### 3.404 $$\int e^{4 \coth ^{-1}(a x)} (c-\frac{c}{a x})^5 \, dx$$

Optimal. Leaf size=64 $-\frac{c^5}{a^3 x^2}-\frac{c^5}{3 a^4 x^3}+\frac{c^5}{4 a^5 x^4}+\frac{2 c^5}{a^2 x}-\frac{c^5 \log (x)}{a}+c^5 x$

[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

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Rubi [A]  time = 0.137443, antiderivative size = 64, 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, 88} $-\frac{c^5}{a^3 x^2}-\frac{c^5}{3 a^4 x^3}+\frac{c^5}{4 a^5 x^4}+\frac{2 c^5}{a^2 x}-\frac{c^5 \log (x)}{a}+c^5 x$

Antiderivative was successfully veriﬁed.

[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 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 88

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]))

Rubi steps

\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^5 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^5 \, dx\\ &=-\frac{c^5 \int \frac{e^{4 \tanh ^{-1}(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]  time = 0.277991, size = 66, normalized size = 1.03 $-\frac{c^5}{a^3 x^2}-\frac{c^5}{3 a^4 x^3}+\frac{c^5}{4 a^5 x^4}+\frac{2 c^5}{a^2 x}-\frac{c^5 \log (a x)}{a}+c^5 x$

Warning: Unable to verify antiderivative.

[In]

Integrate[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[a*x])/a

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Maple [A]  time = 0.045, size = 61, normalized size = 1. \begin{align*}{\frac{{c}^{5}}{4\,{a}^{5}{x}^{4}}}-{\frac{{c}^{5}}{3\,{a}^{4}{x}^{3}}}-{\frac{{c}^{5}}{{x}^{2}{a}^{3}}}+2\,{\frac{{c}^{5}}{{a}^{2}x}}+{c}^{5}x-{\frac{{c}^{5}\ln \left ( x \right ) }{a}} \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)^5,x)

[Out]

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

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Maxima [A]  time = 1.03626, size = 80, normalized size = 1.25 \begin{align*} 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}} \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)^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)

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Fricas [A]  time = 1.45563, size = 150, normalized size = 2.34 \begin{align*} \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}} \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)^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)

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Sympy [A]  time = 0.477993, size = 63, normalized size = 0.98 \begin{align*} \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}} \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)**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

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Giac [B]  time = 1.14656, size = 166, normalized size = 2.59 \begin{align*} \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}} \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)^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)