\(\int e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx\) [238]

Optimal result

Integrand size = 20, antiderivative size = 38 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 38, 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} \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 (5+a x) \sqrt {c-a c x}}{3 a} \]

[In]

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

[Out]

(2*(5 + a*x)*Sqrt[c - a*c*x])/(3*a)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53

method	result	size
gosper	\(\frac {2 \sqrt {-a c x +c}\, \left (a x +5\right )}{3 a}\)	\(20\)
trager	\(\frac {2 \sqrt {-a c x +c}\, \left (a x +5\right )}{3 a}\)	\(20\)
pseudoelliptic	\(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (a x +5\right )}{3 a}\)	\(21\)
risch	\(-\frac {2 c \left (a x +5\right ) \left (a x -1\right )}{3 a \sqrt {-c \left (a x -1\right )}}\)	\(27\)
derivativedivides	\(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {3}{2}}}{3}-2 c \sqrt {-a c x +c}\right )}{c a}\)	\(33\)
default	\(\frac {-\frac {2 \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 c \sqrt {-a c x +c}}{a c}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.50 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, \sqrt {-a c x + c} {\left (a x + 5\right )}}{3 \, a} \]

[In]

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

[Out]

2/3*sqrt(-a*c*x + c)*(a*x + 5)/a

Sympy [A] (verification not implemented)

Time = 2.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.47 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\begin {cases} - \frac {2 \left (- 2 c \sqrt {- a c x + c} + \frac {\left (- a c x + c\right )^{\frac {3}{2}}}{3}\right )}{a c} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\begin {cases} - x & \text {for}\: a = 0 \\\frac {a x + 2 \log {\left (a x - 1 \right )} - 1}{a} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]

[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) + (-a*c*x + c)**(3/2)/3)/(a*c), Ne(a*c, 0)), (sqrt(c)*Piecewise((-x, Eq(a
, 0)), ((a*x + 2*log(a*x - 1) - 1)/a, True)), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} - 6 \, \sqrt {-a c x + c} c\right )}}{3 \, a c} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (3 \, \sqrt {-a c x + c} - \frac {{\left (-a c x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-a c x + c} c}{c}\right )}}{3 \, a} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {c-a\,c\,x}}{a}-\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a\,c} \]

[In]

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

[Out]

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