### 3.388 $$\int e^{2 \coth ^{-1}(a x)} (c-\frac{c}{a x})^4 \, dx$$

Optimal. Leaf size=40 $-\frac{c^4}{a^3 x^2}+\frac{c^4}{3 a^4 x^3}-\frac{2 c^4 \log (x)}{a}+c^4 x$

[Out]

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

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Rubi [A]  time = 0.130371, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {6167, 6131, 6129, 75} $-\frac{c^4}{a^3 x^2}+\frac{c^4}{3 a^4 x^3}-\frac{2 c^4 \log (x)}{a}+c^4 x$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(2*ArcCoth[a*x])*(c - c/(a*x))^4,x]

[Out]

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

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 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

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

Mathematica [A]  time = 0.154748, size = 42, normalized size = 1.05 $-\frac{c^4}{a^3 x^2}+\frac{c^4}{3 a^4 x^3}-\frac{2 c^4 \log (a x)}{a}+c^4 x$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^4,x]

[Out]

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

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Maple [A]  time = 0.043, size = 39, normalized size = 1. \begin{align*}{\frac{{c}^{4}}{3\,{a}^{4}{x}^{3}}}-{\frac{{c}^{4}}{{x}^{2}{a}^{3}}}+{c}^{4}x-2\,{\frac{{c}^{4}\ln \left ( x \right ) }{a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03881, size = 50, normalized size = 1.25 \begin{align*} c^{4} x - \frac{2 \, c^{4} \log \left (x\right )}{a} - \frac{3 \, a c^{4} x - 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)*(a*x+1)*(c-c/a/x)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.4932, size = 97, normalized size = 2.42 \begin{align*} \frac{3 \, a^{4} c^{4} x^{4} - 6 \, a^{3} c^{4} x^{3} \log \left (x\right ) - 3 \, a c^{4} x + 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)*(a*x+1)*(c-c/a/x)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.36418, size = 39, normalized size = 0.98 \begin{align*} \frac{a^{4} c^{4} x - 2 a^{3} c^{4} \log{\left (x \right )} - \frac{3 a c^{4} x - 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)*(a*x+1)*(c-c/a/x)**4,x)

[Out]

(a**4*c**4*x - 2*a**3*c**4*log(x) - (3*a*c**4*x - c**4)/(3*x**3))/a**4

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Giac [A]  time = 1.15485, size = 51, normalized size = 1.27 \begin{align*} c^{4} x - \frac{2 \, c^{4} \log \left ({\left | x \right |}\right )}{a} - \frac{3 \, a c^{4} x - 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)*(a*x+1)*(c-c/a/x)^4,x, algorithm="giac")

[Out]

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