### 3.578 $$\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx$$

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0650561, 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} $\frac{(3-2 a x) e^{3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )}-\frac{2 e^{3 \coth ^{-1}(a x)}}{15 a c^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*E^(3*ArcCoth[a*x]))/(15*a*c^2) + (E^(3*ArcCoth[a*x])*(3 - 2*a*x))/(5*a*c^2*(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 = 0.149389, 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[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)

________________________________________________________________________________________

Maple [A]  time = 0.131, 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 ){c}^{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)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.31571, size = 74, normalized size = 1.35 \begin{align*} \frac{\frac{10 \,{\left (a x - 1\right )}}{a x + 1} - \frac{15 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3}{60 \, a c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} \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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.49653, size = 161, normalized size = 2.93 \begin{align*} -\frac{{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 7\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{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)^2,x, algorithm="fricas")

[Out]

-1/15*(2*a^3*x^3 - 4*a^2*x^2 + a*x + 7)*sqrt((a*x - 1)/(a*x + 1))/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x -
a*c^2)

________________________________________________________________________________________

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)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.14126, size = 93, normalized size = 1.69 \begin{align*} \frac{{\left (a x + 1\right )}^{2}{\left (\frac{10 \,{\left (a x - 1\right )}}{a x + 1} - \frac{15 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3\right )}}{60 \,{\left (a x - 1\right )}^{2} a c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} \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)^2,x, algorithm="giac")

[Out]

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