\(\int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx\) [305]

Optimal result

Integrand size = 23, antiderivative size = 39 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=2 \sqrt {c-a c x}+2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6302, 6265, 21, 81, 65, 214} \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+2 \sqrt {c-a c x} \]

[In]

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

[Out]

2*Sqrt[c - a*c*x] + 2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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}}{x} \, dx \\ & = -\int \frac {(1+a x) \sqrt {c-a c x}}{x (1-a x)} \, dx \\ & = -\left (c \int \frac {1+a x}{x \sqrt {c-a c x}} \, dx\right ) \\ & = 2 \sqrt {c-a c x}-c \int \frac {1}{x \sqrt {c-a c x}} \, dx \\ & = 2 \sqrt {c-a c x}+\frac {2 \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )}{a} \\ & = 2 \sqrt {c-a c x}+2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=2 \sqrt {c-a c x}+2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]

[In]

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

[Out]

2*Sqrt[c - a*c*x] + 2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(2 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right ) \sqrt {c}+2 \sqrt {-a c x +c}\)	\(32\)
default	\(2 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right ) \sqrt {c}+2 \sqrt {-a c x +c}\)	\(32\)
pseudoelliptic	\(2 \,\operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}}{\sqrt {c}}\right ) \sqrt {c}+2 \sqrt {-c \left (a x -1\right )}\)	\(34\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.10 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\left [\sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, \sqrt {-a c x + c}, -2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) + 2 \, \sqrt {-a c x + c}\right ] \]

[In]

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

[Out]

[sqrt(c)*log((a*c*x - 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) + 2*sqrt(-a*c*x + c), -2*sqrt(-c)*arctan(sqrt(-a*c*
x + c)*sqrt(-c)/c) + 2*sqrt(-a*c*x + c)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (34) = 68\).

Time = 3.96 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.05 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\begin {cases} - \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 \sqrt {- a c x + c} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (- \frac {3 a \left (\frac {\log {\left (\frac {2}{x} \right )}}{a} - \frac {\log {\left (2 a - \frac {2}{x} \right )}}{a}\right )}{2} + \frac {\log {\left (\frac {a}{x} - \frac {1}{x^{2}} \right )}}{2}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 2*sqrt(-a*c*x + c), Ne(a*c, 0)), (sqrt(c)*(-3*a*(lo
g(2/x)/a - log(2*a - 2/x)/a)/2 + log(a/x - 1/x**2)/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.26 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=-\sqrt {c} \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right ) + 2 \, \sqrt {-a c x + c} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=-2 \, c {\left (\frac {\arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {\sqrt {-a c x + c}}{c}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=2\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c-a\,c\,x}}{\sqrt {c}}\right )+2\,\sqrt {c-a\,c\,x} \]

[In]

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

[Out]

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