\(\int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx\) [344]

Optimal result

Integrand size = 20, antiderivative size = 76 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6302, 6265, 21, 52, 65, 212} \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}-\frac {4 \sqrt {c-a c x}}{a} \]

[In]

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

[Out]

(-4*Sqrt[c - a*c*x])/a - (2*(c - a*c*x)^(3/2))/(3*a*c) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*S
qrt[c])])/a

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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, 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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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 \\ & = -\frac {\int \frac {(c-a c x)^{3/2}}{1+a x} \, dx}{c} \\ & = -\frac {2 (c-a c x)^{3/2}}{3 a c}-2 \int \frac {\sqrt {c-a c x}}{1+a x} \, dx \\ & = -\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}-(4 c) \int \frac {1}{(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}+\frac {8 \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a} \\ & = -\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 (-7+a x) \sqrt {c-a c x}+12 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{3 a} \]

[In]

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

[Out]

(2*(-7 + a*x)*Sqrt[c - a*c*x] + 12*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(3*a)

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75

method	result	size
risch	\(-\frac {2 \left (a x -7\right ) \left (a x -1\right ) c}{3 a \sqrt {-c \left (a x -1\right )}}+\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}}{a}\)	\(57\)
derivativedivides	\(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {3}{2}}}{3}+2 c \sqrt {-a c x +c}-2 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a}\)	\(59\)
default	\(\frac {-\frac {2 \left (-a c x +c \right )^{\frac {3}{2}}}{3}-4 c \sqrt {-a c x +c}+4 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a c}\)	\(59\)
pseudoelliptic	\(\frac {4 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+\frac {2 a x \sqrt {-c \left (a x -1\right )}}{3}-\frac {14 \sqrt {-c \left (a x -1\right )}}{3}}{a}\)	\(59\)

[In]

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

[Out]

-2/3*(a*x-7)*(a*x-1)/a/(-c*(a*x-1))^(1/2)*c+4*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^(1/2)/a

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.57 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}, -\frac {2 \, {\left (6 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}\right ] \]

[In]

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

[Out]

[2/3*(3*sqrt(2)*sqrt(c)*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + sqrt(-a*c*x + c)*(
a*x - 7))/a, -2/3*(6*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - sqrt(-a*c*x + c)*(a*x
- 7))/a]

Sympy [A] (verification not implemented)

Time = 2.63 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.30 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\begin {cases} - \frac {2 \cdot \left (\frac {2 \sqrt {2} c^{2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 c \sqrt {- a c x + c} + \frac {\left (- a c x + c\right )^{\frac {3}{2}}}{3}\right )}{a c} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\begin {cases} - x & \text {for}\: a = 0 \\\frac {a x - 2 \log {\left (a x + 1 \right )} + 1}{a} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*(2*sqrt(2)*c**2*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) + 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, Tru
e)), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {2 \, {\left (3 \, \sqrt {2} c^{\frac {3}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + {\left (-a c x + c\right )}^{\frac {3}{2}} + 6 \, \sqrt {-a c x + c} c\right )}}{3 \, a c} \]

[In]

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

[Out]

-2/3*(3*sqrt(2)*c^(3/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c) + sqrt(-a*c*x + c))) + (-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 = 77, normalized size of antiderivative = 1.01 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} a^{2} c^{2} + 6 \, \sqrt {-a c x + c} a^{2} c^{3}\right )}}{3 \, a^{3} c^{3}} \]

[In]

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

[Out]

-4*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a*sqrt(-c)) - 2/3*((-a*c*x + c)^(3/2)*a^2*c^2 + 6*
sqrt(-a*c*x + c)*a^2*c^3)/(a^3*c^3)

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {2}\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{a}-\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a\,c}-\frac {4\,\sqrt {c-a\,c\,x}}{a} \]

[In]

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

[Out]

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