\(\int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx\) [713]

Optimal result

Integrand size = 27, antiderivative size = 112 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}+\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}-\frac {7 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3} \]

[Out]

-7/8*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))*c^(1/2)/a^3-2/3*x^2*(-a^2*c*x^2+c)^(1/2)/a+1/4*x^3*(-a^2*c*x^2+c
)^(1/2)-1/24*(-21*a*x+32)*(-a^2*c*x^2+c)^(1/2)/a^3

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6302, 6287, 1823, 847, 794, 223, 209} \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}+\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {7 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3}-\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3} \]

[In]

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

[Out]

(-2*x^2*Sqrt[c - a^2*c*x^2])/(3*a) + (x^3*Sqrt[c - a^2*c*x^2])/4 - ((32 - 21*a*x)*Sqrt[c - a^2*c*x^2])/(24*a^3
) - (7*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^3)

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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 847

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

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 6287

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/c^(n/2), Int[x^m*
((c + d*x^2)^(p + n/2)/(1 - a*x)^n), x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p
] || GtQ[c, 0]) && ILtQ[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)} x^2 \sqrt {c-a^2 c x^2} \, dx \\ & = -\left (c \int \frac {x^2 (1-a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\right ) \\ & = \frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {\int \frac {x^2 \left (-7 a^2 c+8 a^3 c x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{4 a^2} \\ & = -\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}+\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {\int \frac {x \left (-16 a^3 c^2+21 a^4 c^2 x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{12 a^4 c} \\ & = -\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}+\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}-\frac {(7 c) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{8 a^2} \\ & = -\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}+\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}-\frac {(7 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 )}{8 a^2} \\ & = -\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}+\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}-\frac {7 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.79 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (-32+21 a x-16 a^2 x^2+6 a^3 x^3\right )+21 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{24 a^3} \]

[In]

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

[Out]

(Sqrt[c - a^2*c*x^2]*(-32 + 21*a*x - 16*a^2*x^2 + 6*a^3*x^3) + 21*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sq
rt[c]*(-1 + a^2*x^2))])/(24*a^3)

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79

method	result	size
risch	\(-\frac {\left (6 a^{3} x^{3}-16 a^{2} x^{2}+21 a x -32\right ) \left (a^{2} x^{2}-1\right ) c}{24 a^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {7 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c}{8 a^{2} \sqrt {a^{2} c}}\)	\(89\)
default	\(-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 a^{2} c}+\frac {\frac {9 x \sqrt {-a^{2} c \,x^{2}+c}}{8}+\frac {9 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{8 \sqrt {a^{2} c}}}{a^{2}}+\frac {2 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 a^{3} c}-\frac {2 \left (\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}+\frac {a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\right )}{a^{3}}\)	\(176\)

[In]

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

[Out]

-1/24*(6*a^3*x^3-16*a^2*x^2+21*a*x-32)*(a^2*x^2-1)/a^3/(-c*(a^2*x^2-1))^(1/2)*c-7/8/a^2/(a^2*c)^(1/2)*arctan((
a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))*c

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.50 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\left [\frac {2 \, {\left (6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right )} \sqrt {-a^{2} c x^{2} + c} + 21 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{48 \, a^{3}}, \frac {{\left (6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right )} \sqrt {-a^{2} c x^{2} + c} + 21 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{24 \, a^{3}}\right ] \]

[In]

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

[Out]

[1/48*(2*(6*a^3*x^3 - 16*a^2*x^2 + 21*a*x - 32)*sqrt(-a^2*c*x^2 + c) + 21*sqrt(-c)*log(2*a^2*c*x^2 - 2*sqrt(-a
^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a^3, 1/24*((6*a^3*x^3 - 16*a^2*x^2 + 21*a*x - 32)*sqrt(-a^2*c*x^2 + c) + 21*s
qrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))/a^3]

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*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 = 93, normalized size of antiderivative = 0.83 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {9 \, \sqrt {-a^{2} c x^{2} + c} x}{8 \, a^{2}} - \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{4 \, a^{2} c} - \frac {7 \, \sqrt {c} \arcsin \left (a x\right )}{8 \, a^{3}} - \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{3}} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a^{3} c} \]

[In]

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

[Out]

9/8*sqrt(-a^2*c*x^2 + c)*x/a^2 - 1/4*(-a^2*c*x^2 + c)^(3/2)*x/(a^2*c) - 7/8*sqrt(c)*arcsin(a*x)/a^3 - 2*sqrt(-
a^2*c*x^2 + c)/a^3 + 2/3*(-a^2*c*x^2 + c)^(3/2)/(a^3*c)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.75 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {1}{24} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left (3 \, x - \frac {8}{a}\right )} x + \frac {21}{a^{2}}\right )} x - \frac {32}{a^{3}}\right )} + \frac {7 \, c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, a^{2} \sqrt {-c} {\left | a \right |}} \]

[In]

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

[Out]

1/24*sqrt(-a^2*c*x^2 + c)*((2*(3*x - 8/a)*x + 21/a^2)*x - 32/a^3) + 7/8*c*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*
c*x^2 + c)))/(a^2*sqrt(-c)*abs(a))

Mupad [F(-1)]

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

[In]

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

[Out]

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