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

Optimal. Leaf size=85 \[ -\frac{(1-4 a x) e^{\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac{4 (1-2 a x) e^{\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )}+\frac{8 e^{\coth ^{-1}(a x)}}{15 a c^3} \]

[Out]

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

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Rubi [A]  time = 0.0929698, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6185, 6183} \[ -\frac{(1-4 a x) e^{\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac{4 (1-2 a x) e^{\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )}+\frac{8 e^{\coth ^{-1}(a x)}}{15 a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*E^ArcCoth[a*x])/(15*a*c^3) - (E^ArcCoth[a*x]*(1 - 4*a*x))/(15*a*c^3*(1 - a^2*x^2)^2) - (4*E^ArcCoth[a*x]*(1
- 2*a*x))/(15*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^{\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 = 0.166856, size = 66, normalized size = 0.78 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4-8 a^3 x^3-12 a^2 x^2+12 a x+3\right )}{15 c^3 (a x-1)^3 (a x+1)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

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Maple [A]  time = 0.044, size = 65, normalized size = 0.8 \begin{align*}{\frac{8\,{x}^{4}{a}^{4}-8\,{x}^{3}{a}^{3}-12\,{a}^{2}{x}^{2}+12\,ax+3}{15\,{c}^{3} \left ({a}^{2}{x}^{2}-1 \right ) ^{2}a}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(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*x-1)/(a*x+1))^(1/2)/a

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Maxima [A]  time = 0.996608, size = 134, normalized size = 1.58 \begin{align*} -\frac{1}{240} \, a{\left (\frac{5 \,{\left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 12 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2} c^{3}} + \frac{\frac{20 \,{\left (a x - 1\right )}}{a x + 1} - \frac{90 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.63693, size = 181, normalized size = 2.13 \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^{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(1/((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^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(1/((a*x-1)/(a*x+1))**(1/2)/(-a**2*c*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.13373, size = 186, normalized size = 2.19 \begin{align*} -\frac{1}{240} \, a{\left (\frac{{\left (a x + 1\right )}^{2}{\left (\frac{20 \,{\left (a x - 1\right )}}{a x + 1} - \frac{90 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3\right )}}{{\left (a x - 1\right )}^{2} a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{5 \,{\left (\frac{{\left (a x - 1\right )} a^{4} c^{6} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - 12 \, a^{4} c^{6} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{6} c^{9}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/240*a*((a*x + 1)^2*(20*(a*x - 1)/(a*x + 1) - 90*(a*x - 1)^2/(a*x + 1)^2 - 3)/((a*x - 1)^2*a^2*c^3*sqrt((a*x
- 1)/(a*x + 1))) + 5*((a*x - 1)*a^4*c^6*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) - 12*a^4*c^6*sqrt((a*x - 1)/(a*x
+ 1)))/(a^6*c^9))