Integrand size = 24, antiderivative size = 234 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=-\frac {8 c^4 (1-a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}+\frac {32 c^4 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (1-a x)^8 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}+\frac {8 c^4 (1-a x)^9 \sqrt {c-a^2 c x^2}}{9 a \sqrt {1-a^2 x^2}}-\frac {c^4 (1-a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}} \] Output:
-8/3*c^4*(-a*x+1)^6*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)+32/7*c^4*(-a *x+1)^7*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-3*c^4*(-a*x+1)^8*(-a^2*c *x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)+8/9*c^4*(-a*x+1)^9*(-a^2*c*x^2+c)^(1/2) /a/(-a^2*x^2+1)^(1/2)-1/10*c^4*(-a*x+1)^10*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^ 2+1)^(1/2)
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.32 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=-\frac {c^4 (-1+a x)^6 \sqrt {c-a^2 c x^2} \left (193+528 a x+588 a^2 x^2+308 a^3 x^3+63 a^4 x^4\right )}{630 a \sqrt {1-a^2 x^2}} \] Input:
Integrate[(c - a^2*c*x^2)^(9/2)/E^ArcTanh[a*x],x]
Output:
-1/630*(c^4*(-1 + a*x)^6*Sqrt[c - a^2*c*x^2]*(193 + 528*a*x + 588*a^2*x^2 + 308*a^3*x^3 + 63*a^4*x^4))/(a*Sqrt[1 - a^2*x^2])
Time = 0.41 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.46, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6693, 6690, 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.
\(\displaystyle \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx\) |
\(\Big \downarrow \) 6693 |
\(\displaystyle \frac {c^4 \sqrt {c-a^2 c x^2} \int e^{-\text {arctanh}(a x)} \left (1-a^2 x^2\right )^{9/2}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 6690 |
\(\displaystyle \frac {c^4 \sqrt {c-a^2 c x^2} \int (1-a x)^5 (a x+1)^4dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {c^4 \sqrt {c-a^2 c x^2} \int \left ((1-a x)^9-8 (1-a x)^8+24 (1-a x)^7-32 (1-a x)^6+16 (1-a x)^5\right )dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^4 \left (-\frac {(1-a x)^{10}}{10 a}+\frac {8 (1-a x)^9}{9 a}-\frac {3 (1-a x)^8}{a}+\frac {32 (1-a x)^7}{7 a}-\frac {8 (1-a x)^6}{3 a}\right ) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[(c - a^2*c*x^2)^(9/2)/E^ArcTanh[a*x],x]
Output:
(c^4*Sqrt[c - a^2*c*x^2]*((-8*(1 - a*x)^6)/(3*a) + (32*(1 - a*x)^7)/(7*a) - (3*(1 - a*x)^8)/a + (8*(1 - a*x)^9)/(9*a) - (1 - a*x)^10/(10*a)))/Sqrt[1 - a^2*x^2]
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^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a , c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p]) Int [(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.44
method | result | size |
default | \(-\frac {\left (63 a^{9} x^{9}-70 a^{8} x^{8}-315 a^{7} x^{7}+360 x^{6} a^{6}+630 a^{5} x^{5}-756 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}+315 a x -630\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, c^{4} x}{630 \sqrt {-a^{2} x^{2}+1}}\) | \(103\) |
gosper | \(\frac {x \left (63 a^{9} x^{9}-70 a^{8} x^{8}-315 a^{7} x^{7}+360 x^{6} a^{6}+630 a^{5} x^{5}-756 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}+315 a x -630\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}} \sqrt {-a^{2} x^{2}+1}}{630 \left (a x -1\right )^{5} \left (a x +1\right )^{5}}\) | \(113\) |
orering | \(\frac {x \left (63 a^{9} x^{9}-70 a^{8} x^{8}-315 a^{7} x^{7}+360 x^{6} a^{6}+630 a^{5} x^{5}-756 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}+315 a x -630\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}} \sqrt {-a^{2} x^{2}+1}}{630 \left (a x -1\right )^{5} \left (a x +1\right )^{5}}\) | \(113\) |
Input:
int((-a^2*c*x^2+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOS E)
Output:
-1/630*(63*a^9*x^9-70*a^8*x^8-315*a^7*x^7+360*a^6*x^6+630*a^5*x^5-756*a^4* x^4-630*a^3*x^3+840*a^2*x^2+315*a*x-630)/(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1 ))^(1/2)*c^4*x
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.61 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {{\left (63 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} + 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} - 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} + 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} - 630 \, c^{4} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{630 \, {\left (a^{2} x^{2} - 1\right )}} \] Input:
integrate((-a^2*c*x^2+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fr icas")
Output:
1/630*(63*a^9*c^4*x^10 - 70*a^8*c^4*x^9 - 315*a^7*c^4*x^8 + 360*a^6*c^4*x^ 7 + 630*a^5*c^4*x^6 - 756*a^4*c^4*x^5 - 630*a^3*c^4*x^4 + 840*a^2*c^4*x^3 + 315*a*c^4*x^2 - 630*c^4*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2* x^2 - 1)
\[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}}}{a x + 1}\, dx \] Input:
integrate((-a**2*c*x**2+c)**(9/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)
Output:
Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(9/2)/(a*x + 1), x)
\[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} \sqrt {-a^{2} x^{2} + 1}}{a x + 1} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="ma xima")
Output:
integrate((-a^2*c*x^2 + c)^(9/2)*sqrt(-a^2*x^2 + 1)/(a*x + 1), x)
Time = 0.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.47 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=-\frac {1}{630} \, {\left (63 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} + 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} - 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} + 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} - 630 \, c^{4} x\right )} \sqrt {c} \] Input:
integrate((-a^2*c*x^2+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="gi ac")
Output:
-1/630*(63*a^9*c^4*x^10 - 70*a^8*c^4*x^9 - 315*a^7*c^4*x^8 + 360*a^6*c^4*x ^7 + 630*a^5*c^4*x^6 - 756*a^4*c^4*x^5 - 630*a^3*c^4*x^4 + 840*a^2*c^4*x^3 + 315*a*c^4*x^2 - 630*c^4*x)*sqrt(c)
Timed out. \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{9/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \] Input:
int(((c - a^2*c*x^2)^(9/2)*(1 - a^2*x^2)^(1/2))/(a*x + 1),x)
Output:
int(((c - a^2*c*x^2)^(9/2)*(1 - a^2*x^2)^(1/2))/(a*x + 1), x)
Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.33 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {\sqrt {c}\, c^{4} x \left (-63 a^{9} x^{9}+70 a^{8} x^{8}+315 a^{7} x^{7}-360 a^{6} x^{6}-630 a^{5} x^{5}+756 a^{4} x^{4}+630 a^{3} x^{3}-840 a^{2} x^{2}-315 a x +630\right )}{630} \] Input:
int((-a^2*c*x^2+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x)
Output:
(sqrt(c)*c**4*x*( - 63*a**9*x**9 + 70*a**8*x**8 + 315*a**7*x**7 - 360*a**6 *x**6 - 630*a**5*x**5 + 756*a**4*x**4 + 630*a**3*x**3 - 840*a**2*x**2 - 31 5*a*x + 630))/630