3.611 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx$$

Optimal. Leaf size=55 $\frac{2 e^{-3 \coth ^{-1}(a x)}}{15 a c^2}-\frac{(2 a x+3) e^{-3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )}$

[Out]

2/(15*a*c^2*E^(3*ArcCoth[a*x])) - (3 + 2*a*x)/(5*a*c^2*E^(3*ArcCoth[a*x])*(1 - a^2*x^2))

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Rubi [A]  time = 0.0662282, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {6185, 6183} $\frac{2 e^{-3 \coth ^{-1}(a x)}}{15 a c^2}-\frac{(2 a x+3) e^{-3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

2/(15*a*c^2*E^(3*ArcCoth[a*x])) - (3 + 2*a*x)/(5*a*c^2*E^(3*ArcCoth[a*x])*(1 - a^2*x^2))

Rule 6185

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((n + 2*a*(p + 1)*x)*(c + d*x^
2)^(p + 1)*E^(n*ArcCoth[a*x]))/(a*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 - 4*(p + 1)^
2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !In
tegerQ[n/2] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] ||  !IntegerQ[n])

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)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\frac{e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{5 a c^2 \left (1-a^2 x^2\right )}-\frac{2 \int \frac{e^{-3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{5 c}\\ &=\frac{2 e^{-3 \coth ^{-1}(a x)}}{15 a c^2}-\frac{e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{5 a c^2 \left (1-a^2 x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.141018, size = 43, normalized size = 0.78 $\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2+6 a x+7\right )}{15 c^2 (a x+1)^3}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x*(7 + 6*a*x + 2*a^2*x^2))/(15*c^2*(1 + a*x)^3)

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Maple [A]  time = 0.043, size = 49, normalized size = 0.9 \begin{align*}{\frac{2\,{a}^{2}{x}^{2}+6\,ax+7}{ \left ( 15\,{a}^{2}{x}^{2}-15 \right ) a{c}^{2}} \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)^2,x)

[Out]

1/15*((a*x-1)/(a*x+1))^(3/2)*(2*a^2*x^2+6*a*x+7)/(a^2*x^2-1)/a/c^2

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Maxima [A]  time = 1.04872, size = 81, normalized size = 1.47 \begin{align*} \frac{3 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 10 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 15 \, \sqrt{\frac{a x - 1}{a x + 1}}}{60 \, a c^{2}} \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)^2,x, algorithm="maxima")

[Out]

1/60*(3*((a*x - 1)/(a*x + 1))^(5/2) - 10*((a*x - 1)/(a*x + 1))^(3/2) + 15*sqrt((a*x - 1)/(a*x + 1)))/(a*c^2)

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Fricas [A]  time = 1.57377, size = 124, normalized size = 2.25 \begin{align*} \frac{{\left (2 \, a^{2} x^{2} + 6 \, a x + 7\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\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)^2,x, algorithm="fricas")

[Out]

1/15*(2*a^2*x^2 + 6*a*x + 7)*sqrt((a*x - 1)/(a*x + 1))/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2)

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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)/(-a**2*c*x**2+c)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.17797, size = 88, normalized size = 1.6 \begin{align*} -\frac{4 \,{\left (10 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )}^{2} x^{2} + 5 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x + 1\right )}}{15 \,{\left ({\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x + 1\right )}^{5} a c^{2}} \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)^2,x, algorithm="giac")

[Out]

-4/15*(10*(a + sqrt(a^2 - 1/x^2))^2*x^2 + 5*(a + sqrt(a^2 - 1/x^2))*x + 1)/(((a + sqrt(a^2 - 1/x^2))*x + 1)^5*
a*c^2)