### 3.356 $$\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx$$

Optimal. Leaf size=190 $-\frac{47 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{2 x^2 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{47 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}}$

[Out]

(-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x^2) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(2*Sqrt[1
- 1/(a*x)]*x^2) + (47*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x) - (47*a^(3/2)*Sqrt[x^(-1)]
*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(4*Sqrt[1 - 1/(a*x)])

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Rubi [A]  time = 0.231778, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.304, Rules used = {6176, 6181, 89, 80, 50, 54, 215} $-\frac{47 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{2 x^2 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{47 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x^2) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(2*Sqrt[1
- 1/(a*x)]*x^2) + (47*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x) - (47*a^(3/2)*Sqrt[x^(-1)]
*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(4*Sqrt[1 - 1/(a*x)])

Rule 6176

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

Rule 6181

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

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx &=\frac{\sqrt{c-a c x} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x^{5/2}} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} \left (1-\frac{x}{a}\right )^2}{\left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^2}+\frac{\left (2 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} \left (\frac{11}{2 a^2}-\frac{x}{2 a^3}\right )}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^2}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x^2}+\frac{\left (47 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{4 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^2}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x^2}+\frac{47 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}-\frac{\left (47 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^2}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x^2}+\frac{47 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}-\frac{\left (47 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,\sqrt{\frac{1}{x}}\right )}{4 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^2}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x^2}+\frac{47 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}-\frac{47 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0828575, size = 90, normalized size = 0.47 $-\frac{\sqrt{c-a c x} \left (-47 a^2 x^2+\frac{47 a^{5/2} \sqrt{\frac{1}{a x}+1} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\left (\frac{1}{x}\right )^{5/2}}-13 a x+2\right )}{4 a x^3 \sqrt{1-\frac{1}{a^2 x^2}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Sqrt[c - a*c*x]*(2 - 13*a*x - 47*a^2*x^2 + (47*a^(5/2)*Sqrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(
-1))^(5/2)))/(4*a*Sqrt[1 - 1/(a^2*x^2)]*x^3)

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Maple [A]  time = 0.158, size = 103, normalized size = 0.5 \begin{align*}{\frac{ax+1}{4\, \left ( ax-1 \right ) ^{2}{x}^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{-c \left ( ax-1 \right ) } \left ( 47\,\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{2}{a}^{2}\sqrt{-c \left ( ax+1 \right ) }+47\,{x}^{2}{a}^{2}\sqrt{c}+13\,xa\sqrt{c}-2\,\sqrt{c} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^2*(-c*(a*x-1))^(1/2)*(47*arctan((-c*(a*x+1))^(1/2)/c^(1/2))*x^2*a^
2*(-c*(a*x+1))^(1/2)+47*x^2*a^2*c^(1/2)+13*x*a*c^(1/2)-2*c^(1/2))/c^(1/2)/x^2

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

[Out]

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

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Fricas [A]  time = 1.71114, size = 587, normalized size = 3.09 \begin{align*} \left [\frac{47 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \,{\left (47 \, a^{2} x^{2} + 13 \, a x - 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{8 \,{\left (a x^{3} - x^{2}\right )}}, -\frac{47 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) -{\left (47 \, a^{2} x^{2} + 13 \, a x - 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{4 \,{\left (a x^{3} - x^{2}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(47*(a^3*x^3 - a^2*x^2)*sqrt(-c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*
x - 1)/(a*x + 1)) - 2*c)/(a*x^2 - x)) + 2*(47*a^2*x^2 + 13*a*x - 2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))
)/(a*x^3 - x^2), -1/4*(47*(a^3*x^3 - a^2*x^2)*sqrt(c)*arctan(sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1)
)/(a*c*x - c)) - (47*a^2*x^2 + 13*a*x - 2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^3 - x^2)]

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

[Out]

Timed out

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Giac [A]  time = 1.23343, size = 113, normalized size = 0.59 \begin{align*} \frac{1}{4} \, a^{2} c{\left (\frac{47 \, \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{c^{\frac{3}{2}}} + \frac{32}{\sqrt{-a c x - c} c} + \frac{15 \,{\left (-a c x - c\right )}^{\frac{3}{2}} + 17 \, \sqrt{-a c x - c} c}{a^{2} c^{3} x^{2}}\right )}{\left | c \right |} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^2*c*(47*arctan(sqrt(-a*c*x - c)/sqrt(c))/c^(3/2) + 32/(sqrt(-a*c*x - c)*c) + (15*(-a*c*x - c)^(3/2) + 17
*sqrt(-a*c*x - c)*c)/(a^2*c^3*x^2))*abs(c)