### 3.435 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-\frac{c}{a x})^3} \, dx$$

Optimal. Leaf size=45 $\frac{x}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2}{a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}}$

[Out]

-2/(a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x) + x/(c^3*Sqrt[1 - 1/(a^2*x^2)])

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Rubi [A]  time = 0.0494055, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.136, Rules used = {6177, 271, 191} $\frac{x}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2}{a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^3),x]

[Out]

-2/(a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x) + x/(c^3*Sqrt[1 - 1/(a^2*x^2)])

Rule 6177

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c + d*x)^(p -
n)*(1 - x^2/a^2)^(n/2))/x^2, x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)
/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1]) && IntegerQ[2*p]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0220276, size = 33, normalized size = 0.73 $\frac{a^2 x^2-2}{a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^3),x]

[Out]

(-2 + a^2*x^2)/(a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x)

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Maple [A]  time = 0.04, size = 44, normalized size = 1. \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}-2 \right ) \left ( ax+1 \right ) }{a \left ( ax-1 \right ) ^{2}{c}^{3}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.02256, size = 124, normalized size = 2.76 \begin{align*} -\frac{1}{2} \, a{\left (\frac{\frac{5 \,{\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.77781, size = 82, normalized size = 1.82 \begin{align*} \frac{{\left (a^{2} x^{2} - 2\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3} x - a c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.22081, size = 51, normalized size = 1.13 \begin{align*} \frac{{\left (\sqrt{a^{2} x^{2} - 1} - \frac{1}{\sqrt{a^{2} x^{2} - 1}}\right )} \mathrm{sgn}\left (a x + 1\right )}{a c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x, algorithm="giac")

[Out]

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