### 3.610 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx$$

Optimal. Leaf size=18 $-\frac{e^{-3 \coth ^{-1}(a x)}}{3 a c}$

[Out]

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

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Rubi [A]  time = 0.03191, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {6183} $-\frac{e^{-3 \coth ^{-1}(a x)}}{3 a c}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)),x]

[Out]

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

Rule 6183

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcCoth[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rubi steps

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

Mathematica [A]  time = 0.047077, size = 18, normalized size = 1. $-\frac{e^{-3 \coth ^{-1}(a x)}}{3 a c}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.043, size = 24, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,ac} \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^2*c*x^2+c),x)

[Out]

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

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Maxima [A]  time = 1.04226, size = 31, normalized size = 1.72 \begin{align*} -\frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{3 \, a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.51913, size = 78, normalized size = 4.33 \begin{align*} -\frac{{\left (a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{2} c x + a c\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{3} x^{3} + a^{2} x^{2} - a x - 1}\, dx + \int \frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\, dx}{c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.20713, size = 66, normalized size = 3.67 \begin{align*} \frac{2 \,{\left (3 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )}^{2} x^{2} + 1\right )}}{3 \,{\left ({\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x + 1\right )}^{3} a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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