\(\int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx\) [337]

Optimal result

Integrand size = 21, antiderivative size = 104 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=-\frac {8 c \sqrt {1-\frac {1}{a^2 x^2}} x}{5 a \sqrt {c-a c x}}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}}{5 a}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2}}{5 a c} \]

[Out]

-2/5*x*(-a*c*x+c)^(3/2)*(1-1/a^2/x^2)^(1/2)/a/c-8/5*c*x*(1-1/a^2/x^2)^(1/2)/a/(-a*c*x+c)^(1/2)-2/5*x*(1-1/a^2/
x^2)^(1/2)*(-a*c*x+c)^(1/2)/a

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6311, 6316, 79, 47, 37} \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {12 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{5 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {6 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{5 a \sqrt {1-\frac {1}{a x}}} \]

[In]

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

[Out]

(12*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(5*a^2*Sqrt[1 - 1/(a*x)]) - (6*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(5*
a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(5*Sqrt[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 47

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*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 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}} x^{3/2} \, 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^{7/2} \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^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {\left (9 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 a \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {6 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{5 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {\left (6 \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 )}{5 a^2 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {12 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{5 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {6 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{5 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.56 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (6-3 a x+a^2 x^2\right )}{5 a^2 \sqrt {1-\frac {1}{a x}}} \]

[In]

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

[Out]

(2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(6 - 3*a*x + a^2*x^2))/(5*a^2*Sqrt[1 - 1/(a*x)])

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.48

method	result	size
risch	\(-\frac {2 c \sqrt {\frac {a x -1}{a x +1}}\, \left (a^{2} x^{2}-3 a x +6\right ) \left (a x +1\right )}{5 \sqrt {-c \left (a x -1\right )}\, a^{2}}\)	\(50\)
gosper	\(\frac {2 \left (a x +1\right ) \left (a^{2} x^{2}-3 a x +6\right ) \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}}{5 a^{2} \left (a x -1\right )}\)	\(55\)
default	\(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (a^{2} x^{2}-3 a x +6\right )}{5 \left (a x -1\right ) a^{2}}\)	\(56\)

[In]

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

[Out]

-2/5*c*((a*x-1)/(a*x+1))^(1/2)/(-c*(a*x-1))^(1/2)*(a^2*x^2-3*a*x+6)/a^2*(a*x+1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.58 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \, a x + 6\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{3} x - a^{2}\right )}} \]

[In]

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

[Out]

2/5*(a^3*x^3 - 2*a^2*x^2 + 3*a*x + 6)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^3*x - a^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{3} \sqrt {-c} x^{3} - 2 \, a^{2} \sqrt {-c} x^{2} + 3 \, a \sqrt {-c} x + 6 \, \sqrt {-c}\right )} {\left (a x - 1\right )}}{5 \, {\left (a^{3} x - a^{2}\right )} \sqrt {a x + 1}} \]

[In]

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

[Out]

2/5*(a^3*sqrt(-c)*x^3 - 2*a^2*sqrt(-c)*x^2 + 3*a*sqrt(-c)*x + 6*sqrt(-c))*(a*x - 1)/((a^3*x - a^2)*sqrt(a*x +
1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=-\frac {4 \, \sqrt {-a c x - c} {\left | c \right |}}{a^{2} c} - \frac {2 \, {\left ({\left (a c x + c\right )}^{2} \sqrt {-a c x - c} {\left | c \right |} + 5 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c {\left | c \right |}\right )}}{5 \, a^{2} c^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.51 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.55 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (a^3\,x^3-2\,a^2\,x^2+3\,a\,x+6\right )}{5\,a^2\,\left (a\,x-1\right )} \]

[In]

int(x*(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 - 1)/(a*x + 1))^(1/2)*(3*a*x - 2*a^2*x^2 + a^3*x^3 + 6))/(5*a^2*(a*x - 1))