\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx\) [275]

Optimal result

Integrand size = 20, antiderivative size = 85 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {6 \sqrt {1-\frac {1}{a x}}}{a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}+\frac {2 \sqrt {1-\frac {1}{a x}} x}{\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, 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} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 x \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}+\frac {6 \sqrt {1-\frac {1}{a x}}}{a \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}} \]

[In]

Int[1/(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x]),x]

[Out]

(6*Sqrt[1 - 1/(a*x)])/(a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]) + (2*Sqrt[1 - 1/(a*x)]*x)/(Sqrt[1 + 1/(a*x)]*Sqrt[
c - a*c*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 {\left (\sqrt {1-\frac {1}{a x}} \sqrt {x}\right ) \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \, dx}{\sqrt {c-a c x}} \\ & = -\frac {\sqrt {1-\frac {1}{a x}} \text {Subst}\left (\int \frac {1-\frac {x}{a}}{x^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}} \\ & = \frac {2 \sqrt {1-\frac {1}{a x}} x}{\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}+\frac {\left (3 \sqrt {1-\frac {1}{a x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{a \sqrt {\frac {1}{x}} \sqrt {c-a c x}} \\ & = \frac {6 \sqrt {1-\frac {1}{a x}}}{a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}+\frac {2 \sqrt {1-\frac {1}{a x}} x}{\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.56 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 \sqrt {1-\frac {1}{a x}} (3+a x)}{a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}} \]

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.55

method	result	size
gosper	\(\frac {2 \left (a x +1\right ) \left (a x +3\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a \left (a x -1\right ) \sqrt {-a c x +c}}\)	\(47\)
default	\(-\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (a x +3\right )}{\left (a x -1\right )^{2} c a}\)	\(51\)
risch	\(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{\sqrt {-c \left (a x -1\right )}\, a}+\frac {4 \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-c \left (a x -1\right )}}\)	\(67\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.52 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {2 \, \sqrt {-a c x + c} {\left (a x + 3\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.56 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 \, {\left (a^{2} x^{2} + 4 \, a x + 3\right )} {\left (a x - 1\right )}}{{\left (a^{2} \sqrt {-c} x - a \sqrt {-c}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.49 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=2 \, {\left (\frac {\sqrt {-a c x - c}}{a c^{2}} - \frac {2}{\sqrt {-a c x - c} a c}\right )} {\left | c \right |} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.41 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.40 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {\left (2\,x+\frac {6}{a}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\sqrt {c-a\,c\,x}} \]

[In]

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

[Out]

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