\(\int e^{4 \coth ^{-1}(a x)} (c-\frac {c}{a x})^4 \, dx\) [405]

Optimal result

Integrand size = 22, antiderivative size = 30 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x \]

[Out]

-1/3*c^4/a^4/x^3+2*c^4/a^2/x+c^4*x

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6302, 6266, 6264, 74, 276} \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x \]

[In]

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

[Out]

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

Rule 74

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[m] && (NeQ[m, -1] || (EqQ[e, 0] && (EqQ[p, 1] ||  !IntegerQ[p])))

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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 )^4 \, dx \\ & = \frac {c^4 \int \frac {e^{4 \text {arctanh}(a x)} (1-a x)^4}{x^4} \, dx}{a^4} \\ & = \frac {c^4 \int \frac {(1-a x)^2 (1+a x)^2}{x^4} \, dx}{a^4} \\ & = \frac {c^4 \int \frac {\left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4} \\ & = \frac {c^4 \int \left (a^4+\frac {1}{x^4}-\frac {2 a^2}{x^2}\right ) \, dx}{a^4} \\ & = -\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 \left (-\frac {1}{3 x^3}+\frac {2 a^2}{x}+a^4 x\right )}{a^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90

method	result	size
default	\(\frac {c^{4} \left (a^{4} x -\frac {1}{3 x^{3}}+\frac {2 a^{2}}{x}\right )}{a^{4}}\)	\(27\)
gosper	\(\frac {c^{4} \left (3 a^{4} x^{4}+6 a^{2} x^{2}-1\right )}{3 x^{3} a^{4}}\)	\(30\)
risch	\(c^{4} x +\frac {2 a^{2} c^{4} x^{2}-\frac {1}{3} c^{4}}{a^{4} x^{3}}\)	\(31\)
parallelrisch	\(\frac {3 a^{4} c^{4} x^{4}+6 a^{2} c^{4} x^{2}-c^{4}}{3 a^{4} x^{3}}\)	\(37\)
norman	\(\frac {a^{3} c^{4} x^{4}+a^{4} c^{4} x^{5}+\frac {c^{4}}{3 a}-\frac {c^{4} x}{3}-2 a \,c^{4} x^{2}}{\left (a x -1\right ) a^{3} x^{3}}\)	\(59\)
meijerg	\(-\frac {c^{4} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{4} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}-\frac {c^{4} x}{-a x +1}+\frac {4 c^{4} \left (\frac {2 a x}{-2 a x +2}-\ln \left (-a x +1\right )+1+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}+\frac {c^{4} \left (-\frac {3 a x}{-3 a x +3}+2 \ln \left (-a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (-a \right )+\frac {1}{a x}\right )}{a}-\frac {2 c^{4} \left (\frac {4 a x}{-4 a x +4}-3 \ln \left (-a x +1\right )+1+3 \ln \left (x \right )+3 \ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {2}{a x}\right )}{a}-\frac {c^{4} \left (-\frac {5 a x}{-5 a x +5}+4 \ln \left (-a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (-a \right )+\frac {1}{3 x^{3} a^{3}}+\frac {1}{a^{2} x^{2}}+\frac {3}{a x}\right )}{a}\)	\(284\)

[In]

int(1/(a*x-1)^2*(a*x+1)^2*(c-c/a/x)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {3 \, a^{4} c^{4} x^{4} + 6 \, a^{2} c^{4} x^{2} - c^{4}}{3 \, a^{4} x^{3}} \]

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(c-c/a/x)^4,x, algorithm="fricas")

[Out]

1/3*(3*a^4*c^4*x^4 + 6*a^2*c^4*x^2 - c^4)/(a^4*x^3)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {a^{4} c^{4} x + \frac {6 a^{2} c^{4} x^{2} - c^{4}}{3 x^{3}}}{a^{4}} \]

[In]

integrate(1/(a*x-1)**2*(a*x+1)**2*(c-c/a/x)**4,x)

[Out]

(a**4*c**4*x + (6*a**2*c**4*x**2 - c**4)/(3*x**3))/a**4

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=c^{4} x + \frac {6 \, a^{2} c^{4} x^{2} - c^{4}}{3 \, a^{4} x^{3}} \]

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(c-c/a/x)^4,x, algorithm="maxima")

[Out]

c^4*x + 1/3*(6*a^2*c^4*x^2 - c^4)/(a^4*x^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {{\left (a x - 1\right )} c^{4}}{a} - \frac {5 \, c^{4} + \frac {9 \, c^{4}}{a x - 1} + \frac {3 \, c^{4}}{{\left (a x - 1\right )}^{2}}}{3 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{3}} \]

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(c-c/a/x)^4,x, algorithm="giac")

[Out]

(a*x - 1)*c^4/a - 1/3*(5*c^4 + 9*c^4/(a*x - 1) + 3*c^4/(a*x - 1)^2)/(a*(1/(a*x - 1) + 1)^3)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4\,\left (a^4\,x^4+2\,a^2\,x^2-\frac {1}{3}\right )}{a^4\,x^3} \]

[In]

int(((c - c/(a*x))^4*(a*x + 1)^2)/(a*x - 1)^2,x)

[Out]

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