Integrand size = 24, antiderivative size = 142 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{3 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}-\frac {4 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}+\frac {(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7} \] Output:
2/3*(-a*x+1)^6*(-a^2*c*x^2+c)^(7/2)/a^8/(1-1/a^2/x^2)^(7/2)/x^7-4/7*(-a*x+ 1)^7*(-a^2*c*x^2+c)^(7/2)/a^8/(1-1/a^2/x^2)^(7/2)/x^7+1/8*(-a*x+1)^8*(-a^2 *c*x^2+c)^(7/2)/a^8/(1-1/a^2/x^2)^(7/2)/x^7
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.44 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {c^3 (-1+a x)^6 \left (37+54 a x+21 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{168 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \] Input:
Integrate[(c - a^2*c*x^2)^(7/2)/E^(3*ArcCoth[a*x]),x]
Output:
-1/168*(c^3*(-1 + a*x)^6*(37 + 54*a*x + 21*a^2*x^2)*Sqrt[c - a^2*c*x^2])/( a^2*Sqrt[1 - 1/(a^2*x^2)]*x)
Time = 0.79 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.58, 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.
\(\displaystyle \int \left (c-a^2 c x^2\right )^{7/2} e^{-3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6746 |
\(\displaystyle \frac {\left (c-a^2 c x^2\right )^{7/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7dx}{x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {\left (c-a^2 c x^2\right )^{7/2} \int -(1-a x)^5 (a x+1)^2dx}{a^7 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\left (c-a^2 c x^2\right )^{7/2} \int (1-a x)^5 (a x+1)^2dx}{a^7 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {\left (c-a^2 c x^2\right )^{7/2} \int \left ((1-a x)^7-4 (1-a x)^6+4 (1-a x)^5\right )dx}{a^7 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (-\frac {(1-a x)^8}{8 a}+\frac {4 (1-a x)^7}{7 a}-\frac {2 (1-a x)^6}{3 a}\right ) \left (c-a^2 c x^2\right )^{7/2}}{a^7 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\) |
Input:
Int[(c - a^2*c*x^2)^(7/2)/E^(3*ArcCoth[a*x]),x]
Output:
-(((c - a^2*c*x^2)^(7/2)*((-2*(1 - a*x)^6)/(3*a) + (4*(1 - a*x)^7)/(7*a) - (1 - a*x)^8/(8*a)))/(a^7*(1 - 1/(a^2*x^2))^(7/2)*x^7))
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 = 100, normalized size of antiderivative = 0.70
method | result | size |
gosper | \(\frac {x \left (21 a^{7} x^{7}-72 x^{6} a^{6}+28 a^{5} x^{5}+168 a^{4} x^{4}-210 a^{3} x^{3}-56 a^{2} x^{2}+252 a x -168\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{168 \left (a x +1\right )^{2} \left (a x -1\right )^{5}}\) | \(100\) |
orering | \(\frac {x \left (21 a^{7} x^{7}-72 x^{6} a^{6}+28 a^{5} x^{5}+168 a^{4} x^{4}-210 a^{3} x^{3}-56 a^{2} x^{2}+252 a x -168\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{168 \left (a x +1\right )^{2} \left (a x -1\right )^{5}}\) | \(100\) |
default | \(-\frac {\left (21 a^{7} x^{7}-72 x^{6} a^{6}+28 a^{5} x^{5}+168 a^{4} x^{4}-210 a^{3} x^{3}-56 a^{2} x^{2}+252 a x -168\right ) x \,c^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{168 \left (a x -1\right )^{2}}\) | \(102\) |
Input:
int((-a^2*c*x^2+c)^(7/2)*((a*x-1)/(a*x+1))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/168*x*(21*a^7*x^7-72*a^6*x^6+28*a^5*x^5+168*a^4*x^4-210*a^3*x^3-56*a^2*x ^2+252*a*x-168)*(-a^2*c*x^2+c)^(7/2)*((a*x-1)/(a*x+1))^(3/2)/(a*x+1)^2/(a* x-1)^5
Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.67 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {{\left (21 \, a^{7} c^{3} x^{8} - 72 \, a^{6} c^{3} x^{7} + 28 \, a^{5} c^{3} x^{6} + 168 \, a^{4} c^{3} x^{5} - 210 \, a^{3} c^{3} x^{4} - 56 \, a^{2} c^{3} x^{3} + 252 \, a c^{3} x^{2} - 168 \, c^{3} x\right )} \sqrt {-a^{2} c}}{168 \, a} \] Input:
integrate((-a^2*c*x^2+c)^(7/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="frica s")
Output:
-1/168*(21*a^7*c^3*x^8 - 72*a^6*c^3*x^7 + 28*a^5*c^3*x^6 + 168*a^4*c^3*x^5 - 210*a^3*c^3*x^4 - 56*a^2*c^3*x^3 + 252*a*c^3*x^2 - 168*c^3*x)*sqrt(-a^2 *c)/a
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\text {Timed out} \] Input:
integrate((-a**2*c*x**2+c)**(7/2)*((a*x-1)/(a*x+1))**(3/2),x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.21 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {{\left (21 \, a^{9} \sqrt {-c} c^{3} x^{9} - 51 \, a^{8} \sqrt {-c} c^{3} x^{8} - 44 \, a^{7} \sqrt {-c} c^{3} x^{7} + 196 \, a^{6} \sqrt {-c} c^{3} x^{6} - 42 \, a^{5} \sqrt {-c} c^{3} x^{5} - 266 \, a^{4} \sqrt {-c} c^{3} x^{4} + 196 \, a^{3} \sqrt {-c} c^{3} x^{3} + 84 \, a^{2} \sqrt {-c} c^{3} x^{2} + 168 \, \sqrt {-c} c^{3}\right )} {\left (a x - 1\right )}^{2}}{168 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}} \] Input:
integrate((-a^2*c*x^2+c)^(7/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxim a")
Output:
-1/168*(21*a^9*sqrt(-c)*c^3*x^9 - 51*a^8*sqrt(-c)*c^3*x^8 - 44*a^7*sqrt(-c )*c^3*x^7 + 196*a^6*sqrt(-c)*c^3*x^6 - 42*a^5*sqrt(-c)*c^3*x^5 - 266*a^4*s qrt(-c)*c^3*x^4 + 196*a^3*sqrt(-c)*c^3*x^3 + 84*a^2*sqrt(-c)*c^3*x^2 + 168 *sqrt(-c)*c^3)*(a*x - 1)^2/((a^3*x^2 - 2*a^2*x + a)*(a*x + 1))
Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.06 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {1}{168} \, {\left (21 \, a^{7} c^{3} x^{8} \mathrm {sgn}\left (a x + 1\right ) - 72 \, a^{6} c^{3} x^{7} \mathrm {sgn}\left (a x + 1\right ) + 28 \, a^{5} c^{3} x^{6} \mathrm {sgn}\left (a x + 1\right ) + 168 \, a^{4} c^{3} x^{5} \mathrm {sgn}\left (a x + 1\right ) - 210 \, a^{3} c^{3} x^{4} \mathrm {sgn}\left (a x + 1\right ) - 56 \, a^{2} c^{3} x^{3} \mathrm {sgn}\left (a x + 1\right ) + 252 \, a c^{3} x^{2} \mathrm {sgn}\left (a x + 1\right ) - 168 \, c^{3} x \mathrm {sgn}\left (a x + 1\right ) - \frac {219 \, c^{3} \mathrm {sgn}\left (a x + 1\right )}{a}\right )} \sqrt {-c} \] Input:
integrate((-a^2*c*x^2+c)^(7/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac" )
Output:
-1/168*(21*a^7*c^3*x^8*sgn(a*x + 1) - 72*a^6*c^3*x^7*sgn(a*x + 1) + 28*a^5 *c^3*x^6*sgn(a*x + 1) + 168*a^4*c^3*x^5*sgn(a*x + 1) - 210*a^3*c^3*x^4*sgn (a*x + 1) - 56*a^2*c^3*x^3*sgn(a*x + 1) + 252*a*c^3*x^2*sgn(a*x + 1) - 168 *c^3*x*sgn(a*x + 1) - 219*c^3*sgn(a*x + 1)/a)*sqrt(-c)
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int {\left (c-a^2\,c\,x^2\right )}^{7/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \] Input:
int((c - a^2*c*x^2)^(7/2)*((a*x - 1)/(a*x + 1))^(3/2),x)
Output:
int((c - a^2*c*x^2)^(7/2)*((a*x - 1)/(a*x + 1))^(3/2), x)
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.51 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} i \left (21 a^{8} x^{8}-72 a^{7} x^{7}+28 a^{6} x^{6}+168 a^{5} x^{5}-210 a^{4} x^{4}-56 a^{3} x^{3}+252 a^{2} x^{2}-168 a x +37\right )}{168 a} \] Input:
int((-a^2*c*x^2+c)^(7/2)*((a*x-1)/(a*x+1))^(3/2),x)
Output:
(sqrt(c)*c**3*i*(21*a**8*x**8 - 72*a**7*x**7 + 28*a**6*x**6 + 168*a**5*x** 5 - 210*a**4*x**4 - 56*a**3*x**3 + 252*a**2*x**2 - 168*a*x + 37))/(168*a)