60 \, a^{5} c^{3} x^{5} \log \left (x\right ) + 30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^3,x, algorithm="fricas")
Output:
-1/30*(30*a^6*c^3*x^6 + 60*a^5*c^3*x^5*log(x) + 30*a^4*c^3*x^4 + 60*a^3*c^ 3*x^3 + 10*a^2*c^3*x^2 - 15*a*c^3*x - 6*c^3)/(a^6*x^5)
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {- a^{6} c^{3} x - 2 a^{5} c^{3} \log {\left (x \right )} - \frac {30 a^{4} c^{3} x^{4} + 60 a^{3} c^{3} x^{3} + 10 a^{2} c^{3} x^{2} - 15 a c^{3} x - 6 c^{3}}{30 x^{5}}}{a^{6}} \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**3,x)
Output:
(-a**6*c**3*x - 2*a**5*c**3*log(x) - (30*a**4*c**3*x**4 + 60*a**3*c**3*x** 3 + 10*a**2*c**3*x**2 - 15*a*c**3*x - 6*c**3)/(30*x**5))/a**6
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.91 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-c^{3} x - \frac {2 \, c^{3} \log \left (x\right )}{a} - \frac {30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^3,x, algorithm="maxima")
Output:
-c^3*x - 2*c^3*log(x)/a - 1/30*(30*a^4*c^3*x^4 + 60*a^3*c^3*x^3 + 10*a^2*c ^3*x^2 - 15*a*c^3*x - 6*c^3)/(a^6*x^5)
Time = 0.14 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.92 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-c^{3} x - \frac {2 \, c^{3} \log \left ({\left | x \right |}\right )}{a} - \frac {30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^3,x, algorithm="giac")
Output:
-c^3*x - 2*c^3*log(abs(x))/a - 1/30*(30*a^4*c^3*x^4 + 60*a^3*c^3*x^3 + 10* a^2*c^3*x^2 - 15*a*c^3*x - 6*c^3)/(a^6*x^5)
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.73 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {c^3\,\left (\frac {a^2\,x^2}{3}-\frac {a\,x}{2}+2\,a^3\,x^3+a^4\,x^4+a^6\,x^6+2\,a^5\,x^5\,\ln \left (x\right )-\frac {1}{5}\right )}{a^6\,x^5} \] Input:
int(-((c - c/(a^2*x^2))^3*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
-(c^3*((a^2*x^2)/3 - (a*x)/2 + 2*a^3*x^3 + a^4*x^4 + a^6*x^6 + 2*a^5*x^5*l og(x) - 1/5))/(a^6*x^5)
Time = 0.14 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.76 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^{3} \left (-60 \,\mathrm {log}\left (x \right ) a^{5} x^{5}-30 a^{6} x^{6}-30 a^{4} x^{4}-60 a^{3} x^{3}-10 a^{2} x^{2}+15 a x +6\right )}{30 a^{6} x^{5}} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^3,x)
Output:
(c**3*( - 60*log(x)*a**5*x**5 - 30*a**6*x**6 - 30*a**4*x**4 - 60*a**3*x**3 - 10*a**2*x**2 + 15*a*x + 6))/(30*a**6*x**5)