\(\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2)^{9/2} \, dx\) [605]

Optimal result

Integrand size = 24, antiderivative size = 169 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=-\frac {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {2 \left (c-a^2 c x^2\right )^{9/2}}{9 a}+\frac {1}{10} x \left (c-a^2 c x^2\right )^{9/2}-\frac {77 c^{9/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{256 a} \] Output:

-77/256*c^4*x*(-a^2*c*x^2+c)^(1/2)-77/384*c^3*x*(-a^2*c*x^2+c)^(3/2)-77/48 
0*c^2*x*(-a^2*c*x^2+c)^(5/2)-11/80*c*x*(-a^2*c*x^2+c)^(7/2)+2/9*(-a^2*c*x^ 
2+c)^(9/2)/a+1/10*x*(-a^2*c*x^2+c)^(9/2)-77/256*c^(9/2)*arctan(a*c^(1/2)*x 
/(-a^2*c*x^2+c)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {c^4 \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (2560-10615 a x-2185 a^2 x^2+16390 a^3 x^3+9210 a^4 x^4-15048 a^5 x^5-10552 a^6 x^6+7216 a^7 x^7+5584 a^8 x^8-1408 a^9 x^9-1152 a^{10} x^{10}\right )+6930 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{11520 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(c^4*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(2560 - 10615*a*x - 2185*a^2*x^2 + 
 16390*a^3*x^3 + 9210*a^4*x^4 - 15048*a^5*x^5 - 10552*a^6*x^6 + 7216*a^7*x 
^7 + 5584*a^8*x^8 - 1408*a^9*x^9 - 1152*a^10*x^10) + 6930*Sqrt[1 - a*x]*Ar 
cSin[Sqrt[1 - a*x]/Sqrt[2]]))/(11520*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6717, 6691, 469, 455, 211, 211, 211, 211, 224, 216}

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.

\(\displaystyle \int \left (c-a^2 c x^2\right )^{9/2} e^{2 \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6717

\(\displaystyle -\int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{9/2}dx\)

\(\Big \downarrow \) 6691

\(\displaystyle -c \int (a x+1)^2 \left (c-a^2 c x^2\right )^{7/2}dx\)

\(\Big \downarrow \) 469

\(\displaystyle -c \left (\frac {11}{10} \int (a x+1) \left (c-a^2 c x^2\right )^{7/2}dx-\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a c}\right )\)

\(\Big \downarrow \) 455

\(\displaystyle -c \left (\frac {11}{10} \left (\int \left (c-a^2 c x^2\right )^{7/2}dx-\frac {\left (c-a^2 c x^2\right )^{9/2}}{9 a c}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a c}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle -c \left (\frac {11}{10} \left (\frac {7}{8} c \int \left (c-a^2 c x^2\right )^{5/2}dx-\frac {\left (c-a^2 c x^2\right )^{9/2}}{9 a c}+\frac {1}{8} x \left (c-a^2 c x^2\right )^{7/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a c}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle -c \left (\frac {11}{10} \left (\frac {7}{8} c \left (\frac {5}{6} c \int \left (c-a^2 c x^2\right )^{3/2}dx+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )-\frac {\left (c-a^2 c x^2\right )^{9/2}}{9 a c}+\frac {1}{8} x \left (c-a^2 c x^2\right )^{7/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a c}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle -c \left (\frac {11}{10} \left (\frac {7}{8} c \left (\frac {5}{6} c \left (\frac {3}{4} c \int \sqrt {c-a^2 c x^2}dx+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )-\frac {\left (c-a^2 c x^2\right )^{9/2}}{9 a c}+\frac {1}{8} x \left (c-a^2 c x^2\right )^{7/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a c}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle -c \left (\frac {11}{10} \left (\frac {7}{8} c \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )-\frac {\left (c-a^2 c x^2\right )^{9/2}}{9 a c}+\frac {1}{8} x \left (c-a^2 c x^2\right )^{7/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a c}\right )\)

\(\Big \downarrow \) 224

\(\displaystyle -c \left (\frac {11}{10} \left (\frac {7}{8} c \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )-\frac {\left (c-a^2 c x^2\right )^{9/2}}{9 a c}+\frac {1}{8} x \left (c-a^2 c x^2\right )^{7/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a c}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle -c \left (\frac {11}{10} \left (\frac {7}{8} c \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )-\frac {\left (c-a^2 c x^2\right )^{9/2}}{9 a c}+\frac {1}{8} x \left (c-a^2 c x^2\right )^{7/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a c}\right )\)

Input:

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

Output:

-(c*(-1/10*((1 + a*x)*(c - a^2*c*x^2)^(9/2))/(a*c) + (11*((x*(c - a^2*c*x^ 
2)^(7/2))/8 - (c - a^2*c*x^2)^(9/2)/(9*a*c) + (7*c*((x*(c - a^2*c*x^2)^(5/ 
2))/6 + (5*c*((x*(c - a^2*c*x^2)^(3/2))/4 + (3*c*((x*Sqrt[c - a^2*c*x^2])/ 
2 + (Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a)))/4))/6))/8) 
)/10))
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 469

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* 
((n + p)/(n + 2*p + 1))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr 
eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* 
p + 1, 0] && IntegerQ[2*p]
 

rule 6691

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> 
Simp[c^(n/2)   Int[(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c 
, d, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) && IGtQ[n/ 
2, 0]
 

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.82

method	result	size
risch	\(-\frac {\left (1152 a^{9} x^{9}+2560 a^{8} x^{8}-3024 a^{7} x^{7}-10240 x^{6} a^{6}+312 a^{5} x^{5}+15360 a^{4} x^{4}+6150 a^{3} x^{3}-10240 a^{2} x^{2}-8055 a x +2560\right ) \left (a^{2} x^{2}-1\right ) c^{5}}{11520 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {77 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{5}}{256 \sqrt {a^{2} c}}\)	\(138\)
default	\(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}{10}+\frac {9 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8}+\frac {7 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{8}\right )}{10}+\frac {\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {9}{2}}}{9}-2 a c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}{16 a^{2} c}+\frac {7 c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{12 a^{2} c}+\frac {5 c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{8}\right )}{a}\)	\(454\)

Input:

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

Output:

-1/11520*(1152*a^9*x^9+2560*a^8*x^8-3024*a^7*x^7-10240*a^6*x^6+312*a^5*x^5 
+15360*a^4*x^4+6150*a^3*x^3-10240*a^2*x^2-8055*a*x+2560)*(a^2*x^2-1)/a/(-c 
*(a^2*x^2-1))^(1/2)*c^5-77/256/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2* 
c*x^2+c)^(1/2))*c^5
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.95 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\left [\frac {3465 \, \sqrt {-c} c^{4} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (1152 \, a^{9} c^{4} x^{9} + 2560 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} - 10240 \, a^{6} c^{4} x^{6} + 312 \, a^{5} c^{4} x^{5} + 15360 \, a^{4} c^{4} x^{4} + 6150 \, a^{3} c^{4} x^{3} - 10240 \, a^{2} c^{4} x^{2} - 8055 \, a c^{4} x + 2560 \, c^{4}\right )} \sqrt {-a^{2} c x^{2} + c}}{23040 \, a}, \frac {3465 \, c^{\frac {9}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (1152 \, a^{9} c^{4} x^{9} + 2560 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} - 10240 \, a^{6} c^{4} x^{6} + 312 \, a^{5} c^{4} x^{5} + 15360 \, a^{4} c^{4} x^{4} + 6150 \, a^{3} c^{4} x^{3} - 10240 \, a^{2} c^{4} x^{2} - 8055 \, a c^{4} x + 2560 \, c^{4}\right )} \sqrt {-a^{2} c x^{2} + c}}{11520 \, a}\right ] \] Input:

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

Output:

[1/23040*(3465*sqrt(-c)*c^4*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqr 
t(-c)*x - c) + 2*(1152*a^9*c^4*x^9 + 2560*a^8*c^4*x^8 - 3024*a^7*c^4*x^7 - 
 10240*a^6*c^4*x^6 + 312*a^5*c^4*x^5 + 15360*a^4*c^4*x^4 + 6150*a^3*c^4*x^ 
3 - 10240*a^2*c^4*x^2 - 8055*a*c^4*x + 2560*c^4)*sqrt(-a^2*c*x^2 + c))/a, 
1/11520*(3465*c^(9/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - 
 c)) + (1152*a^9*c^4*x^9 + 2560*a^8*c^4*x^8 - 3024*a^7*c^4*x^7 - 10240*a^6 
*c^4*x^6 + 312*a^5*c^4*x^5 + 15360*a^4*c^4*x^4 + 6150*a^3*c^4*x^3 - 10240* 
a^2*c^4*x^2 - 8055*a*c^4*x + 2560*c^4)*sqrt(-a^2*c*x^2 + c))/a]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (155) = 310\).

Time = 3.43 (sec) , antiderivative size = 714, normalized size of antiderivative = 4.22 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise(((-2*c**4*Piecewise(((a**2*x**2/3 - 1/3)*sqrt(-a**2*c*x**2 + c), 
 Ne(c, 0)), (a**2*sqrt(c)*x**2/2, True)) + 6*c**4*Piecewise((sqrt(-a**2*c* 
x**2 + c)*(a**4*x**4/5 - a**2*x**2/15 - 2/15), Ne(c, 0)), (a**4*sqrt(c)*x* 
*4/4, True)) - 6*c**4*Piecewise((sqrt(-a**2*c*x**2 + c)*(a**6*x**6/7 - a** 
4*x**4/35 - 4*a**2*x**2/105 - 8/105), Ne(c, 0)), (a**6*sqrt(c)*x**6/6, Tru 
e)) + 2*c**4*Piecewise((sqrt(-a**2*c*x**2 + c)*(a**8*x**8/9 - a**6*x**6/63 
 - 2*a**4*x**4/105 - 8*a**2*x**2/315 - 16/315), Ne(c, 0)), (a**8*sqrt(c)*x 
**8/8, True)) + c**4*Piecewise((7*c*Piecewise((log(-2*a*c*x + 2*sqrt(-c)*s 
qrt(-a**2*c*x**2 + c))/sqrt(-c), Ne(c, 0)), (a*x*log(a*x)/sqrt(-a**2*c*x** 
2), True))/256 + sqrt(-a**2*c*x**2 + c)*(a**9*x**9/10 - a**7*x**7/80 - 7*a 
**5*x**5/480 - 7*a**3*x**3/384 - 7*a*x/256), Ne(c, 0)), (a**9*sqrt(c)*x**9 
/9, True)) - 2*c**4*Piecewise((5*c*Piecewise((log(-2*a*c*x + 2*sqrt(-c)*sq 
rt(-a**2*c*x**2 + c))/sqrt(-c), Ne(c, 0)), (a*x*log(a*x)/sqrt(-a**2*c*x**2 
), True))/128 + sqrt(-a**2*c*x**2 + c)*(a**7*x**7/8 - a**5*x**5/48 - 5*a** 
3*x**3/192 - 5*a*x/128), Ne(c, 0)), (a**7*sqrt(c)*x**7/7, True)) + 2*c**4* 
Piecewise((c*Piecewise((log(-2*a*c*x + 2*sqrt(-c)*sqrt(-a**2*c*x**2 + c))/ 
sqrt(-c), Ne(c, 0)), (a*x*log(a*x)/sqrt(-a**2*c*x**2), True))/8 + (a**3*x* 
*3/4 - a*x/8)*sqrt(-a**2*c*x**2 + c), Ne(c, 0)), (a**3*sqrt(c)*x**3/3, Tru 
e)) - c**4*Piecewise((a*x*sqrt(-a**2*c*x**2 + c)/2 + c*Piecewise((log(-2*a 
*c*x + 2*sqrt(-c)*sqrt(-a**2*c*x**2 + c))/sqrt(-c), Ne(c, 0)), (a*x*log...
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.14 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {1}{10} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} x - \frac {11}{80} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} c x - \frac {77}{480} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c^{2} x - \frac {77}{384} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{3} x - \frac {35}{64} \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{4} x + \frac {63}{256} \, \sqrt {-a^{2} c x^{2} + c} c^{4} x - \frac {35 \, c^{6} \arcsin \left (a x - 2\right )}{64 \, a \left (-c\right )^{\frac {3}{2}}} + \frac {63 \, c^{\frac {9}{2}} \arcsin \left (a x\right )}{256 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}{9 \, a} + \frac {35 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{4}}{32 \, a} \] Input:

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

Output:

1/10*(-a^2*c*x^2 + c)^(9/2)*x - 11/80*(-a^2*c*x^2 + c)^(7/2)*c*x - 77/480* 
(-a^2*c*x^2 + c)^(5/2)*c^2*x - 77/384*(-a^2*c*x^2 + c)^(3/2)*c^3*x - 35/64 
*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c^4*x + 63/256*sqrt(-a^2*c*x^2 + c)*c^4*x 
 - 35/64*c^6*arcsin(a*x - 2)/(a*(-c)^(3/2)) + 63/256*c^(9/2)*arcsin(a*x)/a 
 + 2/9*(-a^2*c*x^2 + c)^(9/2)/a + 35/32*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c^ 
4/a
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.97 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {77 \, c^{5} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{256 \, \sqrt {-c} {\left | a \right |}} + \frac {1}{11520} \, \sqrt {-a^{2} c x^{2} + c} {\left (\frac {2560 \, c^{4}}{a} - {\left (8055 \, c^{4} + 2 \, {\left (5120 \, a c^{4} - {\left (3075 \, a^{2} c^{4} + 4 \, {\left (1920 \, a^{3} c^{4} + {\left (39 \, a^{4} c^{4} - 2 \, {\left (640 \, a^{5} c^{4} + {\left (189 \, a^{6} c^{4} - 8 \, {\left (9 \, a^{8} c^{4} x + 20 \, a^{7} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \] Input:

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

Output:

77/256*c^5*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(sqrt(-c)*abs( 
a)) + 1/11520*sqrt(-a^2*c*x^2 + c)*(2560*c^4/a - (8055*c^4 + 2*(5120*a*c^4 
 - (3075*a^2*c^4 + 4*(1920*a^3*c^4 + (39*a^4*c^4 - 2*(640*a^5*c^4 + (189*a 
^6*c^4 - 8*(9*a^8*c^4*x + 20*a^7*c^4)*x)*x)*x)*x)*x)*x)*x)*x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.17 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {\sqrt {c}\, c^{4} \left (-3465 \mathit {asin} \left (a x \right )+1152 \sqrt {-a^{2} x^{2}+1}\, a^{9} x^{9}+2560 \sqrt {-a^{2} x^{2}+1}\, a^{8} x^{8}-3024 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}-10240 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+312 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+15360 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+6150 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-10240 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-8055 \sqrt {-a^{2} x^{2}+1}\, a x +2560 \sqrt {-a^{2} x^{2}+1}-2560\right )}{11520 a} \] Input:

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

Output:

(sqrt(c)*c**4*( - 3465*asin(a*x) + 1152*sqrt( - a**2*x**2 + 1)*a**9*x**9 + 
 2560*sqrt( - a**2*x**2 + 1)*a**8*x**8 - 3024*sqrt( - a**2*x**2 + 1)*a**7* 
x**7 - 10240*sqrt( - a**2*x**2 + 1)*a**6*x**6 + 312*sqrt( - a**2*x**2 + 1) 
*a**5*x**5 + 15360*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 6150*sqrt( - a**2*x* 
*2 + 1)*a**3*x**3 - 10240*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 8055*sqrt( - 
a**2*x**2 + 1)*a*x + 2560*sqrt( - a**2*x**2 + 1) - 2560))/(11520*a)