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/(1-1/a^2/x^2)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 0.45, size = 31, normalized size = 1.48 \[ \frac {x \sqrt {\frac {a x - 1}{a x + 1}}}{a c^{3} x - c^{3}} \]

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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giac [A]  time = 0.14, size = 22, normalized size = 1.05 \[ \frac {x \mathrm {sgn}\left (a x + 1\right )}{\sqrt {a^{2} x^{2} - 1} c^{3}} \]

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)

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maple [A]  time = 0.04, size = 33, normalized size = 1.57 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x}{\left (a x -1\right )^{2} c^{3}} \]

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 = 0.31, size = 48, normalized size = 2.29 \[ \frac {1}{2} \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac {1}{a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}\right )} \]

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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mupad [B]  time = 1.17, size = 38, normalized size = 1.81 \[ \frac {\frac {a\,x-1}{a\,x+1}+1}{2\,a\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{3}} \]

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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