Integrand size = 24, antiderivative size = 189 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=-\frac {8 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{2 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {2 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \]
-8/7*(-a*x+1)^7*(-a^2*c*x^2+c)^(9/2)/a^10/(1-1/a^2/x^2)^(9/2)/x^9+3/2*(-a* x+1)^8*(-a^2*c*x^2+c)^(9/2)/a^10/(1-1/a^2/x^2)^(9/2)/x^9-2/3*(-a*x+1)^9*(- a^2*c*x^2+c)^(9/2)/a^10/(1-1/a^2/x^2)^(9/2)/x^9+1/10*(-a*x+1)^10*(-a^2*c*x ^2+c)^(9/2)/a^10/(1-1/a^2/x^2)^(9/2)/x^9
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.38 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {c^4 (-1+a x)^7 \sqrt {c-a^2 c x^2} \left (44+98 a x+77 a^2 x^2+21 a^3 x^3\right )}{210 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
(c^4*(-1 + a*x)^7*Sqrt[c - a^2*c*x^2]*(44 + 98*a*x + 77*a^2*x^2 + 21*a^3*x ^3))/(210*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)
Time = 0.45 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.51, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6746, 6747, 49, 2009}
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.
\(\displaystyle \int \left (c-a^2 c x^2\right )^{9/2} e^{-3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6746 |
\(\displaystyle \frac {\left (c-a^2 c x^2\right )^{9/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9dx}{x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {\left (c-a^2 c x^2\right )^{9/2} \int (1-a x)^6 (a x+1)^3dx}{a^9 x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\left (c-a^2 c x^2\right )^{9/2} \int \left (-(1-a x)^9+6 (1-a x)^8-12 (1-a x)^7+8 (1-a x)^6\right )dx}{a^9 x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (\frac {(1-a x)^{10}}{10 a}-\frac {2 (1-a x)^9}{3 a}+\frac {3 (1-a x)^8}{2 a}-\frac {8 (1-a x)^7}{7 a}\right ) \left (c-a^2 c x^2\right )^{9/2}}{a^9 x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}\) |
((c - a^2*c*x^2)^(9/2)*((-8*(1 - a*x)^7)/(7*a) + (3*(1 - a*x)^8)/(2*a) - ( 2*(1 - a*x)^9)/(3*a) + (1 - a*x)^10/(10*a)))/(a^9*(1 - 1/(a^2*x^2))^(9/2)* x^9)
3.7.59.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p) Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && !IntegerQ[p]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[c^p/a^(2*p) Int[(u/x^(2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !Inte gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
Time = 0.52 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(\frac {x \left (21 a^{9} x^{9}-70 a^{8} x^{8}+240 a^{6} x^{6}-210 a^{5} x^{5}-252 a^{4} x^{4}+420 a^{3} x^{3}-315 a x +210\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{210 \left (a x +1\right )^{3} \left (a x -1\right )^{6}}\) | \(100\) |
default | \(\frac {\left (21 a^{9} x^{9}-70 a^{8} x^{8}+240 a^{6} x^{6}-210 a^{5} x^{5}-252 a^{4} x^{4}+420 a^{3} x^{3}-315 a x +210\right ) x \,c^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{210 \left (a x -1\right )^{2}}\) | \(102\) |
1/210*x*(21*a^9*x^9-70*a^8*x^8+240*a^6*x^6-210*a^5*x^5-252*a^4*x^4+420*a^3 *x^3-315*a*x+210)*(-a^2*c*x^2+c)^(9/2)*((a*x-1)/(a*x+1))^(3/2)/(a*x+1)^3/( a*x-1)^6
Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.50 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {{\left (21 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} + 240 \, a^{6} c^{4} x^{7} - 210 \, a^{5} c^{4} x^{6} - 252 \, a^{4} c^{4} x^{5} + 420 \, a^{3} c^{4} x^{4} - 315 \, a c^{4} x^{2} + 210 \, c^{4} x\right )} \sqrt {-a^{2} c}}{210 \, a} \]
1/210*(21*a^9*c^4*x^10 - 70*a^8*c^4*x^9 + 240*a^6*c^4*x^7 - 210*a^5*c^4*x^ 6 - 252*a^4*c^4*x^5 + 420*a^3*c^4*x^4 - 315*a*c^4*x^2 + 210*c^4*x)*sqrt(-a ^2*c)/a
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.08 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {{\left (21 \, a^{11} \sqrt {-c} c^{4} x^{11} - 49 \, a^{10} \sqrt {-c} c^{4} x^{10} - 70 \, a^{9} \sqrt {-c} c^{4} x^{9} + 240 \, a^{8} \sqrt {-c} c^{4} x^{8} + 30 \, a^{7} \sqrt {-c} c^{4} x^{7} - 462 \, a^{6} \sqrt {-c} c^{4} x^{6} + 168 \, a^{5} \sqrt {-c} c^{4} x^{5} + 420 \, a^{4} \sqrt {-c} c^{4} x^{4} - 315 \, a^{3} \sqrt {-c} c^{4} x^{3} - 105 \, a^{2} \sqrt {-c} c^{4} x^{2} - 210 \, \sqrt {-c} c^{4}\right )} {\left (a x - 1\right )}^{2}}{210 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}} \]
1/210*(21*a^11*sqrt(-c)*c^4*x^11 - 49*a^10*sqrt(-c)*c^4*x^10 - 70*a^9*sqrt (-c)*c^4*x^9 + 240*a^8*sqrt(-c)*c^4*x^8 + 30*a^7*sqrt(-c)*c^4*x^7 - 462*a^ 6*sqrt(-c)*c^4*x^6 + 168*a^5*sqrt(-c)*c^4*x^5 + 420*a^4*sqrt(-c)*c^4*x^4 - 315*a^3*sqrt(-c)*c^4*x^3 - 105*a^2*sqrt(-c)*c^4*x^2 - 210*sqrt(-c)*c^4)*( a*x - 1)^2/((a^3*x^2 - 2*a^2*x + a)*(a*x + 1))
\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int {\left (c-a^2\,c\,x^2\right )}^{9/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]