### 3.222 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx$$

Optimal. Leaf size=21 $\frac{1}{a c^3 \sqrt{1-\frac{1}{a^2 x^2}}}$

[Out]

1/(a*c^3*Sqrt[1 - 1/(a^2*x^2)])

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Rubi [A]  time = 0.100688, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {6175, 6178, 261} $\frac{1}{a c^3 \sqrt{1-\frac{1}{a^2 x^2}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

1/(a*c^3*Sqrt[1 - 1/(a^2*x^2)])

Rule 6175

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

Rule 6178

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c
+ d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0541357, size = 33, normalized size = 1.57 $\frac{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \left (a^2 x^2-1\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.052, size = 33, normalized size = 1.6 \begin{align*}{\frac{ \left ( ax+1 \right ) x}{ \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)/(-a*c*x+c)^3,x)

[Out]

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

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Maxima [B]  time = 1.08119, size = 65, normalized size = 3.1 \begin{align*} \frac{1}{2} \, a{\left (\frac{\sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac{1}{a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.5342, size = 61, normalized size = 2.9 \begin{align*} \frac{x \sqrt{\frac{a x - 1}{a x + 1}}}{a c^{3} x - c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx + \int \frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**4*x**4 - 2*a**3*x**3 + 2*a*x - 1), x) + Integral(a*x*sqrt(a*
x/(a*x + 1) - 1/(a*x + 1))/(a**4*x**4 - 2*a**3*x**3 + 2*a*x - 1), x))/c**3

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Giac [A]  time = 1.17534, size = 30, normalized size = 1.43 \begin{align*} \frac{x \mathrm{sgn}\left (a x + 1\right )}{\sqrt{a^{2} x^{2} - 1} c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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