\(\int e^{4 \coth ^{-1}(a x)} (c-\frac {c}{a x})^5 \, dx\) [404]

Optimal result

Integrand size = 22, antiderivative size = 64 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=\frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}+c^5 x-\frac {c^5 \log (x)}{a} \]

[Out]

1/4*c^5/a^5/x^4-1/3*c^5/a^4/x^3-c^5/a^3/x^2+2*c^5/a^2/x+c^5*x-c^5*ln(x)/a

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 64, 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, 6266, 6264, 90} \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=\frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}-\frac {c^5 \log (x)}{a}+c^5 x \]

[In]

Int[E^(4*ArcCoth[a*x])*(c - c/(a*x))^5,x]

[Out]

c^5/(4*a^5*x^4) - c^5/(3*a^4*x^3) - c^5/(a^3*x^2) + (2*c^5)/(a^2*x) + c^5*x - (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 6264

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[u*(1 + d*(x/c))^
p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 6266

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[u*(1 + c*(x/d))^
p*(E^(n*ArcTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 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^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx \\ & = -\frac {c^5 \int \frac {e^{4 \text {arctanh}(a x)} (1-a x)^5}{x^5} \, dx}{a^5} \\ & = -\frac {c^5 \int \frac {(1-a x)^3 (1+a x)^2}{x^5} \, dx}{a^5} \\ & = -\frac {c^5 \int \left (-a^5+\frac {1}{x^5}-\frac {a}{x^4}-\frac {2 a^2}{x^3}+\frac {2 a^3}{x^2}+\frac {a^4}{x}\right ) \, dx}{a^5} \\ & = \frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}+c^5 x-\frac {c^5 \log (x)}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=\frac {c^5 \left (\frac {1}{4 x^4}-\frac {a}{3 x^3}-\frac {a^2}{x^2}+\frac {2 a^3}{x}+a^5 x-a^4 \log (x)\right )}{a^5} \]

[In]

Integrate[E^(4*ArcCoth[a*x])*(c - c/(a*x))^5,x]

[Out]

(c^5*(1/(4*x^4) - a/(3*x^3) - a^2/x^2 + (2*a^3)/x + a^5*x - a^4*Log[x]))/a^5

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {c^{5} \left (a^{5} x -a^{4} \ln \left (x \right )+\frac {1}{4 x^{4}}-\frac {a}{3 x^{3}}-\frac {a^{2}}{x^{2}}+\frac {2 a^{3}}{x}\right )}{a^{5}}\)	\(48\)
risch	\(c^{5} x +\frac {2 a^{3} c^{5} x^{3}-a^{2} c^{5} x^{2}-\frac {1}{3} a \,c^{5} x +\frac {1}{4} c^{5}}{a^{5} x^{4}}-\frac {c^{5} \ln \left (x \right )}{a}\)	\(59\)
parallelrisch	\(-\frac {-12 a^{5} c^{5} x^{5}+12 c^{5} \ln \left (x \right ) a^{4} x^{4}-24 a^{3} c^{5} x^{3}+12 a^{2} c^{5} x^{2}+4 a \,c^{5} x -3 c^{5}}{12 a^{5} x^{4}}\)	\(68\)
norman	\(\frac {a^{4} c^{5} x^{5}+a^{5} c^{5} x^{6}-\frac {c^{5}}{4 a}+\frac {7 c^{5} x}{12}+\frac {2 a \,c^{5} x^{2}}{3}-3 c^{5} a^{2} x^{3}}{\left (a x -1\right ) a^{4} x^{4}}-\frac {c^{5} \ln \left (x \right )}{a}\)	\(81\)
meijerg	\(-\frac {c^{5} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {3 c^{5} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} x}{-a x +1}+\frac {5 c^{5} \left (\frac {2 a x}{-2 a x +2}-\ln \left (-a x +1\right )+1+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}+\frac {5 c^{5} \left (-\frac {3 a x}{-3 a x +3}+2 \ln \left (-a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (-a \right )+\frac {1}{a x}\right )}{a}-\frac {c^{5} \left (\frac {4 a x}{-4 a x +4}-3 \ln \left (-a x +1\right )+1+3 \ln \left (x \right )+3 \ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {2}{a x}\right )}{a}-\frac {3 c^{5} \left (-\frac {5 a x}{-5 a x +5}+4 \ln \left (-a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (-a \right )+\frac {1}{3 x^{3} a^{3}}+\frac {1}{a^{2} x^{2}}+\frac {3}{a x}\right )}{a}-\frac {c^{5} \left (\frac {6 a x}{-6 a x +6}-5 \ln \left (-a x +1\right )+1+5 \ln \left (x \right )+5 \ln \left (-a \right )-\frac {1}{4 a^{4} x^{4}}-\frac {2}{3 x^{3} a^{3}}-\frac {3}{2 a^{2} x^{2}}-\frac {4}{a x}\right )}{a}\)	\(357\)

[In]

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

[Out]

c^5/a^5*(a^5*x-a^4*ln(x)+1/4/x^4-1/3*a/x^3-a^2/x^2+2*a^3/x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=\frac {12 \, a^{5} c^{5} x^{5} - 12 \, a^{4} c^{5} x^{4} \log \left (x\right ) + 24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}} \]

[In]

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

[Out]

1/12*(12*a^5*c^5*x^5 - 12*a^4*c^5*x^4*log(x) + 24*a^3*c^5*x^3 - 12*a^2*c^5*x^2 - 4*a*c^5*x + 3*c^5)/(a^5*x^4)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=\frac {a^{5} c^{5} x - a^{4} c^{5} \log {\left (x \right )} + \frac {24 a^{3} c^{5} x^{3} - 12 a^{2} c^{5} x^{2} - 4 a c^{5} x + 3 c^{5}}{12 x^{4}}}{a^{5}} \]

[In]

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

[Out]

(a**5*c**5*x - a**4*c**5*log(x) + (24*a**3*c**5*x**3 - 12*a**2*c**5*x**2 - 4*a*c**5*x + 3*c**5)/(12*x**4))/a**
5

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.92 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=c^{5} x - \frac {c^{5} \log \left (x\right )}{a} + \frac {24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}} \]

[In]

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

[Out]

c^5*x - c^5*log(x)/a + 1/12*(24*a^3*c^5*x^3 - 12*a^2*c^5*x^2 - 4*a*c^5*x + 3*c^5)/(a^5*x^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (60) = 120\).

Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.92 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=\frac {c^{5} \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {c^{5} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {{\left (12 \, c^{5} + \frac {37 \, c^{5}}{a x - 1} + \frac {52 \, c^{5}}{{\left (a x - 1\right )}^{2}} + \frac {42 \, c^{5}}{{\left (a x - 1\right )}^{3}} + \frac {12 \, c^{5}}{{\left (a x - 1\right )}^{4}}\right )} {\left (a x - 1\right )}}{12 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{4}} \]

[In]

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

[Out]

c^5*log(abs(a*x - 1)/((a*x - 1)^2*abs(a)))/a - c^5*log(abs(-1/(a*x - 1) - 1))/a + 1/12*(12*c^5 + 37*c^5/(a*x -
1) + 52*c^5/(a*x - 1)^2 + 42*c^5/(a*x - 1)^3 + 12*c^5/(a*x - 1)^4)*(a*x - 1)/(a*(1/(a*x - 1) + 1)^4)

Mupad [B] (verification not implemented)

Time = 3.88 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=-\frac {c^5\,\left (4\,a\,x+12\,a^2\,x^2-24\,a^3\,x^3-12\,a^5\,x^5+12\,a^4\,x^4\,\ln \left (x\right )-3\right )}{12\,a^5\,x^4} \]

[In]

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

[Out]

-(c^5*(4*a*x + 12*a^2*x^2 - 24*a^3*x^3 - 12*a^5*x^5 + 12*a^4*x^4*log(x) - 3))/(12*a^5*x^4)