Integrand size = 24, antiderivative size = 153 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}-\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac {45 c^{7/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a} \]
-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 = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (256-837 a x-187 a^2 x^2+978 a^3 x^3+558 a^4 x^4-600 a^5 x^5-424 a^6 x^6+144 a^7 x^7+112 a^8 x^8\right )+630 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{896 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]
(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])
Time = 0.41 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.12, 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 |
\(\displaystyle -c \left (\frac {9}{8} \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )-\frac {\left (c-a^2 c x^2\right )^{7/2}}{7 a c}+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{7/2}}{8 a c}\right )\) |
-(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))
3.7.24.3.1 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[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]
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]
Time = 0.59 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.80
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 | \(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8}+\frac {7 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{8}+\frac {\frac {2 \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}{7}-2 a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{12 a^{2} c}+\frac {5 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{a}\) | \(375\) |
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 = 0.27 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.87 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\left [\frac {315 \, \sqrt {-c} c^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{1792 \, a}, \frac {315 \, c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{896 \, a}\right ] \]
[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]
Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (139) = 278\).
Time = 2.97 (sec) , antiderivative size = 614, normalized size of antiderivative = 4.01 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\begin {cases} \frac {- 2 c^{3} \left (\begin {cases} \left (\frac {a^{2} x^{2}}{3} - \frac {1}{3}\right ) \sqrt {- a^{2} c x^{2} + c} & \text {for}\: c \neq 0 \\\frac {a^{2} \sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + 4 c^{3} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {a^{4} x^{4}}{5} - \frac {a^{2} x^{2}}{15} - \frac {2}{15}\right ) & \text {for}\: c \neq 0 \\\frac {a^{4} \sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) - 2 c^{3} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {a^{6} x^{6}}{7} - \frac {a^{4} x^{4}}{35} - \frac {4 a^{2} x^{2}}{105} - \frac {8}{105}\right ) & \text {for}\: c \neq 0 \\\frac {a^{6} \sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) - c^{3} \left (\begin {cases} \frac {5 c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{128} + \sqrt {- a^{2} c x^{2} + c} \left (\frac {a^{7} x^{7}}{8} - \frac {a^{5} x^{5}}{48} - \frac {5 a^{3} x^{3}}{192} - \frac {5 a x}{128}\right ) & \text {for}\: c \neq 0 \\\frac {a^{7} \sqrt {c} x^{7}}{7} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{16} + \sqrt {- a^{2} c x^{2} + c} \left (\frac {a^{5} x^{5}}{6} - \frac {a^{3} x^{3}}{24} - \frac {a x}{16}\right ) & \text {for}\: c \neq 0 \\\frac {a^{5} \sqrt {c} x^{5}}{5} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{8} + \left (\frac {a^{3} x^{3}}{4} - \frac {a x}{8}\right ) \sqrt {- a^{2} c x^{2} + c} & \text {for}\: c \neq 0 \\\frac {a^{3} \sqrt {c} x^{3}}{3} & \text {otherwise} \end {cases}\right ) - c^{3} \left (\begin {cases} \frac {a x \sqrt {- a^{2} c x^{2} + c}}{2} + \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{2} & \text {for}\: c \neq 0 \\a \sqrt {c} x & \text {otherwise} \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\- c^{\frac {7}{2}} x & \text {otherwise} \end {cases} \]
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))
Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.13 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {1}{8} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x - \frac {3}{16} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c x - \frac {15}{64} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2} x - \frac {5}{8} \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{3} x + \frac {35}{128} \, \sqrt {-a^{2} c x^{2} + c} c^{3} x - \frac {5 \, c^{5} \arcsin \left (a x - 2\right )}{8 \, a \left (-c\right )^{\frac {3}{2}}} + \frac {35 \, c^{\frac {7}{2}} \arcsin \left (a x\right )}{128 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{7 \, a} + \frac {5 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{3}}{4 \, a} \]
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
Time = 0.32 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.92 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {45 \, c^{4} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{128 \, \sqrt {-c} {\left | a \right |}} + \frac {1}{896} \, \sqrt {-a^{2} c x^{2} + c} {\left (\frac {256 \, c^{3}}{a} - {\left (581 \, c^{3} + 2 \, {\left (384 \, a c^{3} - {\left (105 \, a^{2} c^{3} + 4 \, {\left (96 \, a^{3} c^{3} + {\left (21 \, a^{4} c^{3} - 2 \, {\left (7 \, a^{6} c^{3} x + 16 \, a^{5} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
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)
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]