### 3.539 $$\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx$$

Optimal. Leaf size=150 $-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 c}-\frac{56}{15} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{224 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}{15 \sqrt{c-\frac{c}{a x}}}$

[Out]

(-224*a^2*c*Sqrt[1 - 1/(a^2*x^2)])/(15*Sqrt[c - c/(a*x)]) - (56*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)])/1
5 - (7*a^2*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(3/2))/(5*c) - (a^2*(c - c/(a*x))^(7/2))/(c^3*Sqrt[1 - 1/(a^2*x
^2)])

________________________________________________________________________________________

Rubi [A]  time = 0.266011, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.148, Rules used = {6178, 789, 657, 649} $-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 c}-\frac{56}{15} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{224 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}{15 \sqrt{c-\frac{c}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-224*a^2*c*Sqrt[1 - 1/(a^2*x^2)])/(15*Sqrt[c - c/(a*x)]) - (56*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)])/1
5 - (7*a^2*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(3/2))/(5*c) - (a^2*(c - c/(a*x))^(7/2))/(c^3*Sqrt[1 - 1/(a^2*x
^2)])

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 789

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*(p + 1)), x] - Dist[(e*(m*(d*g + e*f) + 2*e*f*(p + 1)))/(2*c*d*(p + 1)
), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0]
&& LtQ[p, -1] && GtQ[m, 0]

Rule 657

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

Rule 649

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

Rubi steps

\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x \left (c-\frac{c x}{a}\right )^{7/2}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{(7 a) \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{5/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 c^2}\\ &=-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 c}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{(28 a) \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{5 c}\\ &=-\frac{56}{15} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 c}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{1}{15} (112 a) \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{224 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}{15 \sqrt{c-\frac{c}{a x}}}-\frac{56}{15} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 c}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.113466, size = 70, normalized size = 0.47 $-\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (158 a^3 x^3+79 a^2 x^2-16 a x+3\right ) \sqrt{c-\frac{c}{a x}}}{15 x \left (a^2 x^2-1\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(3 - 16*a*x + 79*a^2*x^2 + 158*a^3*x^3))/(15*x*(-1 + a^2*x^2))

________________________________________________________________________________________

Maple [A]  time = 0.128, size = 70, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,ax+2 \right ) \left ( 158\,{x}^{3}{a}^{3}+79\,{a}^{2}{x}^{2}-16\,ax+3 \right ) }{15\,{x}^{2} \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(a*x+1)*(158*a^3*x^3+79*a^2*x^2-16*a*x+3)*(c*(a*x-1)/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^2/(a*x-1)^2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.65063, size = 150, normalized size = 1. \begin{align*} -\frac{2 \,{\left (158 \, a^{3} x^{3} + 79 \, a^{2} x^{2} - 16 \, a x + 3\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{15 \,{\left (a x^{3} - x^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(158*a^3*x^3 + 79*a^2*x^2 - 16*a*x + 3)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(a*x^3 - x^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((c-c/a/x)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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