### 3.534 $$\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx$$

Optimal. Leaf size=251 $\frac{119 x \sqrt{c-\frac{c}{a x}}}{24 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{119 \sqrt{c-\frac{c}{a x}}}{8 a^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{119 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{8 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{x^3 \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{19 x^2 \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}$

[Out]

(119*Sqrt[c - c/(a*x)])/(8*a^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (119*Sqrt[c - c/(a*x)]*x)/(24*a^2*Sqrt[1
- 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (19*Sqrt[c - c/(a*x)]*x^2)/(12*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (Sqrt
[c - c/(a*x)]*x^3)/(3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (119*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]]
)/(8*a^3*Sqrt[1 - 1/(a*x)])

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Rubi [A]  time = 0.292824, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.259, Rules used = {6182, 6180, 89, 78, 51, 63, 208} $\frac{119 x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{119 x \sqrt{c-\frac{c}{a x}}}{12 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{119 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{8 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{x^3 \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{19 x^2 \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-119*Sqrt[c - c/(a*x)]*x)/(12*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (119*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a
*x)]*x)/(8*a^2*Sqrt[1 - 1/(a*x)]) - (19*Sqrt[c - c/(a*x)]*x^2)/(12*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (S
qrt[c - c/(a*x)]*x^3)/(3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (119*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x
)]])/(8*a^3*Sqrt[1 - 1/(a*x)])

Rule 6182

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

Rule 6180

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^p, Subst[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, 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 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 51

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^2 \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^4 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{-\frac{19}{2 a}+\frac{3 x}{a^2}}{x^3 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{19 \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (119 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{24 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{119 \sqrt{c-\frac{c}{a x}} x}{12 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{19 \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (119 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{119 \sqrt{c-\frac{c}{a x}} x}{12 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{119 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{8 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{19 \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\left (119 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a^3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{119 \sqrt{c-\frac{c}{a x}} x}{12 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{119 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{8 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{19 \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\left (119 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{8 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{119 \sqrt{c-\frac{c}{a x}} x}{12 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{119 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{8 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{19 \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{119 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{8 a^3 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.490683, size = 159, normalized size = 0.63 $\frac{\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^3 x^3-38 a^2 x^2+119 a x+357\right ) \sqrt{c-\frac{c}{a x}}}{a^2 x^2-1}-357 \sqrt{c} \log \left (2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )+357 \sqrt{c} \log (1-a x)}{48 a^3}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(357 + 119*a*x - 38*a^2*x^2 + 8*a^3*x^3))/(-1 + a^2*x^2) +
357*Sqrt[c]*Log[1 - a*x] - 357*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1
- a*x + 2*a^2*x^2)])/(48*a^3)

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Maple [A]  time = 0.187, size = 180, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax+1 \right ) x}{48\, \left ( ax-1 \right ) ^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 16\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}-76\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+238\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-357\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) xa+714\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}-357\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^2*(c*(a*x-1)/a/x)^(1/2)*x*(16*a^(7/2)*x^3*((a*x+1)*x)^(1/2)-76*a^
(5/2)*x^2*((a*x+1)*x)^(1/2)+238*a^(3/2)*x*((a*x+1)*x)^(1/2)-357*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a
^(1/2))*x*a+714*((a*x+1)*x)^(1/2)*a^(1/2)-357*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2)))/a^(5/2)/(
(a*x+1)*x)^(1/2)

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

[Out]

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

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Fricas [A]  time = 1.94247, size = 737, normalized size = 2.94 \begin{align*} \left [\frac{357 \,{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{96 \,{\left (a^{4} x - a^{3}\right )}}, \frac{357 \,{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{48 \,{\left (a^{4} x - a^{3}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(357*(a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x -
1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(8*a^4*x^4 - 38*a^3*x^3 + 119*a^2*x^2 + 357*a*x)*sq
rt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3), 1/48*(357*(a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^2
+ a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(8*a^4*x^4 -
38*a^3*x^3 + 119*a^2*x^2 + 357*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3)]

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

[Out]

Timed out

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

[Out]

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