\(\int e^{\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx\) [230]

Optimal result

Integrand size = 18, antiderivative size = 29 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 e^{\coth ^{-1}(a x)} (1+a x) \sqrt {c-a c x}}{3 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6309} \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 (a x+1) \sqrt {c-a c x} e^{\coth ^{-1}(a x)}}{3 a} \]

[In]

Int[E^ArcCoth[a*x]*Sqrt[c - a*c*x],x]

[Out]

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

Rule 6309

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Simp[(1 + a*x)*(c + d*x)^p*(E^(n*Arc
Coth[a*x])/(a*(p + 1))), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a*c + d, 0] && EqQ[p, n/2] &&  !IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = \frac {2 e^{\coth ^{-1}(a x)} (1+a x) \sqrt {c-a c x}}{3 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21

method	result	size
gosper	\(\frac {2 \left (a x +1\right ) \sqrt {-a c x +c}}{3 \sqrt {\frac {a x -1}{a x +1}}\, a}\)	\(35\)
default	\(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (a x +1\right )}{3 \sqrt {\frac {a x -1}{a x +1}}\, a}\)	\(36\)
risch	\(-\frac {2 c \left (a x +1\right ) \left (a x -1\right )}{3 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\)	\(42\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} x - a\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a \sqrt {-c} x + \sqrt {-c}\right )} \sqrt {a x + 1}}{3 \, a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, c^{2} {\left (\frac {2 \, \sqrt {2} \sqrt {-c}}{c} + \frac {{\left (-a c x - c\right )}^{\frac {3}{2}}}{c^{2}}\right )}}{3 \, a {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.39 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a\,\left (a\,x-1\right )} \]

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