\(\int e^{-\coth ^{-1}(a x)} (c-a^2 c x^2)^{9/2} \, dx\) [642]

Optimal result

Integrand size = 24, antiderivative size = 234 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {8 (1-a x)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {32 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {8 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 234, 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} \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {32 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {8 (1-a x)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \]

[In]

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

[Out]

(8*(1 - a*x)^6*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (32*(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))/(a^10*(1 - 1/(a^2*x^2))^(9/
2)*x^9) - (8*(1 - a*x)^9*(c - a^2*c*x^2)^(9/2))/(9*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^{-\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)^5 (1+a x)^4 \, 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 (16 (-1+a x)^5+32 (-1+a x)^6+24 (-1+a x)^7+8 (-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)^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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.34 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {c^4 (-1+a x)^6 \sqrt {c-a^2 c x^2} \left (193+528 a x+588 a^2 x^2+308 a^3 x^3+63 a^4 x^4\right )}{630 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]

[In]

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

[Out]

(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)

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.48

method	result	size
default	\(\frac {\left (63 a^{9} x^{9}-70 a^{8} x^{8}-315 a^{7} x^{7}+360 a^{6} x^{6}+630 a^{5} x^{5}-756 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}+315 a x -630\right ) x \,c^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}{630 a x -630}\)	\(113\)
gosper	\(\frac {x \left (63 a^{9} x^{9}-70 a^{8} x^{8}-315 a^{7} x^{7}+360 a^{6} x^{6}+630 a^{5} x^{5}-756 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}+315 a x -630\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}} \sqrt {\frac {a x -1}{a x +1}}}{630 \left (a x +1\right )^{4} \left (a x -1\right )^{5}}\)	\(116\)

[In]

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

[Out]

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-6
30)*x*c^4*(-c*(a^2*x^2-1))^(1/2)*((a*x-1)/(a*x+1))^(1/2)/(a*x-1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.50 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {{\left (63 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} + 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} - 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} + 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} - 630 \, c^{4} x\right )} \sqrt {-a^{2} c}}{630 \, a} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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