Integrand size = 24, antiderivative size = 154 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {45 c^{7/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a} \]
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)+9/56*(-a^2* c*x^2+c)^(7/2)/a+1/8*(-a*x+1)*(-a^2*c*x^2+c)^(7/2)/a+45/128*c^(7/2)*arctan (a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))/a+45/128*c^3*x*(-a^2*c*x^2+c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.98 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {c^3 \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (-256-325 a x+1349 a^2 x^2-558 a^3 x^3-978 a^4 x^4+936 a^5 x^5+88 a^6 x^6-368 a^7 x^7+112 a^8 x^8\right )+630 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{896 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]
-1/896*(c^3*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(-256 - 325*a*x + 1349*a^2* x^2 - 558*a^3*x^3 - 978*a^4*x^4 + 936*a^5*x^5 + 88*a^6*x^6 - 368*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])
Time = 0.35 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6692, 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.
| \(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 6692 |
| \(\displaystyle c \int (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}dx\) |
| \(\Big \downarrow \) 469 |
| \(\displaystyle c \left (\frac {9}{8} \int (1-a x) \left (c-a^2 c x^2\right )^{5/2}dx+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a c}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c \left (\frac {9}{8} \left (\int \left (c-a^2 c x^2\right )^{5/2}dx+\frac {\left (c-a^2 c x^2\right )^{7/2}}{7 a c}\right )+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c \left (\frac {9}{8} \left (\frac {5}{6} c \int \left (c-a^2 c x^2\right )^{3/2}dx+\frac {\left (c-a^2 c x^2\right )^{7/2}}{7 a c}+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c \left (\frac {9}{8} \left (\frac {5}{6} c \left (\frac {3}{4} c \int \sqrt {c-a^2 c x^2}dx+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {\left (c-a^2 c x^2\right )^{7/2}}{7 a c}+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c \left (\frac {9}{8} \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {\left (c-a^2 c x^2\right )^{7/2}}{7 a c}+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a c}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {9}{8} \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {\left (c-a^2 c x^2\right )^{7/2}}{7 a c}+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a c}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle c \left (\frac {9}{8} \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {\left (c-a^2 c x^2\right )^{7/2}}{7 a c}+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a c}\right )\) |
c*(((1 - a*x)*(c - a^2*c*x^2)^(7/2))/(8*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)
3.13.47.3.1 Defintions of rubi rules used
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])
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])
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]
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]
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]
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]
Time = 0.35 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.79
| method | result | size |
| risch | \(-\frac {\left (112 a^{7} x^{7}-256 a^{6} x^{6}-168 a^{5} x^{5}+768 a^{4} x^{4}-210 a^{3} x^{3}-768 a^{2} x^{2}+581 a x +256\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{896 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {45 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{4}}{128 \sqrt {a^{2} c}}\) | \(122\) |
| default | \(-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8}-\frac {7 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{8}+\frac {\frac {2 \left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}{7}+2 a c \left (-\frac {\left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right ) \left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{12 a^{2} c}+\frac {5 c \left (-\frac {\left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right ) \left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right ) \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{a}\) | \(348\) |
-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
Time = 0.26 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.86 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\left [\frac {315 \, \sqrt {-c} c^{3} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (112 \, a^{7} c^{3} x^{7} - 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} + 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} - 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x + 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{1792 \, a}, -\frac {315 \, c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (112 \, a^{7} c^{3} x^{7} - 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} + 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} - 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x + 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{896 \, a}\right ] \]
[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]
Time = 3.68 (sec) , antiderivative size = 648, normalized size of antiderivative = 4.21 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=a^{6} c^{3} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{7}}{8} - \frac {x^{5}}{48 a^{2}} - \frac {5 x^{3}}{192 a^{4}} - \frac {5 x}{128 a^{6}}\right ) + \frac {5 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 )}{128 a^{6}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{7}}{7} & \text {otherwise} \end {cases}\right ) - 2 a^{5} c^{3} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{6}}{7} - \frac {x^{4}}{35 a^{2}} - \frac {4 x^{2}}{105 a^{4}} - \frac {8}{105 a^{6}}\right ) & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) - a^{4} c^{3} \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 ) + 4 a^{3} c^{3} \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 ) - a^{2} c^{3} \left (\begin {cases} \left (\frac {x^{3}}{4} - \frac {x}{8 a^{2}}\right ) \sqrt {- a^{2} c x^{2} + c} + \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 )}{8 a^{2}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{3}}{3} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \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^{3} \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 ) \]
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))
Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.12 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {1}{8} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x + \frac {3}{16} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c x + \frac {15}{64} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2} x + \frac {5}{8} \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c^{3} x - \frac {35}{128} \, \sqrt {-a^{2} c x^{2} + c} c^{3} x - \frac {5 \, c^{5} \arcsin \left (a x + 2\right )}{8 \, a \left (-c\right )^{\frac {3}{2}}} - \frac {35 \, c^{\frac {7}{2}} \arcsin \left (a x\right )}{128 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{7 \, a} + \frac {5 \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c^{3}}{4 \, a} \]
-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 - 3 5/128*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
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (125) = 250\).
Time = 0.37 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.70 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {{\left (80640 \, a^{9} c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - \frac {{\left (315 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{7} c^{4} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - 2415 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{6} c^{5} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 8043 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{5} c^{6} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 17609 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{4} c^{7} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - 15159 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c^{8} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 8043 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{9} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 315 \, a^{9} c^{11} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 2415 \, a^{9} c^{10} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )\right )} {\left (a x + 1\right )}^{8}}{c^{8}}\right )} {\left | a \right |}}{114688 \, a^{11}} \]
-1/114688*(80640*a^9*c^(7/2)*arctan(sqrt(-c + 2*c/(a*x + 1))/sqrt(c))*sgn( 1/(a*x + 1))*sgn(a) - (315*a^9*(c - 2*c/(a*x + 1))^7*c^4*sqrt(-c + 2*c/(a* x + 1))*sgn(1/(a*x + 1))*sgn(a) - 2415*a^9*(c - 2*c/(a*x + 1))^6*c^5*sqrt( -c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 8043*a^9*(c - 2*c/(a*x + 1)) ^5*c^6*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 17609*a^9*(c - 2 *c/(a*x + 1))^4*c^7*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) - 151 59*a^9*(c - 2*c/(a*x + 1))^3*c^8*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1)) *sgn(a) + 8043*a^9*(c - 2*c/(a*x + 1))^2*c^9*sqrt(-c + 2*c/(a*x + 1))*sgn( 1/(a*x + 1))*sgn(a) + 315*a^9*c^11*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1 ))*sgn(a) + 2415*a^9*c^10*(-c + 2*c/(a*x + 1))^(3/2)*sgn(1/(a*x + 1))*sgn( a))*(a*x + 1)^8/c^8)*abs(a)/a^11
Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\int \frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]