### 3.389 $$\int e^{2 \coth ^{-1}(a x)} (c-\frac{c}{a x})^3 \, dx$$

Optimal. Leaf size=39 $-\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}-\frac{c^3 \log (x)}{a}+c^3 x$

[Out]

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

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Rubi [A]  time = 0.128253, antiderivative size = 39, 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} $-\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}-\frac{c^3 \log (x)}{a}+c^3 x$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(2*ArcCoth[a*x])*(c - c/(a*x))^3,x]

[Out]

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

Mathematica [A]  time = 0.120522, size = 41, normalized size = 1.05 $-\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}-\frac{c^3 \log (a x)}{a}+c^3 x$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^3,x]

[Out]

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

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Maple [A]  time = 0.043, size = 38, normalized size = 1. \begin{align*} -{\frac{{c}^{3}}{2\,{x}^{2}{a}^{3}}}+{\frac{{c}^{3}}{{a}^{2}x}}+{c}^{3}x-{\frac{{c}^{3}\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)^3,x)

[Out]

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

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Maxima [A]  time = 1.04247, size = 50, normalized size = 1.28 \begin{align*} c^{3} x - \frac{c^{3} \log \left (x\right )}{a} + \frac{2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46807, size = 97, normalized size = 2.49 \begin{align*} \frac{2 \, a^{3} c^{3} x^{3} - 2 \, a^{2} c^{3} x^{2} \log \left (x\right ) + 2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.338769, size = 37, normalized size = 0.95 \begin{align*} \frac{a^{3} c^{3} x - a^{2} c^{3} \log{\left (x \right )} + \frac{2 a c^{3} x - c^{3}}{2 x^{2}}}{a^{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)**3,x)

[Out]

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

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Giac [A]  time = 1.12182, size = 51, normalized size = 1.31 \begin{align*} c^{3} x - \frac{c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac{2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \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)^3,x, algorithm="giac")

[Out]

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