Integrand size = 27, antiderivative size = 112 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}+\frac {7 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3} \]
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
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.79 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (32-21 a x+16 a^2 x^2-6 a^3 x^3\right )-21 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{24 a^3} \]
(Sqrt[c - a^2*c*x^2]*(32 - 21*a*x + 16*a^2*x^2 - 6*a^3*x^3) - 21*Sqrt[c]*A rcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(24*a^3)
Time = 0.46 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.43, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {6702, 541, 25, 27, 533, 25, 27, 533, 25, 27, 455, 224, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx\) |
| \(\Big \downarrow \) 6702 |
| \(\displaystyle c \int \frac {x^2 (1-a x)^2}{\sqrt {c-a^2 c x^2}}dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle c \left (-\frac {\int -\frac {a^2 c x^2 (7-8 a x)}{\sqrt {c-a^2 c x^2}}dx}{4 a^2 c}-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {\int \frac {a^2 c x^2 (7-8 a x)}{\sqrt {c-a^2 c x^2}}dx}{4 a^2 c}-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{4} \int \frac {x^2 (7-8 a x)}{\sqrt {c-a^2 c x^2}}dx-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {1}{4} \left (\frac {\int -\frac {a c x (16-21 a x)}{\sqrt {c-a^2 c x^2}}dx}{3 a^2 c}+\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}-\frac {\int \frac {a c x (16-21 a x)}{\sqrt {c-a^2 c x^2}}dx}{3 a^2 c}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}-\frac {\int \frac {x (16-21 a x)}{\sqrt {c-a^2 c x^2}}dx}{3 a}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}-\frac {\frac {\int -\frac {a c (21-32 a x)}{\sqrt {c-a^2 c x^2}}dx}{2 a^2 c}+\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}}{3 a}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}-\frac {\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}-\frac {\int \frac {a c (21-32 a x)}{\sqrt {c-a^2 c x^2}}dx}{2 a^2 c}}{3 a}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}-\frac {\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}-\frac {\int \frac {21-32 a x}{\sqrt {c-a^2 c x^2}}dx}{2 a}}{3 a}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}-\frac {\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}-\frac {21 \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\frac {32 \sqrt {c-a^2 c x^2}}{a c}}{2 a}}{3 a}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}-\frac {\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}-\frac {21 \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}+\frac {32 \sqrt {c-a^2 c x^2}}{a c}}{2 a}}{3 a}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle c \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}-\frac {\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}-\frac {\frac {21 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}+\frac {32 \sqrt {c-a^2 c x^2}}{a c}}{2 a}}{3 a}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
c*(-1/4*(x^3*Sqrt[c - a^2*c*x^2])/c + ((8*x^2*Sqrt[c - a^2*c*x^2])/(3*a*c) - ((21*x*Sqrt[c - a^2*c*x^2])/(2*a*c) - ((32*Sqrt[c - a^2*c*x^2])/(a*c) + (21*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(a*Sqrt[c]))/(2*a))/(3*a)) /4)
3.13.37.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[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]
Time = 0.27 (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 | \(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 a^{2} c}-\frac {9 \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4 a^{2}}-\frac {2 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 a^{3} c}+\frac {2 \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}+\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}}{a^{3}}\) | \(176\) |
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
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.50 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\left [-\frac {2 \, {\left (6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right )} \sqrt {-a^{2} c x^{2} + c} - 21 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{48 \, a^{3}}, -\frac {{\left (6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right )} \sqrt {-a^{2} c x^{2} + c} + 21 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{24 \, a^{3}}\right ] \]
[-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]
\[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=- \int \left (- \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\right )\, dx - \int \frac {a x^{3} \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\, dx \]
-Integral(-x**2*sqrt(-a**2*c*x**2 + c)/(a*x + 1), x) - Integral(a*x**3*sqr t(-a**2*c*x**2 + c)/(a*x + 1), x)
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.83 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=-\frac {9 \, \sqrt {-a^{2} c x^{2} + c} x}{8 \, a^{2}} + \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{4 \, a^{2} c} + \frac {7 \, \sqrt {c} \arcsin \left (a x\right )}{8 \, a^{3}} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{3}} - \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a^{3} c} \]
-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)
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (92) = 184\).
Time = 0.33 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.97 \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=-\frac {{\left (336 \, a^{5} \sqrt {c} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + \frac {{\left (75 \, a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - 83 \, a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{2} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - 21 \, a^{5} c^{4} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - 77 \, a^{5} c^{3} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )\right )} {\left (a x + 1\right )}^{4}}{c^{4}}\right )} {\left | a \right |}}{192 \, a^{9}} \]
-1/192*(336*a^5*sqrt(c)*arctan(sqrt(-c + 2*c/(a*x + 1))/sqrt(c))*sgn(1/(a* x + 1))*sgn(a) + (75*a^5*(c - 2*c/(a*x + 1))^3*c*sqrt(-c + 2*c/(a*x + 1))* sgn(1/(a*x + 1))*sgn(a) - 83*a^5*(c - 2*c/(a*x + 1))^2*c^2*sqrt(-c + 2*c/( a*x + 1))*sgn(1/(a*x + 1))*sgn(a) - 21*a^5*c^4*sqrt(-c + 2*c/(a*x + 1))*sg n(1/(a*x + 1))*sgn(a) - 77*a^5*c^3*(-c + 2*c/(a*x + 1))^(3/2)*sgn(1/(a*x + 1))*sgn(a))*(a*x + 1)^4/c^4)*abs(a)/a^9
Timed out. \[ \int e^{-2 \text {arctanh}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=-\int \frac {x^2\,\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]