Integrand size = 24, antiderivative size = 234 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {8 (1-a x)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {32 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {8 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \] Output:
8/3*(-a*x+1)^6*(-a^2*c*x^2+c)^(9/2)/a^10/(1-1/a^2/x^2)^(9/2)/x^9-32/7*(-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*(-a*x+1)^8*(-a^ 2*c*x^2+c)^(9/2)/a^10/(1-1/a^2/x^2)^(9/2)/x^9-8/9*(-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 = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.34 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {c^4 (-1+a x)^6 \sqrt {c-a^2 c x^2} \left (193+528 a x+588 a^2 x^2+308 a^3 x^3+63 a^4 x^4\right )}{630 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \] Input:
Integrate[(c - a^2*c*x^2)^(9/2)/E^ArcCoth[a*x],x]
Output:
(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))/(630*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)
Time = 0.84 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.47, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6746, 6747, 25, 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.
\(\displaystyle \int \left (c-a^2 c x^2\right )^{9/2} e^{-\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6746 |
\(\displaystyle \frac {\left (c-a^2 c x^2\right )^{9/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9dx}{x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {\left (c-a^2 c x^2\right )^{9/2} \int -(1-a x)^5 (a x+1)^4dx}{a^9 x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\left (c-a^2 c x^2\right )^{9/2} \int (1-a x)^5 (a x+1)^4dx}{a^9 x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {\left (c-a^2 c x^2\right )^{9/2} \int \left ((1-a x)^9-8 (1-a x)^8+24 (1-a x)^7-32 (1-a x)^6+16 (1-a x)^5\right )dx}{a^9 x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (-\frac {(1-a x)^{10}}{10 a}+\frac {8 (1-a x)^9}{9 a}-\frac {3 (1-a x)^8}{a}+\frac {32 (1-a x)^7}{7 a}-\frac {8 (1-a x)^6}{3 a}\right ) \left (c-a^2 c x^2\right )^{9/2}}{a^9 x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}\) |
Input:
Int[(c - a^2*c*x^2)^(9/2)/E^ArcCoth[a*x],x]
Output:
-(((c - a^2*c*x^2)^(9/2)*((-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)))/(a^9*( 1 - 1/(a^2*x^2))^(9/2)*x^9))
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 = 0.14 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.48
method | result | size |
default | \(\frac {\left (63 a^{9} x^{9}-70 a^{8} x^{8}-315 a^{7} x^{7}+360 x^{6} a^{6}+630 a^{5} x^{5}-756 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}+315 a x -630\right ) x \,c^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}{630 a x -630}\) | \(113\) |
gosper | \(\frac {x \left (63 a^{9} x^{9}-70 a^{8} x^{8}-315 a^{7} x^{7}+360 x^{6} a^{6}+630 a^{5} x^{5}-756 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}+315 a x -630\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}} \sqrt {\frac {a x -1}{a x +1}}}{630 \left (a x +1\right )^{4} \left (a x -1\right )^{5}}\) | \(116\) |
orering | \(\frac {x \left (63 a^{9} x^{9}-70 a^{8} x^{8}-315 a^{7} x^{7}+360 x^{6} a^{6}+630 a^{5} x^{5}-756 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}+315 a x -630\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}} \sqrt {\frac {a x -1}{a x +1}}}{630 \left (a x +1\right )^{4} \left (a x -1\right )^{5}}\) | \(116\) |
Input:
int((-a^2*c*x^2+c)^(9/2)*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)
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)*x*c^4*(-c*(a^2*x^2-1))^(1/2)*((a*x -1)/(a*x+1))^(1/2)/(a*x-1)
Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.50 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {{\left (63 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} + 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} - 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} + 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} - 630 \, c^{4} x\right )} \sqrt {-a^{2} c}}{630 \, a} \] Input:
integrate((-a^2*c*x^2+c)^(9/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="frica s")
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)/a
Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\text {Timed out} \] Input:
integrate((-a**2*c*x**2+c)**(9/2)*((a*x-1)/(a*x+1))**(1/2),x)
Output:
Timed out
\[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^(9/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxim a")
Output:
integrate((-a^2*c*x^2 + c)^(9/2)*sqrt((a*x - 1)/(a*x + 1)), x)
Time = 0.13 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.79 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {1}{630} \, {\left (63 \, a^{9} c^{4} x^{10} \mathrm {sgn}\left (a x + 1\right ) - 70 \, a^{8} c^{4} x^{9} \mathrm {sgn}\left (a x + 1\right ) - 315 \, a^{7} c^{4} x^{8} \mathrm {sgn}\left (a x + 1\right ) + 360 \, a^{6} c^{4} x^{7} \mathrm {sgn}\left (a x + 1\right ) + 630 \, a^{5} c^{4} x^{6} \mathrm {sgn}\left (a x + 1\right ) - 756 \, a^{4} c^{4} x^{5} \mathrm {sgn}\left (a x + 1\right ) - 630 \, a^{3} c^{4} x^{4} \mathrm {sgn}\left (a x + 1\right ) + 840 \, a^{2} c^{4} x^{3} \mathrm {sgn}\left (a x + 1\right ) + 315 \, a c^{4} x^{2} \mathrm {sgn}\left (a x + 1\right ) - 630 \, c^{4} x \mathrm {sgn}\left (a x + 1\right ) - \frac {319 \, c^{4} \mathrm {sgn}\left (a x + 1\right )}{a}\right )} \sqrt {-c} \] Input:
integrate((-a^2*c*x^2+c)^(9/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac" )
Output:
1/630*(63*a^9*c^4*x^10*sgn(a*x + 1) - 70*a^8*c^4*x^9*sgn(a*x + 1) - 315*a^ 7*c^4*x^8*sgn(a*x + 1) + 360*a^6*c^4*x^7*sgn(a*x + 1) + 630*a^5*c^4*x^6*sg n(a*x + 1) - 756*a^4*c^4*x^5*sgn(a*x + 1) - 630*a^3*c^4*x^4*sgn(a*x + 1) + 840*a^2*c^4*x^3*sgn(a*x + 1) + 315*a*c^4*x^2*sgn(a*x + 1) - 630*c^4*x*sgn (a*x + 1) - 319*c^4*sgn(a*x + 1)/a)*sqrt(-c)
Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int {\left (c-a^2\,c\,x^2\right )}^{9/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \] Input:
int((c - a^2*c*x^2)^(9/2)*((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
int((c - a^2*c*x^2)^(9/2)*((a*x - 1)/(a*x + 1))^(1/2), x)
Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.38 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {\sqrt {c}\, c^{4} i \left (-63 a^{10} x^{10}+70 a^{9} x^{9}+315 a^{8} x^{8}-360 a^{7} x^{7}-630 a^{6} x^{6}+756 a^{5} x^{5}+630 a^{4} x^{4}-840 a^{3} x^{3}-315 a^{2} x^{2}+630 a x -193\right )}{630 a} \] Input:
int((-a^2*c*x^2+c)^(9/2)*((a*x-1)/(a*x+1))^(1/2),x)
Output:
(sqrt(c)*c**4*i*( - 63*a**10*x**10 + 70*a**9*x**9 + 315*a**8*x**8 - 360*a* *7*x**7 - 630*a**6*x**6 + 756*a**5*x**5 + 630*a**4*x**4 - 840*a**3*x**3 - 315*a**2*x**2 + 630*a*x - 193))/(630*a)