### 3.338 $$\int e^{-\coth ^{-1}(a x)} \sqrt{c-a c x} \, dx$$

Optimal. Leaf size=62 $\frac{2}{3} x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{8 c x \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-a c x}}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.153416, antiderivative size = 89, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 4, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {6176, 6181, 78, 37} $\frac{2 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}-\frac{10 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}$

Warning: Unable to verify antiderivative.

[In]

Int[Sqrt[c - a*c*x]/E^ArcCoth[a*x],x]

[Out]

(-10*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(3*S
qrt[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 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^{-\coth ^{-1}(a x)} \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int 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^{5/2} \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}} x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{\left (5 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{3 a \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{10 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0262458, size = 50, normalized size = 0.81 $\frac{2 \sqrt{\frac{1}{a x}+1} (a x-5) \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[1 + 1/(a*x)]*(-5 + a*x)*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)])

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 47, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( ax-5 \right ) }{ \left ( 3\,ax-3 \right ) a}\sqrt{-acx+c}\sqrt{{\frac{ax-1}{ax+1}}}} \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)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.10216, size = 73, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (a^{2} \sqrt{-c} x^{2} - 4 \, a \sqrt{-c} x - 5 \, \sqrt{-c}\right )}{\left (a x - 1\right )}}{3 \,{\left (a^{2} x - a\right )} \sqrt{a x + 1}} \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, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.53216, size = 111, normalized size = 1.79 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} - 4 \, a x - 5\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))^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 164.195, size = 66, normalized size = 1.06 \begin{align*} \frac{4 i c x \sqrt{\frac{1}{a c x + c}}}{3} + \frac{4 i c \sqrt{\frac{1}{a c x + c}}}{a} - \frac{2 i \left (- a c x + c\right )^{2} \sqrt{\frac{1}{a c x + c}}}{3 a c} \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)

[Out]

4*I*c*x*sqrt(1/(a*c*x + c))/3 + 4*I*c*sqrt(1/(a*c*x + c))/a - 2*I*(-a*c*x + c)**2*sqrt(1/(a*c*x + c))/(3*a*c)

________________________________________________________________________________________

Giac [A]  time = 1.17643, size = 80, normalized size = 1.29 \begin{align*} -\frac{2 \,{\left (\frac{4 \, \sqrt{2} \sqrt{-c} c}{a} - \frac{{\left (-a c x - c\right )}^{\frac{3}{2}} + 6 \, \sqrt{-a c x - c} c}{a}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{3 \, 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, algorithm="giac")

[Out]

-2/3*(4*sqrt(2)*sqrt(-c)*c/a - ((-a*c*x - c)^(3/2) + 6*sqrt(-a*c*x - c)*c)/a)*abs(c)*sgn(a*x + 1)/c^2