### 3.579 $$\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^3} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.100469, 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} $-\frac{(3-4 a x) e^{3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{12 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}-\frac{8 e^{3 \coth ^{-1}(a x)}}{35 a c^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-8*E^(3*ArcCoth[a*x]))/(35*a*c^3) - (E^(3*ArcCoth[a*x])*(3 - 4*a*x))/(7*a*c^3*(1 - a^2*x^2)^2) + (12*E^(3*Arc
Coth[a*x])*(3 - 2*a*x))/(35*a*c^3*(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 = 0.204774, size = 66, normalized size = 0.73 $-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13\right )}{35 c^3 (a x-1)^4 (a x+1)}$

Warning: Unable to verify antiderivative.

[In]

Integrate[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)^4*(1 + a*x))

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Maple [A]  time = 0.122, 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\,{c}^{3} \left ({a}^{2}{x}^{2}-1 \right ) ^{2}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(1/((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c)^3,x)

[Out]

-1/35*(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*x-1)/(a*x+1))^(3/2)/a

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Maxima [A]  time = 1.2443, size = 131, normalized size = 1.44 \begin{align*} -\frac{1}{560} \, a{\left (\frac{35 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac{\frac{28 \,{\left (a x - 1\right )}}{a x + 1} - \frac{70 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{140 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/560*a*(35*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^3) + (28*(a*x - 1)/(a*x + 1) - 70*(a*x - 1)^2/(a*x + 1)^2 + 140*
(a*x - 1)^3/(a*x + 1)^3 - 5)/(a^2*c^3*((a*x - 1)/(a*x + 1))^(7/2)))

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Fricas [A]  time = 1.60854, size = 205, normalized size = 2.25 \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} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((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 - 4*a^4*c^3*x^
3 + 6*a^3*c^3*x^2 - 4*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(1/((a*x-1)/(a*x+1))**(3/2)/(-a**2*c*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.17203, size = 150, normalized size = 1.65 \begin{align*} -\frac{1}{560} \, a{\left (\frac{{\left (a x + 1\right )}^{3}{\left (\frac{28 \,{\left (a x - 1\right )}}{a x + 1} - \frac{70 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{140 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{35 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/560*a*((a*x + 1)^3*(28*(a*x - 1)/(a*x + 1) - 70*(a*x - 1)^2/(a*x + 1)^2 + 140*(a*x - 1)^3/(a*x + 1)^3 - 5)/
((a*x - 1)^3*a^2*c^3*sqrt((a*x - 1)/(a*x + 1))) + 35*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^3))