### 3.387 $$\int e^{2 \coth ^{-1}(a x)} (c-\frac{c}{a x})^5 \, dx$$

Optimal. Leaf size=61 $-\frac{c^5}{a^3 x^2}+\frac{c^5}{a^4 x^3}-\frac{c^5}{4 a^5 x^4}-\frac{2 c^5}{a^2 x}-\frac{3 c^5 \log (x)}{a}+c^5 x$

[Out]

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

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Rubi [A]  time = 0.13671, antiderivative size = 61, 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} $-\frac{c^5}{a^3 x^2}+\frac{c^5}{a^4 x^3}-\frac{c^5}{4 a^5 x^4}-\frac{2 c^5}{a^2 x}-\frac{3 c^5 \log (x)}{a}+c^5 x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-c^5/(4*a^5*x^4) + c^5/(a^4*x^3) - c^5/(a^3*x^2) - (2*c^5)/(a^2*x) + c^5*x - (3*c^5*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 )^5 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^5 \, dx\\ &=\frac{c^5 \int \frac{e^{2 \tanh ^{-1}(a x)} (1-a x)^5}{x^5} \, dx}{a^5}\\ &=\frac{c^5 \int \frac{(1-a x)^4 (1+a x)}{x^5} \, dx}{a^5}\\ &=\frac{c^5 \int \left (a^5+\frac{1}{x^5}-\frac{3 a}{x^4}+\frac{2 a^2}{x^3}+\frac{2 a^3}{x^2}-\frac{3 a^4}{x}\right ) \, dx}{a^5}\\ &=-\frac{c^5}{4 a^5 x^4}+\frac{c^5}{a^4 x^3}-\frac{c^5}{a^3 x^2}-\frac{2 c^5}{a^2 x}+c^5 x-\frac{3 c^5 \log (x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.219055, size = 63, normalized size = 1.03 $-\frac{c^5}{a^3 x^2}+\frac{c^5}{a^4 x^3}-\frac{c^5}{4 a^5 x^4}-\frac{2 c^5}{a^2 x}-\frac{3 c^5 \log (a x)}{a}+c^5 x$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 60, normalized size = 1. \begin{align*} -{\frac{{c}^{5}}{4\,{a}^{5}{x}^{4}}}+{\frac{{c}^{5}}{{a}^{4}{x}^{3}}}-{\frac{{c}^{5}}{{x}^{2}{a}^{3}}}-2\,{\frac{{c}^{5}}{{a}^{2}x}}+{c}^{5}x-3\,{\frac{{c}^{5}\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)^5,x)

[Out]

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

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Maxima [A]  time = 1.01436, size = 77, normalized size = 1.26 \begin{align*} c^{5} x - \frac{3 \, c^{5} \log \left (x\right )}{a} - \frac{8 \, a^{3} c^{5} x^{3} + 4 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + c^{5}}{4 \, a^{5} x^{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)^5,x, algorithm="maxima")

[Out]

c^5*x - 3*c^5*log(x)/a - 1/4*(8*a^3*c^5*x^3 + 4*a^2*c^5*x^2 - 4*a*c^5*x + c^5)/(a^5*x^4)

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Fricas [A]  time = 1.57395, size = 142, normalized size = 2.33 \begin{align*} \frac{4 \, a^{5} c^{5} x^{5} - 12 \, a^{4} c^{5} x^{4} \log \left (x\right ) - 8 \, a^{3} c^{5} x^{3} - 4 \, a^{2} c^{5} x^{2} + 4 \, a c^{5} x - c^{5}}{4 \, a^{5} x^{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)^5,x, algorithm="fricas")

[Out]

1/4*(4*a^5*c^5*x^5 - 12*a^4*c^5*x^4*log(x) - 8*a^3*c^5*x^3 - 4*a^2*c^5*x^2 + 4*a*c^5*x - c^5)/(a^5*x^4)

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Sympy [A]  time = 0.46256, size = 63, normalized size = 1.03 \begin{align*} \frac{a^{5} c^{5} x - 3 a^{4} c^{5} \log{\left (x \right )} - \frac{8 a^{3} c^{5} x^{3} + 4 a^{2} c^{5} x^{2} - 4 a c^{5} x + c^{5}}{4 x^{4}}}{a^{5}} \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)**5,x)

[Out]

(a**5*c**5*x - 3*a**4*c**5*log(x) - (8*a**3*c**5*x**3 + 4*a**2*c**5*x**2 - 4*a*c**5*x + c**5)/(4*x**4))/a**5

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Giac [A]  time = 1.11383, size = 78, normalized size = 1.28 \begin{align*} c^{5} x - \frac{3 \, c^{5} \log \left ({\left | x \right |}\right )}{a} - \frac{8 \, a^{3} c^{5} x^{3} + 4 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + c^{5}}{4 \, a^{5} x^{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)^5,x, algorithm="giac")

[Out]

c^5*x - 3*c^5*log(abs(x))/a - 1/4*(8*a^3*c^5*x^3 + 4*a^2*c^5*x^2 - 4*a*c^5*x + c^5)/(a^5*x^4)