Integrand size = 20, antiderivative size = 127 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {35}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {35 c^4 \arcsin (a x)}{128 a} \] Output:
35/128*c^4*x*(-a^2*x^2+1)^(1/2)+35/192*c^4*x*(-a^2*x^2+1)^(3/2)+7/48*c^4*x *(-a^2*x^2+1)^(5/2)+1/8*c^4*x*(-a^2*x^2+1)^(7/2)-1/9*c^4*(-a^2*x^2+1)^(9/2 )/a+35/128*c^4*arcsin(a*x)/a
Time = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (128-837 a x-512 a^2 x^2+978 a^3 x^3+768 a^4 x^4-600 a^5 x^5-512 a^6 x^6+144 a^7 x^7+128 a^8 x^8\right )+630 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{1152 a} \] Input:
Integrate[E^ArcTanh[a*x]*(c - a^2*c*x^2)^4,x]
Output:
-1/1152*(c^4*(Sqrt[1 - a^2*x^2]*(128 - 837*a*x - 512*a^2*x^2 + 978*a^3*x^3 + 768*a^4*x^4 - 600*a^5*x^5 - 512*a^6*x^6 + 144*a^7*x^7 + 128*a^8*x^8) + 630*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/a
Time = 0.31 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6688, 455, 211, 211, 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.
| \(\displaystyle \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx\) |
| \(\Big \downarrow \) 6688 |
| \(\displaystyle c^4 \int (a x+1) \left (1-a^2 x^2\right )^{7/2}dx\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c^4 \left (\int \left (1-a^2 x^2\right )^{7/2}dx-\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c^4 \left (\frac {7}{8} \int \left (1-a^2 x^2\right )^{5/2}dx-\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} x \left (1-a^2 x^2\right )^{7/2}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c^4 \left (\frac {7}{8} \left (\frac {5}{6} \int \left (1-a^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} x \left (1-a^2 x^2\right )^{7/2}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c^4 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-a^2 x^2}dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} x \left (1-a^2 x^2\right )^{7/2}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c^4 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} x \left (1-a^2 x^2\right )^{7/2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle c^4 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2}+\frac {\arcsin (a x)}{2 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} x \left (1-a^2 x^2\right )^{7/2}\right )\) |
Input:
Int[E^ArcTanh[a*x]*(c - a^2*c*x^2)^4,x]
Output:
c^4*((x*(1 - a^2*x^2)^(7/2))/8 - (1 - a^2*x^2)^(9/2)/(9*a) + (7*((x*(1 - a ^2*x^2)^(5/2))/6 + (5*((x*(1 - a^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 - a^2*x^2 ])/2 + ArcSin[a*x]/(2*a)))/4))/6))/8)
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[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 = 0.35 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {\left (128 a^{8} x^{8}+144 a^{7} x^{7}-512 x^{6} a^{6}-600 a^{5} x^{5}+768 a^{4} x^{4}+978 a^{3} x^{3}-512 a^{2} x^{2}-837 a x +128\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{1152 a \sqrt {-a^{2} x^{2}+1}}+\frac {35 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{128 \sqrt {a^{2}}}\) | \(123\) |
| meijerg | \(-\frac {c^{4} \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (70 a^{8} x^{8}+80 x^{6} a^{6}+96 a^{4} x^{4}+128 a^{2} x^{2}+256\right ) \sqrt {-a^{2} x^{2}+1}}{315}\right )}{2 a \sqrt {\pi }}-\frac {2 c^{4} \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (40 x^{6} a^{6}+48 a^{4} x^{4}+64 a^{2} x^{2}+128\right ) \sqrt {-a^{2} x^{2}+1}}{140}\right )}{a \sqrt {\pi }}-\frac {3 c^{4} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{a \sqrt {\pi }}-\frac {2 c^{4} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{a \sqrt {\pi }}-\frac {c^{4} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}+\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {9}{2}} \left (144 x^{6} a^{6}+168 a^{4} x^{4}+210 a^{2} x^{2}+315\right ) \sqrt {-a^{2} x^{2}+1}}{576 a^{8}}+\frac {35 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {9}{2}} \arcsin \left (a x \right )}{64 a^{9}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {2 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}+70 a^{2} x^{2}+105\right ) \sqrt {-a^{2} x^{2}+1}}{168 a^{6}}+\frac {5 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {2 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{4} \arcsin \left (a x \right )}{a}\) | \(576\) |
| default | \(c^{4} \left (\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {\sqrt {-a^{2} x^{2}+1}}{a}+a^{8} \left (-\frac {x^{7} \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {-\frac {7 x^{5} \sqrt {-a^{2} x^{2}+1}}{48 a^{2}}+\frac {7 \left (-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}\right )}{8 a^{2}}}{a^{2}}\right )+a^{9} \left (-\frac {x^{8} \sqrt {-a^{2} x^{2}+1}}{9 a^{2}}+\frac {-\frac {8 x^{6} \sqrt {-a^{2} x^{2}+1}}{63 a^{2}}+\frac {8 \left (-\frac {6 x^{4} \sqrt {-a^{2} x^{2}+1}}{35 a^{2}}+\frac {6 \left (-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}\right )}{7 a^{2}}\right )}{9 a^{2}}}{a^{2}}\right )-4 a^{3} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )+6 a^{4} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+6 a^{5} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )-4 a^{6} \left (-\frac {x^{5} \sqrt {-a^{2} x^{2}+1}}{6 a^{2}}+\frac {-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}}{a^{2}}\right )-4 a^{7} \left (-\frac {x^{6} \sqrt {-a^{2} x^{2}+1}}{7 a^{2}}+\frac {-\frac {6 x^{4} \sqrt {-a^{2} x^{2}+1}}{35 a^{2}}+\frac {6 \left (-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}\right )}{7 a^{2}}}{a^{2}}\right )-4 a^{2} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )\right )\) | \(747\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)
Output:
1/1152*(128*a^8*x^8+144*a^7*x^7-512*a^6*x^6-600*a^5*x^5+768*a^4*x^4+978*a^ 3*x^3-512*a^2*x^2-837*a*x+128)*(a^2*x^2-1)/a/(-a^2*x^2+1)^(1/2)*c^4+35/128 /(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))*c^4
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.07 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {630 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (128 \, a^{8} c^{4} x^{8} + 144 \, a^{7} c^{4} x^{7} - 512 \, a^{6} c^{4} x^{6} - 600 \, a^{5} c^{4} x^{5} + 768 \, a^{4} c^{4} x^{4} + 978 \, a^{3} c^{4} x^{3} - 512 \, a^{2} c^{4} x^{2} - 837 \, a c^{4} x + 128 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{1152 \, a} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^4,x, algorithm="fricas ")
Output:
-1/1152*(630*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (128*a^8*c^4*x^8 + 144*a^7*c^4*x^7 - 512*a^6*c^4*x^6 - 600*a^5*c^4*x^5 + 768*a^4*c^4*x^4 + 978*a^3*c^4*x^3 - 512*a^2*c^4*x^2 - 837*a*c^4*x + 128*c^4)*sqrt(-a^2*x^2 + 1))/a
Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (110) = 220\).
Time = 0.86 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.25 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\begin {cases} \frac {35 c^{4} \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{128 \sqrt {- a^{2}}} + \sqrt {- a^{2} x^{2} + 1} \left (- \frac {a^{7} c^{4} x^{8}}{9} - \frac {a^{6} c^{4} x^{7}}{8} + \frac {4 a^{5} c^{4} x^{6}}{9} + \frac {25 a^{4} c^{4} x^{5}}{48} - \frac {2 a^{3} c^{4} x^{4}}{3} - \frac {163 a^{2} c^{4} x^{3}}{192} + \frac {4 a c^{4} x^{2}}{9} + \frac {93 c^{4} x}{128} - \frac {c^{4}}{9 a}\right ) & \text {for}\: a^{2} \neq 0 \\\begin {cases} c^{4} x & \text {for}\: a = 0 \\\frac {\frac {a^{10} c^{4} x^{10}}{10} + \frac {a^{9} c^{4} x^{9}}{9} - \frac {a^{8} c^{4} x^{8}}{2} - \frac {4 a^{7} c^{4} x^{7}}{7} + a^{6} c^{4} x^{6} + \frac {6 a^{5} c^{4} x^{5}}{5} - a^{4} c^{4} x^{4} - \frac {4 a^{3} c^{4} x^{3}}{3} + \frac {a^{2} c^{4} x^{2}}{2} + a c^{4} x}{a} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**4,x)
Output:
Piecewise((35*c**4*log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(12 8*sqrt(-a**2)) + sqrt(-a**2*x**2 + 1)*(-a**7*c**4*x**8/9 - a**6*c**4*x**7/ 8 + 4*a**5*c**4*x**6/9 + 25*a**4*c**4*x**5/48 - 2*a**3*c**4*x**4/3 - 163*a **2*c**4*x**3/192 + 4*a*c**4*x**2/9 + 93*c**4*x/128 - c**4/(9*a)), Ne(a**2 , 0)), (Piecewise((c**4*x, Eq(a, 0)), ((a**10*c**4*x**10/10 + a**9*c**4*x* *9/9 - a**8*c**4*x**8/2 - 4*a**7*c**4*x**7/7 + a**6*c**4*x**6 + 6*a**5*c** 4*x**5/5 - a**4*c**4*x**4 - 4*a**3*c**4*x**3/3 + a**2*c**4*x**2/2 + a*c**4 *x)/a, True)), True))
Time = 0.11 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.65 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {1}{9} \, \sqrt {-a^{2} x^{2} + 1} a^{7} c^{4} x^{8} - \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} a^{6} c^{4} x^{7} + \frac {4}{9} \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{4} x^{6} + \frac {25}{48} \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{4} x^{5} - \frac {2}{3} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{4} - \frac {163}{192} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{3} + \frac {4}{9} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{2} + \frac {93}{128} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x + \frac {35 \, c^{4} \arcsin \left (a x\right )}{128 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{9 \, a} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^4,x, algorithm="maxima ")
Output:
-1/9*sqrt(-a^2*x^2 + 1)*a^7*c^4*x^8 - 1/8*sqrt(-a^2*x^2 + 1)*a^6*c^4*x^7 + 4/9*sqrt(-a^2*x^2 + 1)*a^5*c^4*x^6 + 25/48*sqrt(-a^2*x^2 + 1)*a^4*c^4*x^5 - 2/3*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^4 - 163/192*sqrt(-a^2*x^2 + 1)*a^2*c^4 *x^3 + 4/9*sqrt(-a^2*x^2 + 1)*a*c^4*x^2 + 93/128*sqrt(-a^2*x^2 + 1)*c^4*x + 35/128*c^4*arcsin(a*x)/a - 1/9*sqrt(-a^2*x^2 + 1)*c^4/a
Time = 0.16 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {35 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{128 \, {\left | a \right |}} - \frac {1}{1152} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {128 \, c^{4}}{a} - {\left (837 \, c^{4} + 2 \, {\left (256 \, a c^{4} - {\left (489 \, a^{2} c^{4} + 4 \, {\left (96 \, a^{3} c^{4} - {\left (75 \, a^{4} c^{4} + 2 \, {\left (32 \, a^{5} c^{4} - {\left (8 \, a^{7} c^{4} x + 9 \, a^{6} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^4,x, algorithm="giac")
Output:
35/128*c^4*arcsin(a*x)*sgn(a)/abs(a) - 1/1152*sqrt(-a^2*x^2 + 1)*(128*c^4/ a - (837*c^4 + 2*(256*a*c^4 - (489*a^2*c^4 + 4*(96*a^3*c^4 - (75*a^4*c^4 + 2*(32*a^5*c^4 - (8*a^7*c^4*x + 9*a^6*c^4)*x)*x)*x)*x)*x)*x)*x)
Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.93 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {35\,c^4\,x\,\sqrt {1-a^2\,x^2}}{128}+\frac {35\,c^4\,x\,{\left (1-a^2\,x^2\right )}^{3/2}}{192}+\frac {7\,c^4\,x\,{\left (1-a^2\,x^2\right )}^{5/2}}{48}+\frac {c^4\,x\,{\left (1-a^2\,x^2\right )}^{7/2}}{8}-\frac {c^4\,{\left (1-a^2\,x^2\right )}^{9/2}}{9\,a}-\frac {35\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{128\,a^2} \] Input:
int(((c - a^2*c*x^2)^4*(a*x + 1))/(1 - a^2*x^2)^(1/2),x)
Output:
(35*c^4*x*(1 - a^2*x^2)^(1/2))/128 + (35*c^4*x*(1 - a^2*x^2)^(3/2))/192 + (7*c^4*x*(1 - a^2*x^2)^(5/2))/48 + (c^4*x*(1 - a^2*x^2)^(7/2))/8 - (c^4*(1 - a^2*x^2)^(9/2))/(9*a) - (35*c^4*asinh(x*(-a^2)^(1/2))*(-a^2)^(1/2))/(12 8*a^2)
Time = 0.16 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.39 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^{4} \left (315 \mathit {asin} \left (a x \right )-128 \sqrt {-a^{2} x^{2}+1}\, a^{8} x^{8}-144 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+512 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+600 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-768 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-978 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+512 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+837 \sqrt {-a^{2} x^{2}+1}\, a x -128 \sqrt {-a^{2} x^{2}+1}+128\right )}{1152 a} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^4,x)
Output:
(c**4*(315*asin(a*x) - 128*sqrt( - a**2*x**2 + 1)*a**8*x**8 - 144*sqrt( - a**2*x**2 + 1)*a**7*x**7 + 512*sqrt( - a**2*x**2 + 1)*a**6*x**6 + 600*sqrt ( - a**2*x**2 + 1)*a**5*x**5 - 768*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 978* sqrt( - a**2*x**2 + 1)*a**3*x**3 + 512*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 837*sqrt( - a**2*x**2 + 1)*a*x - 128*sqrt( - a**2*x**2 + 1) + 128))/(1152* a)