\(\int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx\) [239]

Optimal result

Integrand size = 20, antiderivative size = 36 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {4}{a \sqrt {c-a c x}}-\frac {2 \sqrt {c-a c x}}{a c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6302, 6265, 21, 45} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {2 \sqrt {c-a c x}}{a c}-\frac {4}{a \sqrt {c-a c x}} \]

[In]

Int[E^(2*ArcCoth[a*x])/Sqrt[c - a*c*x],x]

[Out]

-4/(a*Sqrt[c - a*c*x]) - (2*Sqrt[c - a*c*x])/(a*c)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6265

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Int[u*(c + d*x)^p*((1 + a*x)^(
n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !(IntegerQ[p] || GtQ[c, 0]
)

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx \\ & = -\int \frac {1+a x}{(1-a x) \sqrt {c-a c x}} \, dx \\ & = -\left (c \int \frac {1+a x}{(c-a c x)^{3/2}} \, dx\right ) \\ & = -\left (c \int \left (\frac {2}{(c-a c x)^{3/2}}-\frac {1}{c \sqrt {c-a c x}}\right ) \, dx\right ) \\ & = -\frac {4}{a \sqrt {c-a c x}}-\frac {2 \sqrt {c-a c x}}{a c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.58 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {-6+2 a x}{a \sqrt {c-a c x}} \]

[In]

Integrate[E^(2*ArcCoth[a*x])/Sqrt[c - a*c*x],x]

[Out]

(-6 + 2*a*x)/(a*Sqrt[c - a*c*x])

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.56

method	result	size
gosper	\(\frac {2 a x -6}{a \sqrt {-a c x +c}}\)	\(20\)
pseudoelliptic	\(\frac {2 a x -6}{a \sqrt {-c \left (a x -1\right )}}\)	\(21\)
trager	\(-\frac {2 \left (a x -3\right ) \sqrt {-a c x +c}}{c a \left (a x -1\right )}\)	\(30\)
derivativedivides	\(-\frac {2 \left (\sqrt {-a c x +c}+\frac {2 c}{\sqrt {-a c x +c}}\right )}{c a}\)	\(31\)
default	\(\frac {-2 \sqrt {-a c x +c}-\frac {4 c}{\sqrt {-a c x +c}}}{a c}\)	\(33\)
risch	\(\frac {2 a x -2}{a \sqrt {-c \left (a x -1\right )}}-\frac {4}{a \sqrt {-c \left (a x -1\right )}}\)	\(37\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {2 \, \sqrt {-a c x + c} {\left (a x - 3\right )}}{a^{2} c x - a c} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.72 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\begin {cases} - \frac {2 \cdot \left (\frac {2 c}{\sqrt {- a c x + c}} + \sqrt {- a c x + c}\right )}{a c} & \text {for}\: a c \neq 0 \\\frac {\begin {cases} - x & \text {for}\: a = 0 \\\frac {a x + 2 \log {\left (a x - 1 \right )} - 1}{a} & \text {otherwise} \end {cases}}{\sqrt {c}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*(2*c/sqrt(-a*c*x + c) + sqrt(-a*c*x + c))/(a*c), Ne(a*c, 0)), (Piecewise((-x, Eq(a, 0)), ((a*x +
2*log(a*x - 1) - 1)/a, True))/sqrt(c), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {2 \, {\left (\sqrt {-a c x + c} + \frac {2 \, c}{\sqrt {-a c x + c}}\right )}}{a c} \]

[In]

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

[Out]

-2*(sqrt(-a*c*x + c) + 2*c/sqrt(-a*c*x + c))/(a*c)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {4}{\sqrt {-a c x + c} a} - \frac {2 \, \sqrt {-a c x + c}}{a c} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.53 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2\,a\,x-6}{a\,\sqrt {c-a\,c\,x}} \]

[In]

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

[Out]

(2*a*x - 6)/(a*(c - a*c*x)^(1/2))