Integrand size = 22, antiderivative size = 130 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}-\frac {23 c^3 \left (1-a^2 x^2\right )^{5/2}}{35 a}-\frac {1}{2} c^3 x \left (1-a^2 x^2\right )^{5/2}-\frac {1}{7} a c^3 x^2 \left (1-a^2 x^2\right )^{5/2}+\frac {9 c^3 \arcsin (a x)}{16 a} \] Output:
9/16*c^3*x*(-a^2*x^2+1)^(1/2)+3/8*c^3*x*(-a^2*x^2+1)^(3/2)-23/35*c^3*(-a^2 *x^2+1)^(5/2)/a-1/2*c^3*x*(-a^2*x^2+1)^(5/2)-1/7*a*c^3*x^2*(-a^2*x^2+1)^(5 /2)+9/16*c^3*arcsin(a*x)/a
Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.70 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {c^3 \left (\sqrt {1-a^2 x^2} \left (368-245 a x-656 a^2 x^2-350 a^3 x^3+208 a^4 x^4+280 a^5 x^5+80 a^6 x^6\right )+630 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{560 a} \] Input:
Integrate[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^3,x]
Output:
-1/560*(c^3*(Sqrt[1 - a^2*x^2]*(368 - 245*a*x - 656*a^2*x^2 - 350*a^3*x^3 + 208*a^4*x^4 + 280*a^5*x^5 + 80*a^6*x^6) + 630*ArcSin[Sqrt[1 - a*x]/Sqrt[ 2]]))/a
Time = 0.51 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6688, 469, 469, 455, 211, 211, 223}
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.
| \(\displaystyle \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 6688 |
| \(\displaystyle c^3 \int (a x+1)^3 \left (1-a^2 x^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 469 |
| \(\displaystyle c^3 \left (\frac {9}{7} \int (a x+1)^2 \left (1-a^2 x^2\right )^{3/2}dx-\frac {(a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\) |
| \(\Big \downarrow \) 469 |
| \(\displaystyle c^3 \left (\frac {9}{7} \left (\frac {7}{6} \int (a x+1) \left (1-a^2 x^2\right )^{3/2}dx-\frac {(a x+1) \left (1-a^2 x^2\right )^{5/2}}{6 a}\right )-\frac {(a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c^3 \left (\frac {9}{7} \left (\frac {7}{6} \left (\int \left (1-a^2 x^2\right )^{3/2}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}\right )-\frac {(a x+1) \left (1-a^2 x^2\right )^{5/2}}{6 a}\right )-\frac {(a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c^3 \left (\frac {9}{7} \left (\frac {7}{6} \left (\frac {3}{4} \int \sqrt {1-a^2 x^2}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )-\frac {(a x+1) \left (1-a^2 x^2\right )^{5/2}}{6 a}\right )-\frac {(a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c^3 \left (\frac {9}{7} \left (\frac {7}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )-\frac {(a x+1) \left (1-a^2 x^2\right )^{5/2}}{6 a}\right )-\frac {(a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle c^3 \left (\frac {9}{7} \left (\frac {7}{6} \left (\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2}+\frac {\arcsin (a x)}{2 a}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )-\frac {(a x+1) \left (1-a^2 x^2\right )^{5/2}}{6 a}\right )-\frac {(a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\) |
Input:
Int[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^3,x]
Output:
c^3*(-1/7*((1 + a*x)^2*(1 - a^2*x^2)^(5/2))/a + (9*(-1/6*((1 + a*x)*(1 - a ^2*x^2)^(5/2))/a + (7*((x*(1 - a^2*x^2)^(3/2))/4 - (1 - a^2*x^2)^(5/2)/(5* a) + (3*((x*Sqrt[1 - a^2*x^2])/2 + ArcSin[a*x]/(2*a)))/4))/6))/7)
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
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^p Int[(1 - a^2*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] && IGtQ[(n + 1)/2, 0] && !I ntegerQ[p - n/2]
Time = 0.36 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82
| method | result | size |
| risch | \(\frac {\left (80 x^{6} a^{6}+280 a^{5} x^{5}+208 a^{4} x^{4}-350 a^{3} x^{3}-656 a^{2} x^{2}-245 a x +368\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{560 a \sqrt {-a^{2} x^{2}+1}}+\frac {9 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{3}}{16 \sqrt {a^{2}}}\) | \(107\) |
| meijerg | \(-\frac {8 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{56 a^{6} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {6 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{3} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {c^{3} \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-10 a^{8} x^{8}-16 x^{6} a^{6}-32 a^{4} x^{4}-128 a^{2} x^{2}+256\right )}{70 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {6 c^{3} \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {8 c^{3} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {3 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {9}{2}} \left (-24 x^{6} a^{6}-42 a^{4} x^{4}-105 a^{2} x^{2}+315\right )}{144 a^{8} \sqrt {-a^{2} x^{2}+1}}-\frac {35 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {9}{2}} \arcsin \left (a x \right )}{16 a^{9}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c^{3} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\) | \(457\) |
| default | \(-c^{3} \left (a^{9} \left (-\frac {x^{8}}{7 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {8 x^{6}}{35 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8 \left (-\frac {2 x^{4}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {6 \left (-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}\right )}{5 a^{2}}\right )}{7 a^{2}}}{a^{2}}\right )-\frac {x}{\sqrt {-a^{2} x^{2}+1}}-\frac {3}{a \sqrt {-a^{2} x^{2}+1}}+8 a^{3} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+6 a^{4} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-6 a^{5} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )-8 a^{6} \left (-\frac {x^{5}}{4 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}}{a^{2}}\right )+3 a^{8} \left (-\frac {x^{7}}{6 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {7 x^{5}}{24 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {7 \left (-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}\right )}{6 a^{2}}}{a^{2}}\right )\right )\) | \(586\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
1/560*(80*a^6*x^6+280*a^5*x^5+208*a^4*x^4-350*a^3*x^3-656*a^2*x^2-245*a*x+ 368)*(a^2*x^2-1)/a/(-a^2*x^2+1)^(1/2)*c^3+9/16/(a^2)^(1/2)*arctan((a^2)^(1 /2)*x/(-a^2*x^2+1)^(1/2))*c^3
Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {630 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (80 \, a^{6} c^{3} x^{6} + 280 \, a^{5} c^{3} x^{5} + 208 \, a^{4} c^{3} x^{4} - 350 \, a^{3} c^{3} x^{3} - 656 \, a^{2} c^{3} x^{2} - 245 \, a c^{3} x + 368 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{560 \, a} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^3,x, algorithm="fric as")
Output:
-1/560*(630*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (80*a^6*c^3*x^6 + 280*a^5*c^3*x^5 + 208*a^4*c^3*x^4 - 350*a^3*c^3*x^3 - 656*a^2*c^3*x^2 - 2 45*a*c^3*x + 368*c^3)*sqrt(-a^2*x^2 + 1))/a
Time = 7.61 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.75 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\begin {cases} \frac {9 c^{3} \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{16 \sqrt {- a^{2}}} + \sqrt {- a^{2} x^{2} + 1} \left (- \frac {a^{5} c^{3} x^{6}}{7} - \frac {a^{4} c^{3} x^{5}}{2} - \frac {13 a^{3} c^{3} x^{4}}{35} + \frac {5 a^{2} c^{3} x^{3}}{8} + \frac {41 a c^{3} x^{2}}{35} + \frac {7 c^{3} x}{16} - \frac {23 c^{3}}{35 a}\right ) & \text {for}\: a^{2} \neq 0 \\\frac {a^{7} c^{3} x^{8}}{8} + \frac {3 a^{6} c^{3} x^{7}}{7} + \frac {a^{5} c^{3} x^{6}}{6} - a^{4} c^{3} x^{5} - \frac {5 a^{3} c^{3} x^{4}}{4} + \frac {a^{2} c^{3} x^{3}}{3} + \frac {3 a c^{3} x^{2}}{2} + c^{3} x & \text {otherwise} \end {cases} \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**3,x)
Output:
Piecewise((9*c**3*log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(16* sqrt(-a**2)) + sqrt(-a**2*x**2 + 1)*(-a**5*c**3*x**6/7 - a**4*c**3*x**5/2 - 13*a**3*c**3*x**4/35 + 5*a**2*c**3*x**3/8 + 41*a*c**3*x**2/35 + 7*c**3*x /16 - 23*c**3/(35*a)), Ne(a**2, 0)), (a**7*c**3*x**8/8 + 3*a**6*c**3*x**7/ 7 + a**5*c**3*x**6/6 - a**4*c**3*x**5 - 5*a**3*c**3*x**4/4 + a**2*c**3*x** 3/3 + 3*a*c**3*x**2/2 + c**3*x, True))
Time = 0.12 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.62 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {a^{7} c^{3} x^{8}}{7 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {a^{6} c^{3} x^{7}}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {8 \, a^{5} c^{3} x^{6}}{35 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {9 \, a^{4} c^{3} x^{5}}{8 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {54 \, a^{3} c^{3} x^{4}}{35 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c^{3} x^{3}}{16 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {64 \, a c^{3} x^{2}}{35 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {7 \, c^{3} x}{16 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {9 \, c^{3} \arcsin \left (a x\right )}{16 \, a} - \frac {23 \, c^{3}}{35 \, \sqrt {-a^{2} x^{2} + 1} a} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^3,x, algorithm="maxi ma")
Output:
1/7*a^7*c^3*x^8/sqrt(-a^2*x^2 + 1) + 1/2*a^6*c^3*x^7/sqrt(-a^2*x^2 + 1) + 8/35*a^5*c^3*x^6/sqrt(-a^2*x^2 + 1) - 9/8*a^4*c^3*x^5/sqrt(-a^2*x^2 + 1) - 54/35*a^3*c^3*x^4/sqrt(-a^2*x^2 + 1) + 3/16*a^2*c^3*x^3/sqrt(-a^2*x^2 + 1 ) + 64/35*a*c^3*x^2/sqrt(-a^2*x^2 + 1) + 7/16*c^3*x/sqrt(-a^2*x^2 + 1) + 9 /16*c^3*arcsin(a*x)/a - 23/35*c^3/(sqrt(-a^2*x^2 + 1)*a)
Time = 0.17 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.78 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {9 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, {\left | a \right |}} - \frac {1}{560} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {368 \, c^{3}}{a} - {\left (245 \, c^{3} + 2 \, {\left (328 \, a c^{3} + {\left (175 \, a^{2} c^{3} - 4 \, {\left (26 \, a^{3} c^{3} + 5 \, {\left (2 \, a^{5} c^{3} x + 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^3,x, algorithm="giac ")
Output:
9/16*c^3*arcsin(a*x)*sgn(a)/abs(a) - 1/560*sqrt(-a^2*x^2 + 1)*(368*c^3/a - (245*c^3 + 2*(328*a*c^3 + (175*a^2*c^3 - 4*(26*a^3*c^3 + 5*(2*a^5*c^3*x + 7*a^4*c^3)*x)*x)*x)*x)*x)
Time = 0.05 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.34 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {7\,c^3\,x\,\sqrt {1-a^2\,x^2}}{16}+\frac {9\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,\sqrt {-a^2}}-\frac {23\,c^3\,\sqrt {1-a^2\,x^2}}{35\,a}+\frac {41\,a\,c^3\,x^2\,\sqrt {1-a^2\,x^2}}{35}+\frac {5\,a^2\,c^3\,x^3\,\sqrt {1-a^2\,x^2}}{8}-\frac {13\,a^3\,c^3\,x^4\,\sqrt {1-a^2\,x^2}}{35}-\frac {a^4\,c^3\,x^5\,\sqrt {1-a^2\,x^2}}{2}-\frac {a^5\,c^3\,x^6\,\sqrt {1-a^2\,x^2}}{7} \] Input:
int(((c - a^2*c*x^2)^3*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
(7*c^3*x*(1 - a^2*x^2)^(1/2))/16 + (9*c^3*asinh(x*(-a^2)^(1/2)))/(16*(-a^2 )^(1/2)) - (23*c^3*(1 - a^2*x^2)^(1/2))/(35*a) + (41*a*c^3*x^2*(1 - a^2*x^ 2)^(1/2))/35 + (5*a^2*c^3*x^3*(1 - a^2*x^2)^(1/2))/8 - (13*a^3*c^3*x^4*(1 - a^2*x^2)^(1/2))/35 - (a^4*c^3*x^5*(1 - a^2*x^2)^(1/2))/2 - (a^5*c^3*x^6* (1 - a^2*x^2)^(1/2))/7
Time = 0.15 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.07 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^{3} \left (315 \mathit {asin} \left (a x \right )-80 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-280 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-208 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+350 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+656 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+245 \sqrt {-a^{2} x^{2}+1}\, a x -368 \sqrt {-a^{2} x^{2}+1}+368\right )}{560 a} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^3,x)
Output:
(c**3*(315*asin(a*x) - 80*sqrt( - a**2*x**2 + 1)*a**6*x**6 - 280*sqrt( - a **2*x**2 + 1)*a**5*x**5 - 208*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 350*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 656*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 245*s qrt( - a**2*x**2 + 1)*a*x - 368*sqrt( - a**2*x**2 + 1) + 368))/(560*a)