\(\int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx\) [338]

Optimal result

Integrand size = 20, antiderivative size = 62 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {8 c \sqrt {1-\frac {1}{a^2 x^2}} x}{3 \sqrt {c-a c x}}+\frac {2}{3} \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6311, 6316, 79, 37} \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {10 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}} \]

[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 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]

Rule 79

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] || L
tQ[p, n]))))

Rule 6311

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 6316

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> Dist[(-c^p)*x^m*(1/x)^m, S
ubst[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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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] (verified)

Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.81 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} (-5+a x) \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}} \]

[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] (verified)

Time = 0.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.68

method	result	size
risch	\(-\frac {2 c \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -5\right ) \left (a x +1\right )}{3 \sqrt {-c \left (a x -1\right )}\, a}\)	\(42\)
gosper	\(\frac {2 \left (a x +1\right ) \left (a x -5\right ) \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}}{3 \left (a x -1\right ) a}\)	\(47\)
default	\(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (a x -5\right )}{3 \left (a x -1\right ) a}\)	\(48\)

[In]

int((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.81 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{2} x^{2} - 4 \, a x - 5\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} x - a\right )}} \]

[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 [F]

\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\int \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}\, dx \]

[In]

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

[Out]

Integral(sqrt((a*x - 1)/(a*x + 1))*sqrt(-c*(a*x - 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{2} \sqrt {-c} x^{2} - 4 \, a \sqrt {-c} x - 5 \, \sqrt {-c}\right )} {\left (a x - 1\right )}}{3 \, {\left (a^{2} x - a\right )} \sqrt {a x + 1}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.69 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (-a c x - c\right )}^{\frac {3}{2}} {\left | c \right |}}{3 \, a c^{2}} + \frac {4 \, \sqrt {-a c x - c} {\left | c \right |}}{a c} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.15 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,\left (a\,x-3\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a}-\frac {16\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a\,\left (a\,x-1\right )} \]

[In]

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

[Out]

(2*(c - a*c*x)^(1/2)*(a*x - 3)*((a*x - 1)/(a*x + 1))^(1/2))/(3*a) - (16*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1)
)^(1/2))/(3*a*(a*x - 1))