### 3.353 $$\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx$$

Optimal. Leaf size=137 $-\frac{46 \sqrt{c-a c x}}{3 a^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{20 \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}$

[Out]

(-20*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (46*Sqrt[c - a*c*x])/(3*a^2*Sqrt[1 - 1/(a*x)
]*Sqrt[1 + 1/(a*x)]*x) + (2*x*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])

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Rubi [A]  time = 0.162681, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6176, 6181, 89, 78, 37} $-\frac{46 \sqrt{c-a c x}}{3 a^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{20 \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-20*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (46*Sqrt[c - a*c*x])/(3*a^2*Sqrt[1 - 1/(a*x)
]*Sqrt[1 + 1/(a*x)]*x) + (2*x*Sqrt[c - a*c*x])/(3*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 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 37

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] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{-3 \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{\left (1-\frac{x}{a}\right )^2}{x^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a 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{-\frac{5}{a}+\frac{3 x}{2 a^2}}{x^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{20 \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (23 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{20 \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{46 \sqrt{c-a c x}}{3 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}+\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0258916, size = 48, normalized size = 0.35 $\frac{2 \left (a^2 x^2-10 a x-23\right ) \sqrt{c-a c x}}{3 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[c - a*c*x]*(-23 - 10*a*x + a^2*x^2))/(3*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)

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Maple [A]  time = 0.048, size = 55, normalized size = 0.4 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ({a}^{2}{x}^{2}-10\,ax-23 \right ) }{3\,a \left ( ax-1 \right ) ^{2}}\sqrt{-acx+c} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \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)

[Out]

2/3*(a*x+1)*(a^2*x^2-10*a*x-23)*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/a/(a*x-1)^2

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Maxima [A]  time = 1.07426, size = 101, normalized size = 0.74 \begin{align*} \frac{2 \,{\left (a^{3} \sqrt{-c} x^{3} - 9 \, a^{2} \sqrt{-c} x^{2} - 33 \, a \sqrt{-c} x - 23 \, \sqrt{-c}\right )}{\left (a x - 1\right )}^{2}}{3 \,{\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )}{\left (a x + 1\right )}^{\frac{3}{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))^(3/2),x, algorithm="maxima")

[Out]

2/3*(a^3*sqrt(-c)*x^3 - 9*a^2*sqrt(-c)*x^2 - 33*a*sqrt(-c)*x - 23*sqrt(-c))*(a*x - 1)^2/((a^3*x^2 - 2*a^2*x +
a)*(a*x + 1)^(3/2))

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Fricas [A]  time = 1.18936, size = 113, normalized size = 0.82 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} - 10 \, a x - 23\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{2} x - a\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, algorithm="fricas")

[Out]

2/3*(a^2*x^2 - 10*a*x - 23)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2*x - a)

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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)

[Out]

Timed out

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Giac [A]  time = 1.17272, size = 70, normalized size = 0.51 \begin{align*} \frac{2 \,{\left ({\left (-a c x - c\right )}^{\frac{3}{2}} + 12 \, \sqrt{-a c x - c} c - \frac{12 \, c^{2}}{\sqrt{-a c x - c}}\right )}{\left | c \right |}}{3 \, a 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))^(3/2),x, algorithm="giac")

[Out]

2/3*((-a*c*x - c)^(3/2) + 12*sqrt(-a*c*x - c)*c - 12*c^2/sqrt(-a*c*x - c))*abs(c)/(a*c^2)