\(\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2)^{7/2} \, dx\) [606]

Optimal result

Integrand size = 24, antiderivative size = 146 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}-\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {2 \left (c-a^2 c x^2\right )^{7/2}}{7 a}+\frac {1}{8} x \left (c-a^2 c x^2\right )^{7/2}-\frac {45 c^{7/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a} \] Output:

-45/128*c^3*x*(-a^2*c*x^2+c)^(1/2)-15/64*c^2*x*(-a^2*c*x^2+c)^(3/2)-3/16*c 
*x*(-a^2*c*x^2+c)^(5/2)+2/7*(-a^2*c*x^2+c)^(7/2)/a+1/8*x*(-a^2*c*x^2+c)^(7 
/2)-45/128*c^(7/2)*arctan(a*c^(1/2)*x/(-a^2*c*x^2+c)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (256-837 a x-187 a^2 x^2+978 a^3 x^3+558 a^4 x^4-600 a^5 x^5-424 a^6 x^6+144 a^7 x^7+112 a^8 x^8\right )+630 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{896 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \] Input:

Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2),x]
 

Output:

(c^3*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(256 - 837*a*x - 187*a^2*x^2 + 978 
*a^3*x^3 + 558*a^4*x^4 - 600*a^5*x^5 - 424*a^6*x^6 + 144*a^7*x^7 + 112*a^8 
*x^8) + 630*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(896*a*Sqrt[1 - 
a*x]*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6717, 6691, 469, 455, 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.

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

\(\Big \downarrow \) 6717

\(\displaystyle -\int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2}dx\)

\(\Big \downarrow \) 6691

\(\displaystyle -c \int (a x+1)^2 \left (c-a^2 c x^2\right )^{5/2}dx\)

\(\Big \downarrow \) 469

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

Int[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2),x]
 

Output:

-(c*(-1/8*((1 + a*x)*(c - a^2*c*x^2)^(7/2))/(a*c) + (9*((x*(c - a^2*c*x^2) 
^(5/2))/6 - (c - a^2*c*x^2)^(7/2)/(7*a*c) + (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))
 

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 = 122, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {\left (112 a^{7} x^{7}+256 x^{6} a^{6}-168 a^{5} x^{5}-768 a^{4} x^{4}-210 a^{3} x^{3}+768 a^{2} x^{2}+581 a x -256\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{896 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {45 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{4}}{128 \sqrt {a^{2} c}}\)	\(122\)
default	\(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8}+\frac {7 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{8}+\frac {\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}{7}-2 a c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{12 a^{2} c}+\frac {5 c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{a}\)	\(375\)

Input:

int(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/896*(112*a^7*x^7+256*a^6*x^6-168*a^5*x^5-768*a^4*x^4-210*a^3*x^3+768*a^2 
*x^2+581*a*x-256)*(a^2*x^2-1)/a/(-c*(a^2*x^2-1))^(1/2)*c^4-45/128/(a^2*c)^ 
(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))*c^4
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.96 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\left [\frac {315 \, \sqrt {-c} c^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{1792 \, a}, \frac {315 \, c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{896 \, a}\right ] \] Input:

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")
 

Output:

[1/1792*(315*sqrt(-c)*c^3*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt( 
-c)*x - c) - 2*(112*a^7*c^3*x^7 + 256*a^6*c^3*x^6 - 168*a^5*c^3*x^5 - 768* 
a^4*c^3*x^4 - 210*a^3*c^3*x^3 + 768*a^2*c^3*x^2 + 581*a*c^3*x - 256*c^3)*s 
qrt(-a^2*c*x^2 + c))/a, 1/896*(315*c^(7/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*s 
qrt(c)*x/(a^2*c*x^2 - c)) - (112*a^7*c^3*x^7 + 256*a^6*c^3*x^6 - 168*a^5*c 
^3*x^5 - 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 + 768*a^2*c^3*x^2 + 581*a*c^3*x 
 - 256*c^3)*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. 614 vs. \(2 (133) = 266\).

Time = 3.09 (sec) , antiderivative size = 614, normalized size of antiderivative = 4.21 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx =\text {Too large to display} \] Input:

integrate(1/(a*x-1)*(a*x+1)*(-a**2*c*x**2+c)**(7/2),x)
 

Output:

Piecewise(((-2*c**3*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)) + 4*c**3*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)) - 2*c**3*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)) - c**3*Piecewise((5*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)) 
/128 + sqrt(-a**2*c*x**2 + c)*(a**7*x**7/8 - a**5*x**5/48 - 5*a**3*x**3/19 
2 - 5*a*x/128), Ne(c, 0)), (a**7*sqrt(c)*x**7/7, True)) + c**3*Piecewise(( 
c*Piecewise((log(-2*a*c*x + 2*sqrt(-c)*sqrt(-a**2*c*x**2 + c))/sqrt(-c), N 
e(c, 0)), (a*x*log(a*x)/sqrt(-a**2*c*x**2), True))/16 + sqrt(-a**2*c*x**2 
+ c)*(a**5*x**5/6 - a**3*x**3/24 - a*x/16), Ne(c, 0)), (a**5*sqrt(c)*x**5/ 
5, True)) + c**3*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), Tr 
ue))/8 + (a**3*x**3/4 - a*x/8)*sqrt(-a**2*c*x**2 + c), Ne(c, 0)), (a**3*sq 
rt(c)*x**3/3, True)) - c**3*Piecewise((a*x*sqrt(-a**2*c*x**2 + c)/2 + c*Pi 
ecewise((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))/2, Ne(c, 0)), (a*sqrt(c)*x, 
 True)))/a, Ne(a, 0)), (-c**(7/2)*x, True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.18 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {1}{8} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x - \frac {3}{16} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c x - \frac {15}{64} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2} x - \frac {5}{8} \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{3} x + \frac {35}{128} \, \sqrt {-a^{2} c x^{2} + c} c^{3} x - \frac {5 \, c^{5} \arcsin \left (a x - 2\right )}{8 \, a \left (-c\right )^{\frac {3}{2}}} + \frac {35 \, c^{\frac {7}{2}} \arcsin \left (a x\right )}{128 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{7 \, a} + \frac {5 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{3}}{4 \, a} \] Input:

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")
 

Output:

1/8*(-a^2*c*x^2 + c)^(7/2)*x - 3/16*(-a^2*c*x^2 + c)^(5/2)*c*x - 15/64*(-a 
^2*c*x^2 + c)^(3/2)*c^2*x - 5/8*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c^3*x + 35 
/128*sqrt(-a^2*c*x^2 + c)*c^3*x - 5/8*c^5*arcsin(a*x - 2)/(a*(-c)^(3/2)) + 
 35/128*c^(7/2)*arcsin(a*x)/a + 2/7*(-a^2*c*x^2 + c)^(7/2)/a + 5/4*sqrt(a^ 
2*c*x^2 - 4*a*c*x + 3*c)*c^3/a
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {45 \, c^{4} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{128 \, \sqrt {-c} {\left | a \right |}} + \frac {1}{896} \, \sqrt {-a^{2} c x^{2} + c} {\left (\frac {256 \, c^{3}}{a} - {\left (581 \, c^{3} + 2 \, {\left (384 \, a c^{3} - {\left (105 \, a^{2} c^{3} + 4 \, {\left (96 \, a^{3} c^{3} + {\left (21 \, a^{4} c^{3} - 2 \, {\left (7 \, a^{6} c^{3} x + 16 \, a^{5} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \] Input:

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")
 

Output:

45/128*c^4*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(sqrt(-c)*abs( 
a)) + 1/896*sqrt(-a^2*c*x^2 + c)*(256*c^3/a - (581*c^3 + 2*(384*a*c^3 - (1 
05*a^2*c^3 + 4*(96*a^3*c^3 + (21*a^4*c^3 - 2*(7*a^6*c^3*x + 16*a^5*c^3)*x) 
*x)*x)*x)*x)*x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \] Input:

int(((c - a^2*c*x^2)^(7/2)*(a*x + 1))/(a*x - 1),x)
 

Output:

int(((c - a^2*c*x^2)^(7/2)*(a*x + 1))/(a*x - 1), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.10 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} \left (-315 \mathit {asin} \left (a x \right )-112 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}-256 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+168 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+768 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+210 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-768 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-581 \sqrt {-a^{2} x^{2}+1}\, a x +256 \sqrt {-a^{2} x^{2}+1}-256\right )}{896 a} \] Input:

int(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(7/2),x)
 

Output:

(sqrt(c)*c**3*( - 315*asin(a*x) - 112*sqrt( - a**2*x**2 + 1)*a**7*x**7 - 2 
56*sqrt( - a**2*x**2 + 1)*a**6*x**6 + 168*sqrt( - a**2*x**2 + 1)*a**5*x**5 
 + 768*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 210*sqrt( - a**2*x**2 + 1)*a**3* 
x**3 - 768*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 581*sqrt( - a**2*x**2 + 1)*a 
*x + 256*sqrt( - a**2*x**2 + 1) - 256))/(896*a)