### 3.355 $$\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx$$

Optimal. Leaf size=140 $-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{7 \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\sqrt{1-\frac{1}{a x}}}$

[Out]

(-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(Sqrt[1 - 1
/(a*x)]*x) + (7*Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/Sqrt[1 - 1/(a*x)]

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Rubi [A]  time = 0.222996, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.261, Rules used = {6176, 6181, 89, 80, 54, 215} $-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{7 \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(Sqrt[1 - 1
/(a*x)]*x) + (7*Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/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 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 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^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 )^2}{\sqrt{x} \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}+\frac{\left (2 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\frac{3}{2 a^2}-\frac{x}{2 a^3}}{\sqrt{x} \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}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} x}+\frac{\left (7 \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{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} x}+\frac{\left (7 \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{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} x}+\frac{7 \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}}}\\ \end{align*}

Mathematica [A]  time = 0.077449, size = 79, normalized size = 0.56 $\frac{\sqrt{c-a c x} \left (\frac{7 a^{3/2} \sqrt{\frac{1}{a x}+1} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\left (\frac{1}{x}\right )^{3/2}}-9 a x-1\right )}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[c - a*c*x]*(-1 - 9*a*x + (7*a^(3/2)*Sqrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(3/2)))/(a
*Sqrt[1 - 1/(a^2*x^2)]*x^2)

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Maple [A]  time = 0.144, size = 86, normalized size = 0.6 \begin{align*} -{\frac{ax+1}{ \left ( ax-1 \right ) ^{2}x} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}} \left ( 7\,\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ) xa\sqrt{-c \left ( ax+1 \right ) }+9\,xa\sqrt{c}+\sqrt{c} \right ) \sqrt{-c \left ( ax-1 \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^2,x)

[Out]

-((a*x-1)/(a*x+1))^(3/2)*(a*x+1)*(7*arctan((-c*(a*x+1))^(1/2)/c^(1/2))*x*a*(-c*(a*x+1))^(1/2)+9*x*a*c^(1/2)+c^
(1/2))*(-c*(a*x-1))^(1/2)/(a*x-1)^2/c^(1/2)/x

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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^{2}}\,{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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.38317, size = 524, normalized size = 3.74 \begin{align*} \left [\frac{7 \,{\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 (9 \, a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \,{\left (a x^{2} - x\right )}}, \frac{7 \,{\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 (9 \, 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((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^2,x, algorithm="fricas")

[Out]

[1/2*(7*(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)*(9*a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^2 - x), (7*(
a^2*x^2 - a*x)*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)*(9*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((-a*c*x+c)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**2,x)

[Out]

Timed out

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Giac [A]  time = 1.21293, size = 85, normalized size = 0.61 \begin{align*} -a{\left (\frac{7 \, \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{\sqrt{c}} - \frac{9 \, a c x + c}{{\left (-a c x - c\right )}^{\frac{3}{2}} + \sqrt{-a c x - c} c}\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^2,x, algorithm="giac")

[Out]

-a*(7*arctan(sqrt(-a*c*x - c)/sqrt(c))/sqrt(c) - (9*a*c*x + c)/((-a*c*x - c)^(3/2) + sqrt(-a*c*x - c)*c))*abs(
c)