### 3.203 $$\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx$$

Optimal. Leaf size=62 $\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )^2}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )}$

[Out]

(a*Sqrt[1 - 1/(a^2*x^2)])/(3*c^3*(a - x^(-1))^2) - (2*Sqrt[1 - 1/(a^2*x^2)])/(3*c^3*(a - x^(-1)))

________________________________________________________________________________________

Rubi [A]  time = 0.125238, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {6175, 6178, 793, 651} $\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )^2}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a*Sqrt[1 - 1/(a^2*x^2)])/(3*c^3*(a - x^(-1))^2) - (2*Sqrt[1 - 1/(a^2*x^2)])/(3*c^3*(a - x^(-1)))

Rule 6175

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[u*x^p*(1 + c/(d*
x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !IntegerQ[n/2] && Inte
gerQ[p]

Rule 6178

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c
+ d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rule 793

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d*g - e*f)*(
d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(m + p + 1)), x] + Dist[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d
)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2
+ a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) &&
NeQ[m + p + 1, 0]

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\frac{\int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^3 x^3} \, dx}{a^3 c^3}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{\left (1-\frac{x}{a}\right )^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^3 c^3}\\ &=\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )^2}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 a^2 c^3}\\ &=\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )^2}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )}\\ \end{align*}

Mathematica [A]  time = 0.0549729, size = 34, normalized size = 0.55 $-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x-2)}{3 c^3 (a x-1)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 41, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-2 \right ) \left ( ax+1 \right ) }{3\, \left ( ax-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*c*x+c)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.04444, size = 53, normalized size = 0.85 \begin{align*} -\frac{\frac{3 \,{\left (a x - 1\right )}}{a x + 1} - 1}{6 \, a c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.54921, size = 119, normalized size = 1.92 \begin{align*} -\frac{{\left (a^{2} x^{2} - a x - 2\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c^{3} x^{2} - 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))^(1/2)/(-a*c*x+c)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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^{3} x^{3} - 3 a^{2} x^{2} + 3 a x - 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*c*x+c)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.19694, size = 61, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (3 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x - 1\right )}}{3 \,{\left ({\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x - 1\right )}^{3} a 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*c*x+c)^3,x, algorithm="giac")

[Out]

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