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

Optimal. Leaf size=30 \[ -\frac{c^4}{3 a^4 x^3}+\frac{2 c^4}{a^2 x}+c^4 x \]

[Out]

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

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Rubi [A]  time = 0.126472, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6167, 6131, 6129, 73, 270} \[ -\frac{c^4}{3 a^4 x^3}+\frac{2 c^4}{a^2 x}+c^4 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6167

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]

Rule 6131

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 6129

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 73

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]

Rule 270

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]

Rubi steps

\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx\\ &=\frac{c^4 \int \frac{e^{4 \tanh ^{-1}(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]  time = 0.180698, size = 30, normalized size = 1. \[ -\frac{c^4}{3 a^4 x^3}+\frac{2 c^4}{a^2 x}+c^4 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 27, normalized size = 0.9 \begin{align*}{\frac{{c}^{4}}{{a}^{4}} \left ( x{a}^{4}+2\,{\frac{{a}^{2}}{x}}-{\frac{1}{3\,{x}^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00567, size = 42, normalized size = 1.4 \begin{align*} c^{4} x + \frac{6 \, a^{2} c^{4} x^{2} - c^{4}}{3 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.54152, size = 72, normalized size = 2.4 \begin{align*} \frac{3 \, a^{4} c^{4} x^{4} + 6 \, a^{2} c^{4} x^{2} - c^{4}}{3 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.327616, size = 31, normalized size = 1.03 \begin{align*} \frac{a^{4} c^{4} x + \frac{6 a^{2} c^{4} x^{2} - c^{4}}{3 x^{3}}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [B]  time = 1.13582, size = 80, normalized size = 2.67 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)