### 3.315 $$\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx$$

Optimal. Leaf size=172 $\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}}}+\frac{5 \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{1-\frac{1}{a x}}}$

[Out]

(Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*x) + (5*Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sq
rt[x^(-1)]/Sqrt[a]])/Sqrt[1 - 1/(a*x)] - (4*Sqrt[2]*Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt
[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/Sqrt[1 - 1/(a*x)]

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Rubi [A]  time = 0.251493, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.348, Rules used = {6176, 6181, 102, 157, 54, 215, 93, 206} $\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}}}+\frac{5 \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*x) + (5*Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sq
rt[x^(-1)]/Sqrt[a]])/Sqrt[1 - 1/(a*x)] - (4*Sqrt[2]*Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt
[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/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 102

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

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.076673, size = 120, normalized size = 0.7 $\frac{\sqrt{\frac{1}{x}} \sqrt{c-a c x} \left (\sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1}+5 \sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )-4 \sqrt{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{\sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[x^(-1)]*Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)] + 5*Sqrt[a]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]] - 4*Sq
rt[2]*Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/Sqrt[1 - 1/(a*x)]

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Maple [A]  time = 0.184, size = 119, normalized size = 0.7 \begin{align*} -{\frac{ax-1}{ \left ( ax+1 \right ) x}\sqrt{-c \left ( ax-1 \right ) } \left ( 4\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) xac-5\,\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ) xac-\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{c}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*x-1)*(-c*(a*x-1))^(1/2)*(4*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*x*a*c-5*arctan((-c*(a*x+
1))^(1/2)/c^(1/2))*x*a*c-(-c*(a*x+1))^(1/2)*c^(1/2))/((a*x-1)/(a*x+1))^(3/2)/(a*x+1)/(-c*(a*x+1))^(1/2)/x/c^(1
/2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73131, size = 900, normalized size = 5.23 \begin{align*} \left [\frac{4 \, \sqrt{2}{\left (a^{2} x^{2} - a x\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 5 \,{\left (a^{2} x^{2} - a x\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}}}{2 \,{\left (a x^{2} - x\right )}}, -\frac{4 \, \sqrt{2}{\left (a^{2} x^{2} - a x\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) - 5 \,{\left (a^{2} x^{2} - a x\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}}}{a x^{2} - x}\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*c*x+c)^(1/2)/x^2,x, algorithm="fricas")

[Out]

[1/2*(4*sqrt(2)*(a^2*x^2 - a*x)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt
(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 5*(a^2*x^2 - a*x)*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^2 - x), -(4*sqrt(2)*(a^2*x^2 - a*x)*sqrt(c)*arctan(sqrt(2)*sqrt(-a*
c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 5*(a^2*x^2 - a*x)*sqrt(c)*arctan(sqrt(-a*c*x + c)*sq
rt(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^2 -
x)]

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

[Out]

Timed out

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Giac [C]  time = 1.28979, size = 211, normalized size = 1.23 \begin{align*} -a c{\left (\frac{4 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{\sqrt{c} \mathrm{sgn}\left (-a c x - c\right )} - \frac{5 \, \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{\sqrt{c} \mathrm{sgn}\left (-a c x - c\right )} - \frac{\sqrt{-a c x - c}}{a c x \mathrm{sgn}\left (-a c x - c\right )}\right )} + \frac{-4 i \, \sqrt{2} a \sqrt{-c} \arctan \left (-i\right ) + 5 i \, a \sqrt{-c} \arctan \left (-i \, \sqrt{2}\right ) + \sqrt{2} a \sqrt{-c}}{\mathrm{sgn}\left (c\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*c*x+c)^(1/2)/x^2,x, algorithm="giac")

[Out]

-a*c*(4*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/(sqrt(c)*sgn(-a*c*x - c)) - 5*arctan(sqrt(-a*c*x
- c)/sqrt(c))/(sqrt(c)*sgn(-a*c*x - c)) - sqrt(-a*c*x - c)/(a*c*x*sgn(-a*c*x - c))) + (-4*I*sqrt(2)*a*sqrt(-c)
*arctan(-I) + 5*I*a*sqrt(-c)*arctan(-I*sqrt(2)) + sqrt(2)*a*sqrt(-c))/sgn(c)