### 3.596 $$\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a^2 c x^2)^3} \, dx$$

Optimal. Leaf size=91 $\frac{4 (2 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )}+\frac{(4 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac{8 e^{-\coth ^{-1}(a x)}}{15 a c^3}$

[Out]

-8/(15*a*c^3*E^ArcCoth[a*x]) + (1 + 4*a*x)/(15*a*c^3*E^ArcCoth[a*x]*(1 - a^2*x^2)^2) + (4*(1 + 2*a*x))/(15*a*c
^3*E^ArcCoth[a*x]*(1 - a^2*x^2))

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Rubi [A]  time = 0.0969696, 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} $\frac{4 (2 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )}+\frac{(4 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac{8 e^{-\coth ^{-1}(a x)}}{15 a c^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-8/(15*a*c^3*E^ArcCoth[a*x]) + (1 + 4*a*x)/(15*a*c^3*E^ArcCoth[a*x]*(1 - a^2*x^2)^2) + (4*(1 + 2*a*x))/(15*a*c
^3*E^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^{-\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{e^{-\coth ^{-1}(a x)} (1+4 a x)}{15 a c^3 \left (1-a^2 x^2\right )^2}+\frac{4 \int \frac{e^{-\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{5 c}\\ &=\frac{e^{-\coth ^{-1}(a x)} (1+4 a x)}{15 a c^3 \left (1-a^2 x^2\right )^2}+\frac{4 e^{-\coth ^{-1}(a x)} (1+2 a x)}{15 a c^3 \left (1-a^2 x^2\right )}+\frac{8 \int \frac{e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{15 c^2}\\ &=-\frac{8 e^{-\coth ^{-1}(a x)}}{15 a c^3}+\frac{e^{-\coth ^{-1}(a x)} (1+4 a x)}{15 a c^3 \left (1-a^2 x^2\right )^2}+\frac{4 e^{-\coth ^{-1}(a x)} (1+2 a x)}{15 a c^3 \left (1-a^2 x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.188673, size = 64, normalized size = 0.7 $-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4+8 a^3 x^3-12 a^2 x^2-12 a x+3\right )}{15 (a x-1)^2 (a c x+c)^3}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Sqrt[1 - 1/(a^2*x^2)]*x*(3 - 12*a*x - 12*a^2*x^2 + 8*a^3*x^3 + 8*a^4*x^4))/(15*(-1 + a*x)^2*(c + a*c*x)^3)

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Maple [A]  time = 0.046, size = 65, normalized size = 0.7 \begin{align*} -{\frac{8\,{x}^{4}{a}^{4}+8\,{x}^{3}{a}^{3}-12\,{a}^{2}{x}^{2}-12\,ax+3}{15\, \left ({a}^{2}{x}^{2}-1 \right ) ^{2}{c}^{3}a}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*((a*x-1)/(a*x+1))^(1/2)*(8*a^4*x^4+8*a^3*x^3-12*a^2*x^2-12*a*x+3)/(a^2*x^2-1)^2/c^3/a

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Maxima [A]  time = 0.978571, size = 138, normalized size = 1.52 \begin{align*} -\frac{1}{240} \, a{\left (\frac{3 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 20 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 90 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac{5 \,{\left (\frac{12 \,{\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/240*a*((3*((a*x - 1)/(a*x + 1))^(5/2) - 20*((a*x - 1)/(a*x + 1))^(3/2) + 90*sqrt((a*x - 1)/(a*x + 1)))/(a^2
*c^3) + 5*(12*(a*x - 1)/(a*x + 1) - 1)/(a^2*c^3*((a*x - 1)/(a*x + 1))^(3/2)))

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Fricas [A]  time = 1.51358, size = 163, normalized size = 1.79 \begin{align*} -\frac{{\left (8 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 12 \, a x + 3\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(8*a^4*x^4 + 8*a^3*x^3 - 12*a^2*x^2 - 12*a*x + 3)*sqrt((a*x - 1)/(a*x + 1))/(a^5*c^3*x^4 - 2*a^3*c^3*x^2
+ a*c^3)

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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^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x)/c**3

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

[Out]

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