\(\int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx\) [715]

Optimal result

Integrand size = 24, antiderivative size = 87 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {3 \sqrt {c-a^2 c x^2}}{2 a}-\frac {(1-a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {3 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6302, 6277, 685, 655, 223, 209} \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {3 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}-\frac {(1-a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {3 \sqrt {c-a^2 c x^2}}{2 a} \]

[In]

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

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 655

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

Rule 685

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(m + 2*p + 1))), x] + Dist[2*c*d*((m + p)/(c*(m + 2*p + 1))), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rule 6277

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/c^(n/2), Int[(c + d*x^2)^(p
+ n/2)/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) && I
LtQ[n/2, 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^2 c x^2} \, dx \\ & = -\left (c \int \frac {(1-a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\right ) \\ & = -\frac {(1-a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {1}{2} (3 c) \int \frac {1-a x}{\sqrt {c-a^2 c x^2}} \, dx \\ & = -\frac {3 \sqrt {c-a^2 c x^2}}{2 a}-\frac {(1-a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {1}{2} (3 c) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx \\ & = -\frac {3 \sqrt {c-a^2 c x^2}}{2 a}-\frac {(1-a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {1}{2} (3 c) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right ) \\ & = -\frac {3 \sqrt {c-a^2 c x^2}}{2 a}-\frac {(1-a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {3 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.15 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (-\sqrt {1+a x} \left (4-5 a x+a^2 x^2\right )+6 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]

[In]

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

[Out]

(Sqrt[c - a^2*c*x^2]*(-(Sqrt[1 + a*x]*(4 - 5*a*x + a^2*x^2)) + 6*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))
/(2*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.79

method	result	size
risch	\(-\frac {\left (a x -4\right ) \left (a^{2} x^{2}-1\right ) c}{2 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {3 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\)	\(69\)
default	\(\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}-\frac {2 \left (\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}+\frac {a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\right )}{a}\)	\(127\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.54 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\left [\frac {2 \, \sqrt {-a^{2} c x^{2} + c} {\left (a x - 4\right )} + 3 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{4 \, a}, \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - 4\right )} + 3 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{2 \, a}\right ] \]

[In]

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

[Out]

[1/4*(2*sqrt(-a^2*c*x^2 + c)*(a*x - 4) + 3*sqrt(-c)*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c)
)/a, 1/2*(sqrt(-a^2*c*x^2 + c)*(a*x - 4) + 3*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))
/a]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.54 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} x - \frac {3 \, \sqrt {c} \arcsin \left (a x\right )}{2 \, a} - \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} {\left (x - \frac {4}{a}\right )} + \frac {3 \, c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{2 \, \sqrt {-c} {\left | a \right |}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]

[In]

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

[Out]

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