3.339 \(\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx\)

Optimal. Leaf size=94 \[ \frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{1-\frac{1}{a x}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.208993, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6176, 6181, 78, 54, 215} \[ \frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/Sqrt[1 - 1/(a*x)] + (2*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]
/Sqrt[a]])/(Sqrt[a]*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 78

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

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

Mathematica [A]  time = 0.0488794, size = 74, normalized size = 0.79 \[ \frac{2 \sqrt{c-a c x} \left (\sqrt{a} \sqrt{\frac{1}{a x}+1}+\sqrt{\frac{1}{x}} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )\right )}{\sqrt{a} \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.155, size = 80, normalized size = 0.9 \begin{align*} 2\,{\frac{ \left ( ax+1 \right ) \sqrt{-c \left ( ax-1 \right ) }}{ \left ( ax-1 \right ) \sqrt{-c \left ( ax+1 \right ) }}\sqrt{{\frac{ax-1}{ax+1}}} \left ( \sqrt{c}\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ) +\sqrt{-c \left ( ax+1 \right ) } \right ) } \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))^(1/2)/x,x)

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.64288, size = 489, normalized size = 5.2 \begin{align*} \left [\frac{{\left (a x - 1\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 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a x - 1}, \frac{2 \,{\left ({\left (a x - 1\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 ) + \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a x - 1}\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))^(1/2)/x,x, algorithm="fricas")

[Out]

[((a*x - 1)*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*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x - 1), 2*((a*x - 1)*sqrt(c)
*arctan(sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) + sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x
- 1)/(a*x + 1)))/(a*x - 1)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}} \sqrt{- c \left (a x - 1\right )}}{x}\, dx \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))**(1/2)/x,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.19514, size = 103, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{2} \sqrt{-c}}{\sqrt{c}}\right ) -{\left (\sqrt{c} \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right ) + \sqrt{-a c x - c}\right )} c + \sqrt{2} \sqrt{-c} c\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{c^{2}} \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))^(1/2)/x,x, algorithm="giac")

[Out]

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