\(\int e^{2 \text {arctanh}(a x)} (c-a^2 c x^2)^{7/2} \, dx\) [1144]

Optimal result

Integrand size = 24, antiderivative size = 146 \[ \int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}-\frac {2 \left (c-a^2 c x^2\right )^{7/2}}{7 a}-\frac {1}{8} x \left (c-a^2 c x^2\right )^{7/2}+\frac {45 c^{7/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a} \] 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 = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03 \[ \int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {c^3 \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (256-837 a x-187 a^2 x^2+978 a^3 x^3+558 a^4 x^4-600 a^5 x^5-424 a^6 x^6+144 a^7 x^7+112 a^8 x^8\right )+630 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{896 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \] Input:

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

Output:

-1/896*(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]]))/(a*Sqrt[1 
 - a*x]*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.

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

\(\Big \downarrow \) 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*ArcTanh[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]
 

Maple [A] (verified)

Time = 0.16 (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	\(-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8}-\frac {7 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{8}-\frac {2 \left (\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}{7}-a c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{12 a^{2} c}+\frac {5 c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )\right )}{a}\)	\(375\)

Input:

int((a*x+1)^2/(-a^2*x^2+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 \[ \int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\left [\frac {315 \, \sqrt {-c} c^{3} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{1792 \, a}, -\frac {315 \, c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{896 \, a}\right ] \] Input:

integrate((a*x+1)^2/(-a^2*x^2+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* 
sqrt(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 [A] (verification not implemented)

Time = 5.00 (sec) , antiderivative size = 648, normalized size of antiderivative = 4.44 \[ \int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx =\text {Too large to display} \] Input:

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

Output:

a**6*c**3*Piecewise((sqrt(-a**2*c*x**2 + c)*(x**7/8 - x**5/(48*a**2) - 5*x 
**3/(192*a**4) - 5*x/(128*a**6)) + 5*c*Piecewise((log(-2*a**2*c*x + 2*sqrt 
(-a**2*c)*sqrt(-a**2*c*x**2 + c))/sqrt(-a**2*c), Ne(c, 0)), (x*log(x)/sqrt 
(-a**2*c*x**2), True))/(128*a**6), Ne(a**2*c, 0)), (sqrt(c)*x**7/7, True)) 
 + 2*a**5*c**3*Piecewise((sqrt(-a**2*c*x**2 + c)*(x**6/7 - x**4/(35*a**2) 
- 4*x**2/(105*a**4) - 8/(105*a**6)), Ne(a**2*c, 0)), (sqrt(c)*x**6/6, True 
)) - a**4*c**3*Piecewise((sqrt(-a**2*c*x**2 + c)*(x**5/6 - x**3/(24*a**2) 
- x/(16*a**4)) + c*Piecewise((log(-2*a**2*c*x + 2*sqrt(-a**2*c)*sqrt(-a**2 
*c*x**2 + c))/sqrt(-a**2*c), Ne(c, 0)), (x*log(x)/sqrt(-a**2*c*x**2), True 
))/(16*a**4), Ne(a**2*c, 0)), (sqrt(c)*x**5/5, True)) - 4*a**3*c**3*Piecew 
ise((sqrt(-a**2*c*x**2 + c)*(x**4/5 - x**2/(15*a**2) - 2/(15*a**4)), Ne(a* 
*2*c, 0)), (sqrt(c)*x**4/4, True)) - a**2*c**3*Piecewise(((x**3/4 - x/(8*a 
**2))*sqrt(-a**2*c*x**2 + c) + c*Piecewise((log(-2*a**2*c*x + 2*sqrt(-a**2 
*c)*sqrt(-a**2*c*x**2 + c))/sqrt(-a**2*c), Ne(c, 0)), (x*log(x)/sqrt(-a**2 
*c*x**2), True))/(8*a**2), Ne(a**2*c, 0)), (sqrt(c)*x**3/3, True)) + 2*a*c 
**3*Piecewise(((x**2/3 - 1/(3*a**2))*sqrt(-a**2*c*x**2 + c), Ne(a**2*c, 0) 
), (sqrt(c)*x**2/2, True)) + c**3*Piecewise((c*Piecewise((log(-2*a**2*c*x 
+ 2*sqrt(-a**2*c)*sqrt(-a**2*c*x**2 + c))/sqrt(-a**2*c), Ne(c, 0)), (x*log 
(x)/sqrt(-a**2*c*x**2), True))/2 + x*sqrt(-a**2*c*x**2 + c)/2, Ne(a**2*c, 
0)), (sqrt(c)*x, True))
 

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.33 \[ \int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {1}{896} \, {\left (\frac {112 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x}{a} - \frac {168 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c x}{a} - \frac {210 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2} x}{a} - \frac {560 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{3} x}{a} + \frac {245 \, \sqrt {-a^{2} c x^{2} + c} c^{3} x}{a} + \frac {245 \, c^{\frac {7}{2}} \arcsin \left (a x\right )}{a^{2}} + \frac {256 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{a^{2}} + \frac {1120 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{3}}{a^{2}} - \frac {560 \, c^{5} \arcsin \left (a x - 2\right )}{a^{5} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} a \] Input:

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

Output:

-1/896*(112*(-a^2*c*x^2 + c)^(7/2)*x/a - 168*(-a^2*c*x^2 + c)^(5/2)*c*x/a 
- 210*(-a^2*c*x^2 + c)^(3/2)*c^2*x/a - 560*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c) 
*c^3*x/a + 245*sqrt(-a^2*c*x^2 + c)*c^3*x/a + 245*c^(7/2)*arcsin(a*x)/a^2 
+ 256*(-a^2*c*x^2 + c)^(7/2)/a^2 + 1120*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c^ 
3/a^2 - 560*c^5*arcsin(a*x - 2)/(a^5*(-c/a^2)^(3/2)))*a
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97 \[ \int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {45 \, c^{4} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{128 \, \sqrt {-c} {\left | a \right |}} - \frac {1}{896} \, \sqrt {-a^{2} c x^{2} + c} {\left (\frac {256 \, c^{3}}{a} - {\left (581 \, c^{3} + 2 \, {\left (384 \, a c^{3} - {\left (105 \, a^{2} c^{3} + 4 \, {\left (96 \, a^{3} c^{3} + {\left (21 \, a^{4} c^{3} - 2 \, {\left (7 \, a^{6} c^{3} x + 16 \, a^{5} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \] Input:

integrate((a*x+1)^2/(-a^2*x^2+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 - ( 
105*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. \[ \int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int -\frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

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

int((a*x+1)^2/(-a^2*x^2+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 + 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))/(896*a)