\(\int e^{-2 \text {arctanh}(a x)} (c-a^2 c x^2)^{5/2} \, dx\) [1283]

Optimal result

Integrand size = 24, antiderivative size = 135 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {7}{16} c^2 x \sqrt {c-a^2 c x^2}+\frac {7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac {7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac {\left (c-a^2 c x^2\right )^{7/2}}{6 a c (1+a x)}+\frac {7 c^{5/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{16 a} \] Output:

7/16*c^2*x*(-a^2*c*x^2+c)^(1/2)+7/24*c*x*(-a^2*c*x^2+c)^(3/2)+7/30*(-a^2*c 
*x^2+c)^(5/2)/a+1/6*(-a^2*c*x^2+c)^(7/2)/a/c/(a*x+1)+7/16*c^(5/2)*arctan(a 
*c^(1/2)*x/(-a^2*c*x^2+c)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {c^2 \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (96+39 a x-327 a^2 x^2+202 a^3 x^3+86 a^4 x^4-136 a^5 x^5+40 a^6 x^6\right )-210 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{240 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \] Input:

Integrate[(c - a^2*c*x^2)^(5/2)/E^(2*ArcTanh[a*x]),x]
 

Output:

(c^2*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(96 + 39*a*x - 327*a^2*x^2 + 202*a 
^3*x^3 + 86*a^4*x^4 - 136*a^5*x^5 + 40*a^6*x^6) - 210*Sqrt[1 - a*x]*ArcSin 
[Sqrt[1 - a*x]/Sqrt[2]]))/(240*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6692, 469, 455, 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.

\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx\)

\(\Big \downarrow \) 6692

\(\displaystyle c \int (1-a x)^2 \left (c-a^2 c x^2\right )^{3/2}dx\)

\(\Big \downarrow \) 469

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

Int[(c - a^2*c*x^2)^(5/2)/E^(2*ArcTanh[a*x]),x]
 

Output:

c*(((1 - a*x)*(c - a^2*c*x^2)^(5/2))/(6*a*c) + (7*((x*(c - a^2*c*x^2)^(3/2 
))/4 + (c - a^2*c*x^2)^(5/2)/(5*a*c) + (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)
 

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 6692

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> 
Simp[1/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]) && ILtQ[ 
n/2, 0]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {\left (40 a^{5} x^{5}-96 a^{4} x^{4}-10 a^{3} x^{3}+192 a^{2} x^{2}-135 a x -96\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{240 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {7 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{3}}{16 \sqrt {a^{2} c}}\)	\(106\)
default	\(-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}-\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}+\frac {\frac {2 \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{5}+2 a c \left (-\frac {\left (-2 \left (x +\frac {1}{a}\right ) a^{2} c +2 a c \right ) \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 \left (x +\frac {1}{a}\right ) a^{2} c +2 a c \right ) \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{a}\)	\(275\)

Input:

int((-a^2*c*x^2+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)
 

Output:

1/240*(40*a^5*x^5-96*a^4*x^4-10*a^3*x^3+192*a^2*x^2-135*a*x-96)*(a^2*x^2-1 
)/a/(-c*(a^2*x^2-1))^(1/2)*c^3+7/16/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/( 
-a^2*c*x^2+c)^(1/2))*c^3
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.79 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\left [\frac {105 \, \sqrt {-c} c^{2} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (40 \, a^{5} c^{2} x^{5} - 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} + 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x - 96 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{480 \, a}, -\frac {105 \, c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (40 \, a^{5} c^{2} x^{5} - 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} + 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x - 96 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{240 \, a}\right ] \] Input:

integrate((-a^2*c*x^2+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas 
")
 

Output:

[1/480*(105*sqrt(-c)*c^2*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(- 
c)*x - c) - 2*(40*a^5*c^2*x^5 - 96*a^4*c^2*x^4 - 10*a^3*c^2*x^3 + 192*a^2* 
c^2*x^2 - 135*a*c^2*x - 96*c^2)*sqrt(-a^2*c*x^2 + c))/a, -1/240*(105*c^(5/ 
2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (40*a^5*c^2* 
x^5 - 96*a^4*c^2*x^4 - 10*a^3*c^2*x^3 + 192*a^2*c^2*x^2 - 135*a*c^2*x - 96 
*c^2)*sqrt(-a^2*c*x^2 + c))/a]
 

Sympy [A] (verification not implemented)

Time = 2.97 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.41 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=- a^{4} c^{2} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{5}}{6} - \frac {x^{3}}{24 a^{2}} - \frac {x}{16 a^{4}}\right ) + \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{16 a^{4}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{5}}{5} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{2} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{4}}{5} - \frac {x^{2}}{15 a^{2}} - \frac {2}{15 a^{4}}\right ) & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) - 2 a c^{2} \left (\begin {cases} \left (\frac {x^{2}}{3} - \frac {1}{3 a^{2}}\right ) \sqrt {- a^{2} c x^{2} + c} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {- a^{2} c x^{2} + c}}{2} & \text {for}\: a^{2} c \neq 0 \\\sqrt {c} x & \text {otherwise} \end {cases}\right ) \] Input:

integrate((-a**2*c*x**2+c)**(5/2)/(a*x+1)**2*(-a**2*x**2+1),x)
 

Output:

-a**4*c**2*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)) + 2*a**3*c**2*Piecewise( 
(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)) - 2*a*c**2*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**2*Pi 
ecewise((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 = 0.11 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.14 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=-\frac {1}{6} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x + \frac {7}{24} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c x + \frac {3}{4} \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c^{2} x - \frac {5}{16} \, \sqrt {-a^{2} c x^{2} + c} c^{2} x - \frac {3 \, c^{4} \arcsin \left (a x + 2\right )}{4 \, a \left (-c\right )^{\frac {3}{2}}} - \frac {5 \, c^{\frac {5}{2}} \arcsin \left (a x\right )}{16 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{5 \, a} + \frac {3 \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c^{2}}{2 \, a} \] Input:

integrate((-a^2*c*x^2+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima 
")
 

Output:

-1/6*(-a^2*c*x^2 + c)^(5/2)*x + 7/24*(-a^2*c*x^2 + c)^(3/2)*c*x + 3/4*sqrt 
(a^2*c*x^2 + 4*a*c*x + 3*c)*c^2*x - 5/16*sqrt(-a^2*c*x^2 + c)*c^2*x - 3/4* 
c^4*arcsin(a*x + 2)/(a*(-c)^(3/2)) - 5/16*c^(5/2)*arcsin(a*x)/a + 2/5*(-a^ 
2*c*x^2 + c)^(5/2)/a + 3/2*sqrt(a^2*c*x^2 + 4*a*c*x + 3*c)*c^2/a
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (111) = 222\).

Time = 0.17 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.37 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=-\frac {{\left (6720 \, a^{7} c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - \frac {{\left (105 \, a^{7} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{5} c^{3} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - 595 \, a^{7} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{4} c^{4} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - 1686 \, a^{7} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c^{5} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 1386 \, a^{7} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{6} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 105 \, a^{7} c^{8} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 595 \, a^{7} c^{7} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )\right )} {\left (a x + 1\right )}^{6}}{c^{6}}\right )} {\left | a \right |}}{7680 \, a^{9}} \] Input:

integrate((-a^2*c*x^2+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")
 

Output:

-1/7680*(6720*a^7*c^(5/2)*arctan(sqrt(-c + 2*c/(a*x + 1))/sqrt(c))*sgn(1/( 
a*x + 1))*sgn(a) - (105*a^7*(c - 2*c/(a*x + 1))^5*c^3*sqrt(-c + 2*c/(a*x + 
 1))*sgn(1/(a*x + 1))*sgn(a) - 595*a^7*(c - 2*c/(a*x + 1))^4*c^4*sqrt(-c + 
 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) - 1686*a^7*(c - 2*c/(a*x + 1))^3*c 
^5*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 1386*a^7*(c - 2*c/(a 
*x + 1))^2*c^6*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 105*a^7* 
c^8*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 595*a^7*c^7*(-c + 2 
*c/(a*x + 1))^(3/2)*sgn(1/(a*x + 1))*sgn(a))*(a*x + 1)^6/c^6)*abs(a)/a^9
 

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=-\int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \] Input:

int(-((c - a^2*c*x^2)^(5/2)*(a^2*x^2 - 1))/(a*x + 1)^2,x)
 

Output:

-int(((c - a^2*c*x^2)^(5/2)*(a^2*x^2 - 1))/(a*x + 1)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.90 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {\sqrt {c}\, c^{2} \left (105 \mathit {asin} \left (a x \right )-40 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+96 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+10 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-192 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+135 \sqrt {-a^{2} x^{2}+1}\, a x +96 \sqrt {-a^{2} x^{2}+1}-96\right )}{240 a} \] Input:

int((-a^2*c*x^2+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x)
 

Output:

(sqrt(c)*c**2*(105*asin(a*x) - 40*sqrt( - a**2*x**2 + 1)*a**5*x**5 + 96*sq 
rt( - a**2*x**2 + 1)*a**4*x**4 + 10*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 192 
*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 135*sqrt( - a**2*x**2 + 1)*a*x + 96*sq 
rt( - a**2*x**2 + 1) - 96))/(240*a)