\(\int e^{-3 \coth ^{-1}(a x)} (c-a^2 c x^2)^{9/2} \, dx\) [659]

Optimal result

Integrand size = 24, antiderivative size = 189 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=-\frac {8 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{2 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {2 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6327, 6328, 45} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {2 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{2 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \]

[In]

Int[(c - a^2*c*x^2)^(9/2)/E^(3*ArcCoth[a*x]),x]

[Out]

(-8*(1 - a*x)^7*(c - a^2*c*x^2)^(9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + (3*(1 - a*x)^8*(c - a^2*c*x^2)^(
9/2))/(2*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (2*(1 - a*x)^9*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(
9/2)*x^9) + ((1 - a*x)^10*(c - a^2*c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6327

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(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]

Rule 6328

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[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] &&  !
IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]

Rubi steps \begin{align*} \text {integral}& = \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^9 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \\ & = \frac {\left (c-a^2 c x^2\right )^{9/2} \int (-1+a x)^6 (1+a x)^3 \, dx}{a^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \\ & = \frac {\left (c-a^2 c x^2\right )^{9/2} \int \left (8 (-1+a x)^6+12 (-1+a x)^7+6 (-1+a x)^8+(-1+a x)^9\right ) \, dx}{a^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.38 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {c^4 (-1+a x)^7 \sqrt {c-a^2 c x^2} \left (44+98 a x+77 a^2 x^2+21 a^3 x^3\right )}{210 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]

[In]

Integrate[(c - a^2*c*x^2)^(9/2)/E^(3*ArcCoth[a*x]),x]

[Out]

(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)

Maple [A] (verified)

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	\(\frac {\left (21 a^{9} x^{9}-70 a^{8} x^{8}+240 a^{6} x^{6}-210 a^{5} x^{5}-252 a^{4} x^{4}+420 a^{3} x^{3}-315 a x +210\right ) x \,c^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{210 \left (a x -1\right )^{2}}\)	\(102\)

[In]

int((-a^2*c*x^2+c)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.50 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {{\left (21 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} + 240 \, a^{6} c^{4} x^{7} - 210 \, a^{5} c^{4} x^{6} - 252 \, a^{4} c^{4} x^{5} + 420 \, a^{3} c^{4} x^{4} - 315 \, a c^{4} x^{2} + 210 \, c^{4} x\right )} \sqrt {-a^{2} c}}{210 \, a} \]

[In]

integrate((-a^2*c*x^2+c)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\text {Timed out} \]

[In]

integrate((-a**2*c*x**2+c)**(9/2)*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.08 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {{\left (21 \, a^{11} \sqrt {-c} c^{4} x^{11} - 49 \, a^{10} \sqrt {-c} c^{4} x^{10} - 70 \, a^{9} \sqrt {-c} c^{4} x^{9} + 240 \, a^{8} \sqrt {-c} c^{4} x^{8} + 30 \, a^{7} \sqrt {-c} c^{4} x^{7} - 462 \, a^{6} \sqrt {-c} c^{4} x^{6} + 168 \, a^{5} \sqrt {-c} c^{4} x^{5} + 420 \, a^{4} \sqrt {-c} c^{4} x^{4} - 315 \, a^{3} \sqrt {-c} c^{4} x^{3} - 105 \, a^{2} \sqrt {-c} c^{4} x^{2} - 210 \, \sqrt {-c} c^{4}\right )} {\left (a x - 1\right )}^{2}}{210 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}} \]

[In]

integrate((-a^2*c*x^2+c)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

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))

Giac [F]

\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((-a^2*c*x^2+c)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^(9/2)*((a*x - 1)/(a*x + 1))^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int {\left (c-a^2\,c\,x^2\right )}^{9/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]

[In]

int((c - a^2*c*x^2)^(9/2)*((a*x - 1)/(a*x + 1))^(3/2),x)

[Out]

int((c - a^2*c*x^2)^(9/2)*((a*x - 1)/(a*x + 1))^(3/2), x)