\(\int e^{2 \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^5 \, dx\) [756]

Optimal result

Integrand size = 22, antiderivative size = 127 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-\frac {c^5}{9 a^{10} x^9}-\frac {c^5}{4 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}+\frac {4 c^5}{3 a^7 x^6}-\frac {2 c^5}{5 a^6 x^5}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}+\frac {3 c^5}{a^2 x}+c^5 x+\frac {2 c^5 \log (x)}{a} \] 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
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-\frac {c^5}{9 a^{10} x^9}-\frac {c^5}{4 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}+\frac {4 c^5}{3 a^7 x^6}-\frac {2 c^5}{5 a^6 x^5}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}+\frac {3 c^5}{a^2 x}+c^5 x+\frac {2 c^5 \log (x)}{a} \] Input:

Integrate[E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^5,x]
 

Output:

-1/9*c^5/(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
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.78, 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.

\(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^5 e^{2 \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6717

\(\displaystyle -\int \frac {c^5 e^{2 \text {arctanh}(a x)} \left (a^2-\frac {1}{x^2}\right )^5}{a^{10}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c^5 \int e^{2 \text {arctanh}(a x)} \left (a^2-\frac {1}{x^2}\right )^5dx}{a^{10}}\)

\(\Big \downarrow \) 6707

\(\displaystyle \frac {c^5 \int \frac {e^{2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^5}{x^{10}}dx}{a^{10}}\)

\(\Big \downarrow \) 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*ArcCoth[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^10* 
x + 2*a^9*Log[x]))/a^10
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 99

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]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6700

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])
 

rule 6707

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]
 

rule 6717

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]
 

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.69

method	result	size
default	\(\frac {c^{5} \left (x \,a^{10}+\frac {4 a^{3}}{3 x^{6}}+\frac {4 a^{7}}{x^{2}}+\frac {3 a^{8}}{x}+\frac {3 a^{2}}{7 x^{7}}-\frac {1}{9 x^{9}}-\frac {3 a^{5}}{x^{4}}+2 a^{9} \ln \left (x \right )-\frac {2 a^{6}}{3 x^{3}}-\frac {2 a^{4}}{5 x^{5}}-\frac {a}{4 x^{8}}\right )}{a^{10}}\)	\(88\)
risch	\(c^{5} x +\frac {3 a^{8} c^{5} x^{8}+4 a^{7} c^{5} x^{7}-\frac {2}{3} a^{6} c^{5} x^{6}-3 a^{5} c^{5} x^{5}-\frac {2}{5} a^{4} c^{5} x^{4}+\frac {4}{3} a^{3} c^{5} x^{3}+\frac {3}{7} a^{2} c^{5} x^{2}-\frac {1}{4} a \,c^{5} x -\frac {1}{9} c^{5}}{a^{10} x^{9}}+\frac {2 c^{5} \ln \left (x \right )}{a}\)	\(114\)
norman	\(\frac {a^{9} c^{5} x^{10}-\frac {c^{5}}{9 a}-\frac {c^{5} x}{4}+\frac {4 a^{2} c^{5} x^{3}}{3}-3 a^{4} c^{5} x^{5}-\frac {2 a^{5} c^{5} x^{6}}{3}+4 a^{6} c^{5} x^{7}+3 a^{7} c^{5} x^{8}+\frac {3 c^{5} a \,x^{2}}{7}-\frac {2 c^{5} a^{3} x^{4}}{5}}{a^{9} x^{9}}+\frac {2 c^{5} \ln \left (x \right )}{a}\)	\(119\)
parallelrisch	\(\frac {1260 a^{10} c^{5} x^{10}+2520 c^{5} \ln \left (x \right ) a^{9} x^{9}+3780 a^{8} c^{5} x^{8}+5040 a^{7} c^{5} x^{7}-840 a^{6} c^{5} x^{6}-3780 a^{5} c^{5} x^{5}-504 a^{4} c^{5} x^{4}+1680 a^{3} c^{5} x^{3}+540 a^{2} c^{5} x^{2}-315 a \,c^{5} x -140 c^{5}}{1260 a^{10} x^{9}}\)	\(123\)
meijerg	\(-\frac {c^{5} \left (-a x -\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (\ln \left (x \right )+\ln \left (-a \right )-\ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}+\ln \left (x \right )+\ln \left (-a \right )-\ln \left (-a x +1\right )\right )}{a}+\frac {10 c^{5} \left (-\frac {1}{4 a^{4} x^{4}}-\frac {1}{3 x^{3} a^{3}}-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}+\ln \left (x \right )+\ln \left (-a \right )-\ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (-\frac {1}{6 a^{6} x^{6}}-\frac {1}{5 x^{5} a^{5}}-\frac {1}{4 a^{4} x^{4}}-\frac {1}{3 x^{3} a^{3}}-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}+\ln \left (x \right )+\ln \left (-a \right )-\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} \left (-\frac {1}{8 a^{8} x^{8}}-\frac {1}{7 x^{7} a^{7}}-\frac {1}{6 a^{6} x^{6}}-\frac {1}{5 x^{5} a^{5}}-\frac {1}{4 a^{4} x^{4}}-\frac {1}{3 x^{3} a^{3}}-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}+\ln \left (x \right )+\ln \left (-a \right )-\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} \ln \left (-a x +1\right )}{a}-\frac {5 c^{5} \left (\frac {1}{a x}-\ln \left (x \right )-\ln \left (-a \right )+\ln \left (-a x +1\right )\right )}{a}+\frac {10 c^{5} \left (\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}-\ln \left (x \right )-\ln \left (-a \right )+\ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (\frac {1}{5 x^{5} a^{5}}+\frac {1}{4 a^{4} x^{4}}+\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}-\ln \left (x \right )-\ln \left (-a \right )+\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (\frac {1}{7 x^{7} a^{7}}+\frac {1}{6 a^{6} x^{6}}+\frac {1}{5 x^{5} a^{5}}+\frac {1}{4 a^{4} x^{4}}+\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}-\ln \left (x \right )-\ln \left (-a \right )+\ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (\frac {1}{9 x^{9} a^{9}}+\frac {1}{8 a^{8} x^{8}}+\frac {1}{7 x^{7} a^{7}}+\frac {1}{6 a^{6} x^{6}}+\frac {1}{5 x^{5} a^{5}}+\frac {1}{4 a^{4} x^{4}}+\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}-\ln \left (x \right )-\ln \left (-a \right )+\ln \left (-a x +1\right )\right )}{a}\)	\(642\)

Input:

int(1/(a*x-1)*(a*x+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)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.96 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {1260 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \left (x\right ) + 3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \] Input:

integrate(1/(a*x-1)*(a*x+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)
 

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {a^{10} c^{5} x + 2 a^{9} c^{5} \log {\left (x \right )} + \frac {3780 a^{8} c^{5} x^{8} + 5040 a^{7} c^{5} x^{7} - 840 a^{6} c^{5} x^{6} - 3780 a^{5} c^{5} x^{5} - 504 a^{4} c^{5} x^{4} + 1680 a^{3} c^{5} x^{3} + 540 a^{2} c^{5} x^{2} - 315 a c^{5} x - 140 c^{5}}{1260 x^{9}}}{a^{10}} \] Input:

integrate(1/(a*x-1)*(a*x+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 + 16 
80*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
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=c^{5} x + \frac {2 \, c^{5} \log \left (x\right )}{a} + \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \] Input:

integrate(1/(a*x-1)*(a*x+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 - 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)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.91 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=c^{5} x + \frac {2 \, c^{5} \log \left ({\left | x \right |}\right )}{a} + \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \] Input:

integrate(1/(a*x-1)*(a*x+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)
 

Mupad [B] (verification not implemented)

Time = 13.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.70 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {c^5\,\left (\frac {3\,a^2\,x^2}{7}-\frac {a\,x}{4}+\frac {4\,a^3\,x^3}{3}-\frac {2\,a^4\,x^4}{5}-3\,a^5\,x^5-\frac {2\,a^6\,x^6}{3}+4\,a^7\,x^7+3\,a^8\,x^8+a^{10}\,x^{10}+2\,a^9\,x^9\,\ln \left (x\right )-\frac {1}{9}\right )}{a^{10}\,x^9} \] Input:

int(((c - c/(a^2*x^2))^5*(a*x + 1))/(a*x - 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)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.72 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {c^{5} \left (2520 \,\mathrm {log}\left (x \right ) a^{9} x^{9}+1260 a^{10} x^{10}+3780 a^{8} x^{8}+5040 a^{7} x^{7}-840 a^{6} x^{6}-3780 a^{5} x^{5}-504 a^{4} x^{4}+1680 a^{3} x^{3}+540 a^{2} x^{2}-315 a x -140\right )}{1260 a^{10} x^{9}} \] Input:

int(1/(a*x-1)*(a*x+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)