\(\int e^{2 \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^5 \, dx\) [780]

Optimal result

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

[Out]

-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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6302, 6292, 6285, 90} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-\frac {c^5}{9 a^{10} x^9}-\frac {c^5}{4 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}+\frac {4 c^5}{3 a^7 x^6}-\frac {2 c^5}{5 a^6 x^5}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}+\frac {3 c^5}{a^2 x}+\frac {2 c^5 \log (x)}{a}+c^5 x \]

[In]

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

[Out]

-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

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(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 6285

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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 6292

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[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 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx \\ & = \frac {c^5 \int \frac {e^{2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^5}{x^{10}} \, dx}{a^{10}} \\ & = \frac {c^5 \int \frac {(1-a x)^4 (1+a x)^6}{x^{10}} \, dx}{a^{10}} \\ & = \frac {c^5 \int \left (a^{10}+\frac {1}{x^{10}}+\frac {2 a}{x^9}-\frac {3 a^2}{x^8}-\frac {8 a^3}{x^7}+\frac {2 a^4}{x^6}+\frac {12 a^5}{x^5}+\frac {2 a^6}{x^4}-\frac {8 a^7}{x^3}-\frac {3 a^8}{x^2}+\frac {2 a^9}{x}\right ) \, dx}{a^{10}} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-\frac {c^5}{9 a^{10} x^9}-\frac {c^5}{4 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}+\frac {4 c^5}{3 a^7 x^6}-\frac {2 c^5}{5 a^6 x^5}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}+\frac {3 c^5}{a^2 x}+c^5 x+\frac {2 c^5 \log (x)}{a} \]

[In]

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

[Out]

-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

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.69

method	result	size
default	\(\frac {c^{5} \left (a^{10} x -\frac {1}{9 x^{9}}+2 a^{9} \ln \left (x \right )-\frac {3 a^{5}}{x^{4}}-\frac {2 a^{6}}{3 x^{3}}+\frac {4 a^{3}}{3 x^{6}}+\frac {4 a^{7}}{x^{2}}+\frac {3 a^{8}}{x}+\frac {3 a^{2}}{7 x^{7}}-\frac {a}{4 x^{8}}-\frac {2 a^{4}}{5 x^{5}}\right )}{a^{10}}\)	\(88\)
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	\(\frac {a^{9} c^{5} x^{10}-\frac {c^{5}}{9 a}-\frac {c^{5} x}{4}+\frac {3 a \,c^{5} x^{2}}{7}-\frac {2 a^{3} c^{5} x^{4}}{5}-3 a^{4} c^{5} x^{5}-\frac {2 a^{5} c^{5} x^{6}}{3}+4 a^{6} c^{5} x^{7}+3 a^{7} c^{5} x^{8}+\frac {4 c^{5} a^{2} x^{3}}{3}}{a^{9} x^{9}}+\frac {2 c^{5} \ln \left (x \right )}{a}\)	\(119\)
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 (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}-\frac {10 c^{5} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}+\frac {10 c^{5} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\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}\right )}{a}-\frac {5 c^{5} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\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}\right )}{a}+\frac {c^{5} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\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}\right )}{a}+\frac {c^{5} \ln \left (-a x +1\right )}{a}-\frac {5 c^{5} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{a x}\right )}{a}+\frac {10 c^{5} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}\right )}{a}-\frac {10 c^{5} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\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}\right )}{a}+\frac {5 c^{5} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\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}\right )}{a}-\frac {c^{5} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\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}\right )}{a}\)	\(642\)

[In]

int(1/(a*x-1)*(a*x+1)*(c-c/a^2/x^2)^5,x,method=_RETURNVERBOSE)

[Out]

c^5/a^10*(a^10*x-1/9/x^9+2*a^9*ln(x)-3*a^5/x^4-2/3*a^6/x^3+4/3*a^3/x^6+4*a^7/x^2+3*a^8/x+3/7*a^2/x^7-1/4*a/x^8
-2/5*a^4/x^5)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.96 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {1260 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \left (x\right ) + 3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a^2/x^2)^5,x, algorithm="fricas")

[Out]

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 = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {a^{10} c^{5} x + 2 a^{9} c^{5} \log {\left (x \right )} + \frac {3780 a^{8} c^{5} x^{8} + 5040 a^{7} c^{5} x^{7} - 840 a^{6} c^{5} x^{6} - 3780 a^{5} c^{5} x^{5} - 504 a^{4} c^{5} x^{4} + 1680 a^{3} c^{5} x^{3} + 540 a^{2} c^{5} x^{2} - 315 a c^{5} x - 140 c^{5}}{1260 x^{9}}}{a^{10}} \]

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a**2/x**2)**5,x)

[Out]

(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 + 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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=c^{5} x + \frac {2 \, c^{5} \log \left (x\right )}{a} + \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a^2/x^2)^5,x, algorithm="maxima")

[Out]

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 - 50
4*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)

none

Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.91 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=c^{5} x + \frac {2 \, c^{5} \log \left ({\left | x \right |}\right )}{a} + \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a^2/x^2)^5,x, algorithm="giac")

[Out]

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 = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.70 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {c^5\,\left (\frac {3\,a^2\,x^2}{7}-\frac {a\,x}{4}+\frac {4\,a^3\,x^3}{3}-\frac {2\,a^4\,x^4}{5}-3\,a^5\,x^5-\frac {2\,a^6\,x^6}{3}+4\,a^7\,x^7+3\,a^8\,x^8+a^{10}\,x^{10}+2\,a^9\,x^9\,\ln \left (x\right )-\frac {1}{9}\right )}{a^{10}\,x^9} \]

[In]

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

[Out]

(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)