### 3.612 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^3} \, dx$$

Optimal. Leaf size=91 $-\frac{12 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}+\frac{(4 a x+3) e^{-3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{8 e^{-3 \coth ^{-1}(a x)}}{35 a c^3}$

[Out]

8/(35*a*c^3*E^(3*ArcCoth[a*x])) + (3 + 4*a*x)/(7*a*c^3*E^(3*ArcCoth[a*x])*(1 - a^2*x^2)^2) - (12*(3 + 2*a*x))/
(35*a*c^3*E^(3*ArcCoth[a*x])*(1 - a^2*x^2))

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Rubi [A]  time = 0.100999, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {6185, 6183} $-\frac{12 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}+\frac{(4 a x+3) e^{-3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{8 e^{-3 \coth ^{-1}(a x)}}{35 a c^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

8/(35*a*c^3*E^(3*ArcCoth[a*x])) + (3 + 4*a*x)/(7*a*c^3*E^(3*ArcCoth[a*x])*(1 - a^2*x^2)^2) - (12*(3 + 2*a*x))/
(35*a*c^3*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 )^3} \, dx &=\frac{e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{12 \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{7 c}\\ &=\frac{e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}-\frac{12 e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{35 a c^3 \left (1-a^2 x^2\right )}-\frac{24 \int \frac{e^{-3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{35 c^2}\\ &=\frac{8 e^{-3 \coth ^{-1}(a x)}}{35 a c^3}+\frac{e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}-\frac{12 e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{35 a c^3 \left (1-a^2 x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.185284, size = 66, normalized size = 0.73 $\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4+24 a^3 x^3+20 a^2 x^2-4 a x-13\right )}{35 c^3 (a x-1) (a x+1)^4}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x*(-13 - 4*a*x + 20*a^2*x^2 + 24*a^3*x^3 + 8*a^4*x^4))/(35*c^3*(-1 + a*x)*(1 + a*x)^4)

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Maple [A]  time = 0.047, size = 65, normalized size = 0.7 \begin{align*}{\frac{8\,{x}^{4}{a}^{4}+24\,{x}^{3}{a}^{3}+20\,{a}^{2}{x}^{2}-4\,ax-13}{35\, \left ({a}^{2}{x}^{2}-1 \right ) ^{2}{c}^{3}a} \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)^3,x)

[Out]

1/35*((a*x-1)/(a*x+1))^(3/2)*(8*a^4*x^4+24*a^3*x^3+20*a^2*x^2-4*a*x-13)/(a^2*x^2-1)^2/c^3/a

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Maxima [A]  time = 1.03213, size = 139, normalized size = 1.53 \begin{align*} -\frac{1}{560} \, a{\left (\frac{5 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 28 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 70 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 140 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}} - \frac{35}{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^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/560*a*((5*((a*x - 1)/(a*x + 1))^(7/2) - 28*((a*x - 1)/(a*x + 1))^(5/2) + 70*((a*x - 1)/(a*x + 1))^(3/2) - 1
40*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^3) - 35/(a^2*c^3*sqrt((a*x - 1)/(a*x + 1))))

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Fricas [A]  time = 1.53686, size = 182, normalized size = 2. \begin{align*} \frac{{\left (8 \, a^{4} x^{4} + 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} - 4 \, a x - 13\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{35 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a 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)/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/35*(8*a^4*x^4 + 24*a^3*x^3 + 20*a^2*x^2 - 4*a*x - 13)*sqrt((a*x - 1)/(a*x + 1))/(a^5*c^3*x^4 + 2*a^4*c^3*x^3
- 2*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)/(-a**2*c*x**2+c)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (a^{2} c x^{2} - c\right )}^{3}}\,{d x} \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)^3,x, algorithm="giac")

[Out]

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