Integrand size = 19, antiderivative size = 146 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^4 \, dx=\frac {5 c^4 x \sqrt {1-a^2 x^2}}{16 a^2}+\frac {5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}-\frac {1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3}+\frac {5 c^4 \arcsin (a x)}{16 a^3} \]
5/7*c^4*x^2*(-a^2*x^2+1)^(3/2)/a-1/2*c^4*x^3*(-a^2*x^2+1)^(3/2)+1/7*a*c^4* x^4*(-a^2*x^2+1)^(3/2)+5/168*c^4*(-21*a*x+16)*(-a^2*x^2+1)^(3/2)/a^3+5/16* c^4*arcsin(a*x)/a^3+5/16*c^4*x*(-a^2*x^2+1)^(1/2)/a^2
Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.62 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^4 \, dx=-\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (-160+105 a x-80 a^2 x^2-42 a^3 x^3+192 a^4 x^4-168 a^5 x^5+48 a^6 x^6\right )+210 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{336 a^3} \]
-1/336*(c^4*(Sqrt[1 - a^2*x^2]*(-160 + 105*a*x - 80*a^2*x^2 - 42*a^3*x^3 + 192*a^4*x^4 - 168*a^5*x^5 + 48*a^6*x^6) + 210*ArcSin[Sqrt[1 - a*x]/Sqrt[2 ]]))/a^3
Time = 0.49 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.23, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {6678, 27, 541, 25, 2340, 27, 533, 27, 533, 25, 27, 455, 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.
| \(\displaystyle \int x^2 e^{\text {arctanh}(a x)} (c-a c x)^4 \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int c^3 x^2 (1-a x)^3 \sqrt {1-a^2 x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int x^2 (1-a x)^3 \sqrt {1-a^2 x^2}dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle c^4 \left (\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}-\frac {\int -x^2 \sqrt {1-a^2 x^2} \left (21 x^2 a^4-25 x a^3+7 a^2\right )dx}{7 a^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c^4 \left (\frac {\int x^2 \sqrt {1-a^2 x^2} \left (21 x^2 a^4-25 x a^3+7 a^2\right )dx}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle c^4 \left (\frac {-\frac {\int -15 a^4 x^2 (7-10 a x) \sqrt {1-a^2 x^2}dx}{6 a^2}-\frac {7}{2} a^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \left (\frac {\frac {5}{2} a^2 \int x^2 (7-10 a x) \sqrt {1-a^2 x^2}dx-\frac {7}{2} a^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c^4 \left (\frac {\frac {5}{2} a^2 \left (\frac {\int -5 a x (4-7 a x) \sqrt {1-a^2 x^2}dx}{5 a^2}+\frac {2 x^2 \left (1-a^2 x^2\right )^{3/2}}{a}\right )-\frac {7}{2} a^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \left (\frac {\frac {5}{2} a^2 \left (\frac {2 x^2 \left (1-a^2 x^2\right )^{3/2}}{a}-\frac {\int x (4-7 a x) \sqrt {1-a^2 x^2}dx}{a}\right )-\frac {7}{2} a^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c^4 \left (\frac {\frac {5}{2} a^2 \left (\frac {2 x^2 \left (1-a^2 x^2\right )^{3/2}}{a}-\frac {\frac {\int -a (7-16 a x) \sqrt {1-a^2 x^2}dx}{4 a^2}+\frac {7 x \left (1-a^2 x^2\right )^{3/2}}{4 a}}{a}\right )-\frac {7}{2} a^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c^4 \left (\frac {\frac {5}{2} a^2 \left (\frac {2 x^2 \left (1-a^2 x^2\right )^{3/2}}{a}-\frac {\frac {7 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {\int a (7-16 a x) \sqrt {1-a^2 x^2}dx}{4 a^2}}{a}\right )-\frac {7}{2} a^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \left (\frac {\frac {5}{2} a^2 \left (\frac {2 x^2 \left (1-a^2 x^2\right )^{3/2}}{a}-\frac {\frac {7 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {\int (7-16 a x) \sqrt {1-a^2 x^2}dx}{4 a}}{a}\right )-\frac {7}{2} a^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c^4 \left (\frac {\frac {5}{2} a^2 \left (\frac {2 x^2 \left (1-a^2 x^2\right )^{3/2}}{a}-\frac {\frac {7 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {7 \int \sqrt {1-a^2 x^2}dx+\frac {16 \left (1-a^2 x^2\right )^{3/2}}{3 a}}{4 a}}{a}\right )-\frac {7}{2} a^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c^4 \left (\frac {\frac {5}{2} a^2 \left (\frac {2 x^2 \left (1-a^2 x^2\right )^{3/2}}{a}-\frac {\frac {7 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {7 \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2}\right )+\frac {16 \left (1-a^2 x^2\right )^{3/2}}{3 a}}{4 a}}{a}\right )-\frac {7}{2} a^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle c^4 \left (\frac {\frac {5}{2} a^2 \left (\frac {2 x^2 \left (1-a^2 x^2\right )^{3/2}}{a}-\frac {\frac {7 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {7 \left (\frac {1}{2} x \sqrt {1-a^2 x^2}+\frac {\arcsin (a x)}{2 a}\right )+\frac {16 \left (1-a^2 x^2\right )^{3/2}}{3 a}}{4 a}}{a}\right )-\frac {7}{2} a^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{7 a^2}+\frac {1}{7} a x^4 \left (1-a^2 x^2\right )^{3/2}\right )\) |
c^4*((a*x^4*(1 - a^2*x^2)^(3/2))/7 + ((-7*a^2*x^3*(1 - a^2*x^2)^(3/2))/2 + (5*a^2*((2*x^2*(1 - a^2*x^2)^(3/2))/a - ((7*x*(1 - a^2*x^2)^(3/2))/(4*a) - ((16*(1 - a^2*x^2)^(3/2))/(3*a) + 7*((x*Sqrt[1 - a^2*x^2])/2 + ArcSin[a* x]/(2*a)))/(4*a))/a))/2)/(7*a^2))
3.4.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(\frac {\left (48 a^{6} x^{6}-168 a^{5} x^{5}+192 a^{4} x^{4}-42 a^{3} x^{3}-80 a^{2} x^{2}+105 a x -160\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{336 a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{16 a^{2} \sqrt {a^{2}}}\) | \(110\) |
| meijerg | \(\frac {c^{4} \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (40 a^{6} x^{6}+48 a^{4} x^{4}+64 a^{2} x^{2}+128\right ) \sqrt {-a^{2} x^{2}+1}}{140}\right )}{2 a^{3} \sqrt {\pi }}+\frac {3 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}+70 a^{2} x^{2}+105\right ) \sqrt {-a^{2} x^{2}+1}}{168 a^{6}}+\frac {5 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{2 a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{4} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{a^{3} \sqrt {\pi }}+\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c^{4} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 a^{3} \sqrt {\pi }}-\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}\) | \(382\) |
| default | \(c^{4} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}+a^{5} \left (-\frac {x^{6} \sqrt {-a^{2} x^{2}+1}}{7 a^{2}}+\frac {-\frac {6 x^{4} \sqrt {-a^{2} x^{2}+1}}{35 a^{2}}+\frac {6 \left (-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}\right )}{7 a^{2}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{5} \sqrt {-a^{2} x^{2}+1}}{6 a^{2}}+\frac {-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )+2 a^{2} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-3 a \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )\right )\) | \(444\) |
1/336*(48*a^6*x^6-168*a^5*x^5+192*a^4*x^4-42*a^3*x^3-80*a^2*x^2+105*a*x-16 0)*(a^2*x^2-1)/a^3/(-a^2*x^2+1)^(1/2)*c^4+5/16/a^2/(a^2)^(1/2)*arctan((a^2 )^(1/2)*x/(-a^2*x^2+1)^(1/2))*c^4
Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.78 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^4 \, dx=-\frac {210 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (48 \, a^{6} c^{4} x^{6} - 168 \, a^{5} c^{4} x^{5} + 192 \, a^{4} c^{4} x^{4} - 42 \, a^{3} c^{4} x^{3} - 80 \, a^{2} c^{4} x^{2} + 105 \, a c^{4} x - 160 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{336 \, a^{3}} \]
-1/336*(210*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (48*a^6*c^4*x^6 - 168*a^5*c^4*x^5 + 192*a^4*c^4*x^4 - 42*a^3*c^4*x^3 - 80*a^2*c^4*x^2 + 105 *a*c^4*x - 160*c^4)*sqrt(-a^2*x^2 + 1))/a^3
Time = 0.79 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.45 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^4 \, dx=\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (- \frac {a^{3} c^{4} x^{6}}{7} + \frac {a^{2} c^{4} x^{5}}{2} - \frac {4 a c^{4} x^{4}}{7} + \frac {c^{4} x^{3}}{8} + \frac {5 c^{4} x^{2}}{21 a} - \frac {5 c^{4} x}{16 a^{2}} + \frac {10 c^{4}}{21 a^{3}}\right ) + \frac {5 c^{4} \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{16 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {a^{5} c^{4} x^{8}}{8} - \frac {3 a^{4} c^{4} x^{7}}{7} + \frac {a^{3} c^{4} x^{6}}{3} + \frac {2 a^{2} c^{4} x^{5}}{5} - \frac {3 a c^{4} x^{4}}{4} + \frac {c^{4} x^{3}}{3} & \text {otherwise} \end {cases} \]
Piecewise((sqrt(-a**2*x**2 + 1)*(-a**3*c**4*x**6/7 + a**2*c**4*x**5/2 - 4* a*c**4*x**4/7 + c**4*x**3/8 + 5*c**4*x**2/(21*a) - 5*c**4*x/(16*a**2) + 10 *c**4/(21*a**3)) + 5*c**4*log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(16*a**2*sqrt(-a**2)), Ne(a**2, 0)), (a**5*c**4*x**8/8 - 3*a**4*c**4*x **7/7 + a**3*c**4*x**6/3 + 2*a**2*c**4*x**5/5 - 3*a*c**4*x**4/4 + c**4*x** 3/3, True))
Time = 0.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.12 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^4 \, dx=-\frac {1}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{6} + \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{5} - \frac {4}{7} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{4} + \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{3} + \frac {5 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{2}}{21 \, a} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x}{16 \, a^{2}} + \frac {5 \, c^{4} \arcsin \left (a x\right )}{16 \, a^{3}} + \frac {10 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{21 \, a^{3}} \]
-1/7*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^6 + 1/2*sqrt(-a^2*x^2 + 1)*a^2*c^4*x^5 - 4/7*sqrt(-a^2*x^2 + 1)*a*c^4*x^4 + 1/8*sqrt(-a^2*x^2 + 1)*c^4*x^3 + 5/21* sqrt(-a^2*x^2 + 1)*c^4*x^2/a - 5/16*sqrt(-a^2*x^2 + 1)*c^4*x/a^2 + 5/16*c^ 4*arcsin(a*x)/a^3 + 10/21*sqrt(-a^2*x^2 + 1)*c^4/a^3
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.71 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^4 \, dx=\frac {5 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, a^{2} {\left | a \right |}} - \frac {1}{336} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (\frac {105 \, c^{4}}{a^{2}} - 2 \, {\left (\frac {40 \, c^{4}}{a} + 3 \, {\left (7 \, c^{4} - 4 \, {\left (8 \, a c^{4} + {\left (2 \, a^{3} c^{4} x - 7 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x - \frac {160 \, c^{4}}{a^{3}}\right )} \]
5/16*c^4*arcsin(a*x)*sgn(a)/(a^2*abs(a)) - 1/336*sqrt(-a^2*x^2 + 1)*((105* c^4/a^2 - 2*(40*c^4/a + 3*(7*c^4 - 4*(8*a*c^4 + (2*a^3*c^4*x - 7*a^2*c^4)* x)*x)*x)*x)*x - 160*c^4/a^3)
Time = 3.45 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.21 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^4 \, dx=\frac {10\,c^4\,\sqrt {1-a^2\,x^2}}{21\,a^3}+\frac {c^4\,x^3\,\sqrt {1-a^2\,x^2}}{8}-\frac {5\,c^4\,x\,\sqrt {1-a^2\,x^2}}{16\,a^2}-\frac {4\,a\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{7}+\frac {5\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a^2\,\sqrt {-a^2}}+\frac {5\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{21\,a}+\frac {a^2\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{2}-\frac {a^3\,c^4\,x^6\,\sqrt {1-a^2\,x^2}}{7} \]
(10*c^4*(1 - a^2*x^2)^(1/2))/(21*a^3) + (c^4*x^3*(1 - a^2*x^2)^(1/2))/8 - (5*c^4*x*(1 - a^2*x^2)^(1/2))/(16*a^2) - (4*a*c^4*x^4*(1 - a^2*x^2)^(1/2)) /7 + (5*c^4*asinh(x*(-a^2)^(1/2)))/(16*a^2*(-a^2)^(1/2)) + (5*c^4*x^2*(1 - a^2*x^2)^(1/2))/(21*a) + (a^2*c^4*x^5*(1 - a^2*x^2)^(1/2))/2 - (a^3*c^4*x ^6*(1 - a^2*x^2)^(1/2))/7