Integrand size = 27, antiderivative size = 60 \[ \int \frac {e^{-3 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx=\frac {(1+3 a x) \sqrt {1-a^2 x^2}}{24 a^3 c^5 (1-a x)^3 (1+a x)^6 \sqrt {c-a^2 c x^2}} \] Output:
1/24*(3*a*x+1)*(-a^2*x^2+1)^(1/2)/a^3/c^5/(-a*x+1)^3/(a*x+1)^6/(-a^2*c*x^2 +c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-3 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx=-\frac {(1+3 a x) \sqrt {1-a^2 x^2}}{24 a^3 c^5 (-1+a x)^3 (1+a x)^6 \sqrt {c-a^2 c x^2}} \] Input:
Integrate[x^2/(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(11/2)),x]
Output:
-1/24*((1 + 3*a*x)*Sqrt[1 - a^2*x^2])/(a^3*c^5*(-1 + a*x)^3*(1 + a*x)^6*Sq rt[c - a^2*c*x^2])
Time = 0.75 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6703, 6700, 91}
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.
\(\displaystyle \int \frac {x^2 e^{-3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{11/2}} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{-3 \text {arctanh}(a x)} x^2}{\left (1-a^2 x^2\right )^{11/2}}dx}{c^5 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {x^2}{(1-a x)^4 (a x+1)^7}dx}{c^5 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 91 |
\(\displaystyle \frac {(3 a x+1) \sqrt {1-a^2 x^2}}{24 a^3 c^5 (1-a x)^3 (a x+1)^6 \sqrt {c-a^2 c x^2}}\) |
Input:
Int[x^2/(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(11/2)),x]
Output:
((1 + 3*a*x)*Sqrt[1 - a^2*x^2])/(24*a^3*c^5*(1 - a*x)^3*(1 + a*x)^6*Sqrt[c - a^2*c*x^2])
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)*((2*a*d*f*(n + p + 3 ) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x)/(d^2*f^2*(n + p + 2)*(n + p + 3))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2 , 0] && NeQ[n + p + 3, 0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*( b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1) + c*f*(p + 1))*(a* d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar t[p]) Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !I ntegerQ[n/2]
Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(-\frac {\left (a x -1\right ) \left (3 a x +1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{24 \left (a x +1\right )^{2} a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {11}{2}}}\) | \(49\) |
default | \(-\frac {\left (3 a x +1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{24 \sqrt {-a^{2} x^{2}+1}\, \left (a x +1\right )^{6} \left (a x -1\right )^{3} c^{6} a^{3}}\) | \(55\) |
orering | \(-\frac {x^{3} \left (x^{6} a^{6}+3 a^{5} x^{5}-8 a^{3} x^{3}-6 a^{2} x^{2}+6 a x +8\right ) \left (a x -1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{24 \left (a x +1\right )^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {11}{2}}}\) | \(80\) |
Input:
int(x^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(11/2),x,method=_RETUR NVERBOSE)
Output:
-1/24*(a*x-1)*(3*a*x+1)*(-a^2*x^2+1)^(3/2)/(a*x+1)^2/a^3/(-a^2*c*x^2+c)^(1 1/2)
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (53) = 106\).
Time = 0.14 (sec) , antiderivative size = 192, normalized size of antiderivative = 3.20 \[ \int \frac {e^{-3 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx=\frac {{\left (a^{6} x^{9} + 3 \, a^{5} x^{8} - 8 \, a^{3} x^{6} - 6 \, a^{2} x^{5} + 6 \, a x^{4} + 8 \, x^{3}\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{24 \, {\left (a^{11} c^{6} x^{11} + 3 \, a^{10} c^{6} x^{10} - a^{9} c^{6} x^{9} - 11 \, a^{8} c^{6} x^{8} - 6 \, a^{7} c^{6} x^{7} + 14 \, a^{6} c^{6} x^{6} + 14 \, a^{5} c^{6} x^{5} - 6 \, a^{4} c^{6} x^{4} - 11 \, a^{3} c^{6} x^{3} - a^{2} c^{6} x^{2} + 3 \, a c^{6} x + c^{6}\right )}} \] Input:
integrate(x^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(11/2),x, algori thm="fricas")
Output:
1/24*(a^6*x^9 + 3*a^5*x^8 - 8*a^3*x^6 - 6*a^2*x^5 + 6*a*x^4 + 8*x^3)*sqrt( -a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^11*c^6*x^11 + 3*a^10*c^6*x^10 - a^9* c^6*x^9 - 11*a^8*c^6*x^8 - 6*a^7*c^6*x^7 + 14*a^6*c^6*x^6 + 14*a^5*c^6*x^5 - 6*a^4*c^6*x^4 - 11*a^3*c^6*x^3 - a^2*c^6*x^2 + 3*a*c^6*x + c^6)
Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx=\text {Timed out} \] Input:
integrate(x**2/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(11/2),x )
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.55 \[ \int \frac {e^{-3 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx=-\frac {3 \, a x + 1}{24 \, {\left (a^{12} c^{\frac {11}{2}} x^{9} + 3 \, a^{11} c^{\frac {11}{2}} x^{8} - 8 \, a^{9} c^{\frac {11}{2}} x^{6} - 6 \, a^{8} c^{\frac {11}{2}} x^{5} + 6 \, a^{7} c^{\frac {11}{2}} x^{4} + 8 \, a^{6} c^{\frac {11}{2}} x^{3} - 3 \, a^{4} c^{\frac {11}{2}} x - a^{3} c^{\frac {11}{2}}\right )}} \] Input:
integrate(x^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(11/2),x, algori thm="maxima")
Output:
-1/24*(3*a*x + 1)/(a^12*c^(11/2)*x^9 + 3*a^11*c^(11/2)*x^8 - 8*a^9*c^(11/2 )*x^6 - 6*a^8*c^(11/2)*x^5 + 6*a^7*c^(11/2)*x^4 + 8*a^6*c^(11/2)*x^3 - 3*a ^4*c^(11/2)*x - a^3*c^(11/2))
\[ \int \frac {e^{-3 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {11}{2}} {\left (a x + 1\right )}^{3}} \,d x } \] Input:
integrate(x^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(11/2),x, algori thm="giac")
Output:
integrate((-a^2*x^2 + 1)^(3/2)*x^2/((-a^2*c*x^2 + c)^(11/2)*(a*x + 1)^3), x)
Time = 23.90 (sec) , antiderivative size = 187, normalized size of antiderivative = 3.12 \[ \int \frac {e^{-3 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx=\frac {\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}+3\,a\,x\,\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}}{24\,a^{14}\,c^6\,x^{11}+72\,a^{13}\,c^6\,x^{10}-24\,a^{12}\,c^6\,x^9-264\,a^{11}\,c^6\,x^8-144\,a^{10}\,c^6\,x^7+336\,a^9\,c^6\,x^6+336\,a^8\,c^6\,x^5-144\,a^7\,c^6\,x^4-264\,a^6\,c^6\,x^3-24\,a^5\,c^6\,x^2+72\,a^4\,c^6\,x+24\,a^3\,c^6} \] Input:
int((x^2*(1 - a^2*x^2)^(3/2))/((c - a^2*c*x^2)^(11/2)*(a*x + 1)^3),x)
Output:
((c - a^2*c*x^2)^(1/2)*(1 - a^2*x^2)^(1/2) + 3*a*x*(c - a^2*c*x^2)^(1/2)*( 1 - a^2*x^2)^(1/2))/(24*a^3*c^6 + 72*a^4*c^6*x - 24*a^5*c^6*x^2 - 264*a^6* c^6*x^3 - 144*a^7*c^6*x^4 + 336*a^8*c^6*x^5 + 336*a^9*c^6*x^6 - 144*a^10*c ^6*x^7 - 264*a^11*c^6*x^8 - 24*a^12*c^6*x^9 + 72*a^13*c^6*x^10 + 24*a^14*c ^6*x^11)
Time = 0.16 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-3 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx=\frac {\sqrt {c}\, \left (-3 a x -1\right )}{24 a^{3} c^{6} \left (a^{9} x^{9}+3 a^{8} x^{8}-8 a^{6} x^{6}-6 a^{5} x^{5}+6 a^{4} x^{4}+8 a^{3} x^{3}-3 a x -1\right )} \] Input:
int(x^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(11/2),x)
Output:
(sqrt(c)*( - 3*a*x - 1))/(24*a**3*c**6*(a**9*x**9 + 3*a**8*x**8 - 8*a**6*x **6 - 6*a**5*x**5 + 6*a**4*x**4 + 8*a**3*x**3 - 3*a*x - 1))