Integrand size = 25, antiderivative size = 120 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^3 x^{1+m}}{1+m}+\frac {2 a c^3 x^{2+m}}{2+m}-\frac {a^2 c^3 x^{3+m}}{3+m}-\frac {4 a^3 c^3 x^{4+m}}{4+m}-\frac {a^4 c^3 x^{5+m}}{5+m}+\frac {2 a^5 c^3 x^{6+m}}{6+m}+\frac {a^6 c^3 x^{7+m}}{7+m} \] Output:
c^3*x^(1+m)/(1+m)+2*a*c^3*x^(2+m)/(2+m)-a^2*c^3*x^(3+m)/(3+m)-4*a^3*c^3*x^ (4+m)/(4+m)-a^4*c^3*x^(5+m)/(5+m)+2*a^5*c^3*x^(6+m)/(6+m)+a^6*c^3*x^(7+m)/ (7+m)
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.73 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^3 \, dx=c^3 x^{1+m} \left (\frac {1}{1+m}+\frac {2 a x}{2+m}-\frac {a^2 x^2}{3+m}-\frac {4 a^3 x^3}{4+m}-\frac {a^4 x^4}{5+m}+\frac {2 a^5 x^5}{6+m}+\frac {a^6 x^6}{7+m}\right ) \] Input:
Integrate[E^(2*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^3,x]
Output:
c^3*x^(1 + m)*((1 + m)^(-1) + (2*a*x)/(2 + m) - (a^2*x^2)/(3 + m) - (4*a^3 *x^3)/(4 + m) - (a^4*x^4)/(5 + m) + (2*a^5*x^5)/(6 + m) + (a^6*x^6)/(7 + m ))
Time = 0.56 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6700, 99, 2009}
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.
| \(\displaystyle \int x^m e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 6700 |
| \(\displaystyle c^3 \int x^m (1-a x)^2 (a x+1)^4dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle c^3 \int \left (x^m+2 a x^{m+1}-a^2 x^{m+2}-4 a^3 x^{m+3}-a^4 x^{m+4}+2 a^5 x^{m+5}+a^6 x^{m+6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c^3 \left (\frac {a^6 x^{m+7}}{m+7}+\frac {2 a^5 x^{m+6}}{m+6}-\frac {a^4 x^{m+5}}{m+5}-\frac {4 a^3 x^{m+4}}{m+4}-\frac {a^2 x^{m+3}}{m+3}+\frac {2 a x^{m+2}}{m+2}+\frac {x^{m+1}}{m+1}\right )\) |
Input:
Int[E^(2*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^3,x]
Output:
c^3*(x^(1 + m)/(1 + m) + (2*a*x^(2 + m))/(2 + m) - (a^2*x^(3 + m))/(3 + m) - (4*a^3*x^(4 + m))/(4 + m) - (a^4*x^(5 + m))/(5 + m) + (2*a^5*x^(6 + m)) /(6 + m) + (a^6*x^(7 + m))/(7 + m))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(474\) vs. \(2(120)=240\).
Time = 0.27 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.96
| method | result | size |
| risch | \(\frac {c^{3} x^{m} \left (a^{6} m^{6} x^{6}+21 a^{6} m^{5} x^{6}+175 a^{6} m^{4} x^{6}+2 a^{5} m^{6} x^{5}+735 a^{6} m^{3} x^{6}+44 a^{5} m^{5} x^{5}+1624 a^{6} m^{2} x^{6}+380 a^{5} m^{4} x^{5}-a^{4} x^{4} m^{6}+1764 a^{6} x^{6} m +1640 a^{5} m^{3} x^{5}-23 a^{4} x^{4} m^{5}+720 x^{6} a^{6}+3698 a^{5} m^{2} x^{5}-207 a^{4} x^{4} m^{4}-4 a^{3} m^{6} x^{3}+4076 a^{5} m \,x^{5}-925 a^{4} x^{4} m^{3}-96 a^{3} m^{5} x^{3}+1680 a^{5} x^{5}-2144 a^{4} x^{4} m^{2}-904 a^{3} m^{4} x^{3}-a^{2} x^{2} m^{6}-2412 a^{4} x^{4} m -4224 a^{3} m^{3} x^{3}-25 a^{2} x^{2} m^{5}-1008 a^{4} x^{4}-10180 a^{3} m^{2} x^{3}-247 a^{2} x^{2} m^{4}+2 a \,m^{6} x -11808 a^{3} m \,x^{3}-1219 a^{2} x^{2} m^{3}+52 a \,m^{5} x -5040 a^{3} x^{3}-3112 a^{2} m^{2} x^{2}+540 a \,m^{4} x +m^{6}-3796 a^{2} m \,x^{2}+2840 a \,m^{3} x +27 m^{5}-1680 a^{2} x^{2}+7858 a x \,m^{2}+295 m^{4}+10548 a m x +1665 m^{3}+5040 a x +5104 m^{2}+8028 m +5040\right ) x}{\left (m +7\right ) \left (m +6\right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(475\) |
| gosper | \(\frac {x^{1+m} c^{3} \left (a^{6} m^{6} x^{6}+21 a^{6} m^{5} x^{6}+175 a^{6} m^{4} x^{6}+2 a^{5} m^{6} x^{5}+735 a^{6} m^{3} x^{6}+44 a^{5} m^{5} x^{5}+1624 a^{6} m^{2} x^{6}+380 a^{5} m^{4} x^{5}-a^{4} x^{4} m^{6}+1764 a^{6} x^{6} m +1640 a^{5} m^{3} x^{5}-23 a^{4} x^{4} m^{5}+720 x^{6} a^{6}+3698 a^{5} m^{2} x^{5}-207 a^{4} x^{4} m^{4}-4 a^{3} m^{6} x^{3}+4076 a^{5} m \,x^{5}-925 a^{4} x^{4} m^{3}-96 a^{3} m^{5} x^{3}+1680 a^{5} x^{5}-2144 a^{4} x^{4} m^{2}-904 a^{3} m^{4} x^{3}-a^{2} x^{2} m^{6}-2412 a^{4} x^{4} m -4224 a^{3} m^{3} x^{3}-25 a^{2} x^{2} m^{5}-1008 a^{4} x^{4}-10180 a^{3} m^{2} x^{3}-247 a^{2} x^{2} m^{4}+2 a \,m^{6} x -11808 a^{3} m \,x^{3}-1219 a^{2} x^{2} m^{3}+52 a \,m^{5} x -5040 a^{3} x^{3}-3112 a^{2} m^{2} x^{2}+540 a \,m^{4} x +m^{6}-3796 a^{2} m \,x^{2}+2840 a \,m^{3} x +27 m^{5}-1680 a^{2} x^{2}+7858 a x \,m^{2}+295 m^{4}+10548 a m x +1665 m^{3}+5040 a x +5104 m^{2}+8028 m +5040\right )}{\left (1+m \right ) \left (2+m \right ) \left (3+m \right ) \left (4+m \right ) \left (5+m \right ) \left (m +6\right ) \left (m +7\right )}\) | \(476\) |
| orering | \(\frac {\left (a^{6} m^{6} x^{6}+21 a^{6} m^{5} x^{6}+175 a^{6} m^{4} x^{6}+2 a^{5} m^{6} x^{5}+735 a^{6} m^{3} x^{6}+44 a^{5} m^{5} x^{5}+1624 a^{6} m^{2} x^{6}+380 a^{5} m^{4} x^{5}-a^{4} x^{4} m^{6}+1764 a^{6} x^{6} m +1640 a^{5} m^{3} x^{5}-23 a^{4} x^{4} m^{5}+720 x^{6} a^{6}+3698 a^{5} m^{2} x^{5}-207 a^{4} x^{4} m^{4}-4 a^{3} m^{6} x^{3}+4076 a^{5} m \,x^{5}-925 a^{4} x^{4} m^{3}-96 a^{3} m^{5} x^{3}+1680 a^{5} x^{5}-2144 a^{4} x^{4} m^{2}-904 a^{3} m^{4} x^{3}-a^{2} x^{2} m^{6}-2412 a^{4} x^{4} m -4224 a^{3} m^{3} x^{3}-25 a^{2} x^{2} m^{5}-1008 a^{4} x^{4}-10180 a^{3} m^{2} x^{3}-247 a^{2} x^{2} m^{4}+2 a \,m^{6} x -11808 a^{3} m \,x^{3}-1219 a^{2} x^{2} m^{3}+52 a \,m^{5} x -5040 a^{3} x^{3}-3112 a^{2} m^{2} x^{2}+540 a \,m^{4} x +m^{6}-3796 a^{2} m \,x^{2}+2840 a \,m^{3} x +27 m^{5}-1680 a^{2} x^{2}+7858 a x \,m^{2}+295 m^{4}+10548 a m x +1665 m^{3}+5040 a x +5104 m^{2}+8028 m +5040\right ) x \,x^{m} \left (-a^{2} c \,x^{2}+c \right )^{3}}{\left (m +7\right ) \left (m +6\right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (-a^{2} x^{2}+1\right )}\) | \(511\) |
| parallelrisch | \(\frac {720 x^{7} x^{m} a^{6} c^{3}+1680 x^{6} x^{m} a^{5} c^{3}-1008 x^{5} x^{m} a^{4} c^{3}-5040 x^{4} x^{m} a^{3} c^{3}+x \,x^{m} c^{3} m^{6}+27 x \,x^{m} c^{3} m^{5}-1680 x^{3} x^{m} a^{2} c^{3}+295 x \,x^{m} c^{3} m^{4}+1665 x \,x^{m} c^{3} m^{3}+5040 x^{2} x^{m} a \,c^{3}+5104 x \,x^{m} c^{3} m^{2}+8028 x \,x^{m} c^{3} m +2 x^{6} x^{m} a^{5} c^{3} m^{6}+735 x^{7} x^{m} a^{6} c^{3} m^{3}+44 x^{6} x^{m} a^{5} c^{3} m^{5}+1624 x^{7} x^{m} a^{6} c^{3} m^{2}+380 x^{6} x^{m} a^{5} c^{3} m^{4}-x^{5} x^{m} a^{4} c^{3} m^{6}+1764 x^{7} x^{m} a^{6} c^{3} m +1640 x^{6} x^{m} a^{5} c^{3} m^{3}-23 x^{5} x^{m} a^{4} c^{3} m^{5}+3698 x^{6} x^{m} a^{5} c^{3} m^{2}-207 x^{5} x^{m} a^{4} c^{3} m^{4}-4 x^{4} x^{m} a^{3} c^{3} m^{6}+4076 x^{6} x^{m} a^{5} c^{3} m -925 x^{5} x^{m} a^{4} c^{3} m^{3}-96 x^{4} x^{m} a^{3} c^{3} m^{5}-2144 x^{5} x^{m} a^{4} c^{3} m^{2}-904 x^{4} x^{m} a^{3} c^{3} m^{4}-x^{3} x^{m} a^{2} c^{3} m^{6}-2412 x^{5} x^{m} a^{4} c^{3} m -4224 x^{4} x^{m} a^{3} c^{3} m^{3}-25 x^{3} x^{m} a^{2} c^{3} m^{5}-10180 x^{4} x^{m} a^{3} c^{3} m^{2}-247 x^{3} x^{m} a^{2} c^{3} m^{4}+2 x^{2} x^{m} a \,c^{3} m^{6}-11808 x^{4} x^{m} a^{3} c^{3} m -1219 x^{3} x^{m} a^{2} c^{3} m^{3}+52 x^{2} x^{m} a \,c^{3} m^{5}-3112 x^{3} x^{m} a^{2} c^{3} m^{2}+540 x^{2} x^{m} a \,c^{3} m^{4}-3796 x^{3} x^{m} a^{2} c^{3} m +2840 x^{2} x^{m} a \,c^{3} m^{3}+7858 x^{2} x^{m} a \,c^{3} m^{2}+10548 x^{2} x^{m} a \,c^{3} m +x^{7} x^{m} a^{6} c^{3} m^{6}+21 x^{7} x^{m} a^{6} c^{3} m^{5}+175 x^{7} x^{m} a^{6} c^{3} m^{4}+5040 c^{3} x^{m} x}{\left (m +7\right ) \left (m +6\right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(785\) |
| meijerg | \(-\frac {\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} c^{3} \left (-\frac {2 x^{1+m} \left (-a^{2}\right )^{\frac {9}{2}+\frac {m}{2}} \left (a^{6} m^{3} x^{6}+9 a^{6} m^{2} x^{6}+23 a^{6} x^{6} m +15 x^{6} a^{6}+a^{4} x^{4} m^{3}+11 a^{4} x^{4} m^{2}+31 a^{4} x^{4} m +21 a^{4} x^{4}+a^{2} x^{2} m^{3}+13 a^{2} m^{2} x^{2}+47 a^{2} m \,x^{2}+35 a^{2} x^{2}+m^{3}+15 m^{2}+71 m +105\right )}{a^{8} \left (m +7\right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}+\frac {x^{1+m} \left (-a^{2}\right )^{\frac {9}{2}+\frac {m}{2}} \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{a^{8}}\right )}{2}-\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} c^{3} \left (-\frac {2 x^{1+m} \left (-a^{2}\right )^{\frac {7}{2}+\frac {m}{2}} \left (a^{4} x^{4} m^{2}+4 a^{4} x^{4} m +3 a^{4} x^{4}+a^{2} m^{2} x^{2}+6 a^{2} m \,x^{2}+5 a^{2} x^{2}+m^{2}+8 m +15\right )}{a^{6} \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}+\frac {x^{1+m} \left (-a^{2}\right )^{\frac {7}{2}+\frac {m}{2}} \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{a^{6}}\right )+\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} c^{3} \left (\frac {2 x^{1+m} \left (-a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (-3-m \right )}{\left (3+m \right ) \left (1+m \right ) a^{2}}+\frac {x^{1+m} \left (-a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{a^{2}}\right )-\frac {\left (-a^{2}\right )^{-\frac {m}{2}} c^{3} \left (-\frac {2 x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (a^{6} m^{3} x^{6}+6 a^{6} m^{2} x^{6}+8 a^{6} x^{6} m +a^{4} x^{4} m^{3}+8 a^{4} x^{4} m^{2}+12 a^{4} x^{4} m +a^{2} x^{2} m^{3}+10 a^{2} m^{2} x^{2}+24 a^{2} m \,x^{2}+m^{3}+12 m^{2}+44 m +48\right )}{\left (m +6\right ) \left (4+m \right ) \left (2+m \right ) m}+x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}\right )\right )}{a}-\frac {3 \left (-a^{2}\right )^{-\frac {m}{2}} c^{3} \left (\frac {2 x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (a^{4} x^{4} m^{2}+2 a^{4} x^{4} m +a^{2} m^{2} x^{2}+4 a^{2} m \,x^{2}+m^{2}+6 m +8\right )}{\left (4+m \right ) \left (2+m \right ) m}-x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}\right )\right )}{a}-\frac {3 \left (-a^{2}\right )^{-\frac {m}{2}} c^{3} \left (-\frac {2 x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (a^{2} m \,x^{2}+m +2\right )}{\left (2+m \right ) m}+x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}\right )\right )}{a}-\frac {\left (-a^{2}\right )^{-\frac {m}{2}} c^{3} \left (-\frac {2 x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (-2-m \right )}{\left (2+m \right ) m}-x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}\right )\right )}{a}+\frac {c^{3} x^{1+m} \left (\frac {m}{2}+\frac {1}{2}\right ) \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{1+m}\) | \(931\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
c^3*x^m*(a^6*m^6*x^6+21*a^6*m^5*x^6+175*a^6*m^4*x^6+2*a^5*m^6*x^5+735*a^6* m^3*x^6+44*a^5*m^5*x^5+1624*a^6*m^2*x^6+380*a^5*m^4*x^5-a^4*m^6*x^4+1764*a ^6*m*x^6+1640*a^5*m^3*x^5-23*a^4*m^5*x^4+720*a^6*x^6+3698*a^5*m^2*x^5-207* a^4*m^4*x^4-4*a^3*m^6*x^3+4076*a^5*m*x^5-925*a^4*m^3*x^4-96*a^3*m^5*x^3+16 80*a^5*x^5-2144*a^4*m^2*x^4-904*a^3*m^4*x^3-a^2*m^6*x^2-2412*a^4*m*x^4-422 4*a^3*m^3*x^3-25*a^2*m^5*x^2-1008*a^4*x^4-10180*a^3*m^2*x^3-247*a^2*m^4*x^ 2+2*a*m^6*x-11808*a^3*m*x^3-1219*a^2*m^3*x^2+52*a*m^5*x-5040*a^3*x^3-3112* a^2*m^2*x^2+540*a*m^4*x+m^6-3796*a^2*m*x^2+2840*a*m^3*x+27*m^5-1680*a^2*x^ 2+7858*a*m^2*x+295*m^4+10548*a*m*x+1665*m^3+5040*a*x+5104*m^2+8028*m+5040) *x/(m+7)/(m+6)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)
Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (120) = 240\).
Time = 0.13 (sec) , antiderivative size = 540, normalized size of antiderivative = 4.50 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^3 \, dx=\frac {{\left ({\left (a^{6} c^{3} m^{6} + 21 \, a^{6} c^{3} m^{5} + 175 \, a^{6} c^{3} m^{4} + 735 \, a^{6} c^{3} m^{3} + 1624 \, a^{6} c^{3} m^{2} + 1764 \, a^{6} c^{3} m + 720 \, a^{6} c^{3}\right )} x^{7} + 2 \, {\left (a^{5} c^{3} m^{6} + 22 \, a^{5} c^{3} m^{5} + 190 \, a^{5} c^{3} m^{4} + 820 \, a^{5} c^{3} m^{3} + 1849 \, a^{5} c^{3} m^{2} + 2038 \, a^{5} c^{3} m + 840 \, a^{5} c^{3}\right )} x^{6} - {\left (a^{4} c^{3} m^{6} + 23 \, a^{4} c^{3} m^{5} + 207 \, a^{4} c^{3} m^{4} + 925 \, a^{4} c^{3} m^{3} + 2144 \, a^{4} c^{3} m^{2} + 2412 \, a^{4} c^{3} m + 1008 \, a^{4} c^{3}\right )} x^{5} - 4 \, {\left (a^{3} c^{3} m^{6} + 24 \, a^{3} c^{3} m^{5} + 226 \, a^{3} c^{3} m^{4} + 1056 \, a^{3} c^{3} m^{3} + 2545 \, a^{3} c^{3} m^{2} + 2952 \, a^{3} c^{3} m + 1260 \, a^{3} c^{3}\right )} x^{4} - {\left (a^{2} c^{3} m^{6} + 25 \, a^{2} c^{3} m^{5} + 247 \, a^{2} c^{3} m^{4} + 1219 \, a^{2} c^{3} m^{3} + 3112 \, a^{2} c^{3} m^{2} + 3796 \, a^{2} c^{3} m + 1680 \, a^{2} c^{3}\right )} x^{3} + 2 \, {\left (a c^{3} m^{6} + 26 \, a c^{3} m^{5} + 270 \, a c^{3} m^{4} + 1420 \, a c^{3} m^{3} + 3929 \, a c^{3} m^{2} + 5274 \, a c^{3} m + 2520 \, a c^{3}\right )} x^{2} + {\left (c^{3} m^{6} + 27 \, c^{3} m^{5} + 295 \, c^{3} m^{4} + 1665 \, c^{3} m^{3} + 5104 \, c^{3} m^{2} + 8028 \, c^{3} m + 5040 \, c^{3}\right )} x\right )} x^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^3,x, algorithm="fricas ")
Output:
((a^6*c^3*m^6 + 21*a^6*c^3*m^5 + 175*a^6*c^3*m^4 + 735*a^6*c^3*m^3 + 1624* a^6*c^3*m^2 + 1764*a^6*c^3*m + 720*a^6*c^3)*x^7 + 2*(a^5*c^3*m^6 + 22*a^5* c^3*m^5 + 190*a^5*c^3*m^4 + 820*a^5*c^3*m^3 + 1849*a^5*c^3*m^2 + 2038*a^5* c^3*m + 840*a^5*c^3)*x^6 - (a^4*c^3*m^6 + 23*a^4*c^3*m^5 + 207*a^4*c^3*m^4 + 925*a^4*c^3*m^3 + 2144*a^4*c^3*m^2 + 2412*a^4*c^3*m + 1008*a^4*c^3)*x^5 - 4*(a^3*c^3*m^6 + 24*a^3*c^3*m^5 + 226*a^3*c^3*m^4 + 1056*a^3*c^3*m^3 + 2545*a^3*c^3*m^2 + 2952*a^3*c^3*m + 1260*a^3*c^3)*x^4 - (a^2*c^3*m^6 + 25* a^2*c^3*m^5 + 247*a^2*c^3*m^4 + 1219*a^2*c^3*m^3 + 3112*a^2*c^3*m^2 + 3796 *a^2*c^3*m + 1680*a^2*c^3)*x^3 + 2*(a*c^3*m^6 + 26*a*c^3*m^5 + 270*a*c^3*m ^4 + 1420*a*c^3*m^3 + 3929*a*c^3*m^2 + 5274*a*c^3*m + 2520*a*c^3)*x^2 + (c ^3*m^6 + 27*c^3*m^5 + 295*c^3*m^4 + 1665*c^3*m^3 + 5104*c^3*m^2 + 8028*c^3 *m + 5040*c^3)*x)*x^m/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 1313 2*m^2 + 13068*m + 5040)
Leaf count of result is larger than twice the leaf count of optimal. 3009 vs. \(2 (105) = 210\).
Time = 0.88 (sec) , antiderivative size = 3009, normalized size of antiderivative = 25.08 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^3 \, dx=\text {Too large to display} \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*x**m*(-a**2*c*x**2+c)**3,x)
Output:
Piecewise((a**6*c**3*log(x) - 2*a**5*c**3/x + a**4*c**3/(2*x**2) + 4*a**3* c**3/(3*x**3) + a**2*c**3/(4*x**4) - 2*a*c**3/(5*x**5) - c**3/(6*x**6), Eq (m, -7)), (a**6*c**3*x + 2*a**5*c**3*log(x) + a**4*c**3/x + 2*a**3*c**3/x* *2 + a**2*c**3/(3*x**3) - a*c**3/(2*x**4) - c**3/(5*x**5), Eq(m, -6)), (a* *6*c**3*x**2/2 + 2*a**5*c**3*x - a**4*c**3*log(x) + 4*a**3*c**3/x + a**2*c **3/(2*x**2) - 2*a*c**3/(3*x**3) - c**3/(4*x**4), Eq(m, -5)), (a**6*c**3*x **3/3 + a**5*c**3*x**2 - a**4*c**3*x - 4*a**3*c**3*log(x) + a**2*c**3/x - a*c**3/x**2 - c**3/(3*x**3), Eq(m, -4)), (a**6*c**3*x**4/4 + 2*a**5*c**3*x **3/3 - a**4*c**3*x**2/2 - 4*a**3*c**3*x - a**2*c**3*log(x) - 2*a*c**3/x - c**3/(2*x**2), Eq(m, -3)), (a**6*c**3*x**5/5 + a**5*c**3*x**4/2 - a**4*c* *3*x**3/3 - 2*a**3*c**3*x**2 - a**2*c**3*x + 2*a*c**3*log(x) - c**3/x, Eq( m, -2)), (a**6*c**3*x**6/6 + 2*a**5*c**3*x**5/5 - a**4*c**3*x**4/4 - 4*a** 3*c**3*x**3/3 - a**2*c**3*x**2/2 + 2*a*c**3*x + c**3*log(x), Eq(m, -1)), ( a**6*c**3*m**6*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m** 3 + 13132*m**2 + 13068*m + 5040) + 21*a**6*c**3*m**5*x**7*x**m/(m**7 + 28* m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1 75*a**6*c**3*m**4*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769* m**3 + 13132*m**2 + 13068*m + 5040) + 735*a**6*c**3*m**3*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1624*a**6*c**3*m**2*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4...
Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (120) = 240\).
Time = 0.07 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.53 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^3 \, dx=\frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} a^{6} c^{3} x^{7} + 2 \, {\left (m^{6} + 22 \, m^{5} + 190 \, m^{4} + 820 \, m^{3} + 1849 \, m^{2} + 2038 \, m + 840\right )} a^{5} c^{3} x^{6} - {\left (m^{6} + 23 \, m^{5} + 207 \, m^{4} + 925 \, m^{3} + 2144 \, m^{2} + 2412 \, m + 1008\right )} a^{4} c^{3} x^{5} - 4 \, {\left (m^{6} + 24 \, m^{5} + 226 \, m^{4} + 1056 \, m^{3} + 2545 \, m^{2} + 2952 \, m + 1260\right )} a^{3} c^{3} x^{4} - {\left (m^{6} + 25 \, m^{5} + 247 \, m^{4} + 1219 \, m^{3} + 3112 \, m^{2} + 3796 \, m + 1680\right )} a^{2} c^{3} x^{3} + 2 \, {\left (m^{6} + 26 \, m^{5} + 270 \, m^{4} + 1420 \, m^{3} + 3929 \, m^{2} + 5274 \, m + 2520\right )} a c^{3} x^{2} + {\left (m^{6} + 27 \, m^{5} + 295 \, m^{4} + 1665 \, m^{3} + 5104 \, m^{2} + 8028 \, m + 5040\right )} c^{3} x\right )} x^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^3,x, algorithm="maxima ")
Output:
((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*a^6*c^3*x^7 + 2*(m^6 + 22*m^5 + 190*m^4 + 820*m^3 + 1849*m^2 + 2038*m + 840)*a^5*c^3*x ^6 - (m^6 + 23*m^5 + 207*m^4 + 925*m^3 + 2144*m^2 + 2412*m + 1008)*a^4*c^3 *x^5 - 4*(m^6 + 24*m^5 + 226*m^4 + 1056*m^3 + 2545*m^2 + 2952*m + 1260)*a^ 3*c^3*x^4 - (m^6 + 25*m^5 + 247*m^4 + 1219*m^3 + 3112*m^2 + 3796*m + 1680) *a^2*c^3*x^3 + 2*(m^6 + 26*m^5 + 270*m^4 + 1420*m^3 + 3929*m^2 + 5274*m + 2520)*a*c^3*x^2 + (m^6 + 27*m^5 + 295*m^4 + 1665*m^3 + 5104*m^2 + 8028*m + 5040)*c^3*x)*x^m/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^ 2 + 13068*m + 5040)
\[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^3 \, dx=\int { \frac {{\left (a^{2} c x^{2} - c\right )}^{3} {\left (a x + 1\right )}^{2} x^{m}}{a^{2} x^{2} - 1} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^3,x, algorithm="giac")
Output:
integrate((a^2*c*x^2 - c)^3*(a*x + 1)^2*x^m/(a^2*x^2 - 1), x)
Time = 24.34 (sec) , antiderivative size = 531, normalized size of antiderivative = 4.42 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^3\,x\,x^m\,\left (m^6+27\,m^5+295\,m^4+1665\,m^3+5104\,m^2+8028\,m+5040\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {2\,a\,c^3\,x^m\,x^2\,\left (m^6+26\,m^5+270\,m^4+1420\,m^3+3929\,m^2+5274\,m+2520\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {a^6\,c^3\,x^m\,x^7\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {2\,a^5\,c^3\,x^m\,x^6\,\left (m^6+22\,m^5+190\,m^4+820\,m^3+1849\,m^2+2038\,m+840\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}-\frac {a^4\,c^3\,x^m\,x^5\,\left (m^6+23\,m^5+207\,m^4+925\,m^3+2144\,m^2+2412\,m+1008\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}-\frac {4\,a^3\,c^3\,x^m\,x^4\,\left (m^6+24\,m^5+226\,m^4+1056\,m^3+2545\,m^2+2952\,m+1260\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}-\frac {a^2\,c^3\,x^m\,x^3\,\left (m^6+25\,m^5+247\,m^4+1219\,m^3+3112\,m^2+3796\,m+1680\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040} \] Input:
int(-(x^m*(c - a^2*c*x^2)^3*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
(c^3*x*x^m*(8028*m + 5104*m^2 + 1665*m^3 + 295*m^4 + 27*m^5 + m^6 + 5040)) /(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 504 0) + (2*a*c^3*x^m*x^2*(5274*m + 3929*m^2 + 1420*m^3 + 270*m^4 + 26*m^5 + m ^6 + 2520))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) + (a^6*c^3*x^m*x^7*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) + (2*a^5*c^3*x^m*x^6*(2038*m + 1849*m^2 + 820*m^3 + 190*m^4 + 22*m^5 + m^6 + 840))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) - (a^4*c^3*x^m*x^5*(2412*m + 2144*m^2 + 925*m^3 + 207*m^4 + 23*m^5 + m^6 + 1008))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) - (4*a^3*c^3*x^m*x^4*(2952*m + 2545*m^2 + 1056*m^3 + 226*m^4 + 24*m^5 + m^6 + 1260))/(13068*m + 13132*m^ 2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) - (a^2*c^3*x^m*x^ 3*(3796*m + 3112*m^2 + 1219*m^3 + 247*m^4 + 25*m^5 + m^6 + 1680))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)
Time = 0.15 (sec) , antiderivative size = 474, normalized size of antiderivative = 3.95 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^3 \, dx=\frac {x^{m} c^{3} x \left (a^{6} m^{6} x^{6}+21 a^{6} m^{5} x^{6}+175 a^{6} m^{4} x^{6}+2 a^{5} m^{6} x^{5}+735 a^{6} m^{3} x^{6}+44 a^{5} m^{5} x^{5}+1624 a^{6} m^{2} x^{6}+380 a^{5} m^{4} x^{5}-a^{4} m^{6} x^{4}+1764 a^{6} m \,x^{6}+1640 a^{5} m^{3} x^{5}-23 a^{4} m^{5} x^{4}+720 a^{6} x^{6}+3698 a^{5} m^{2} x^{5}-207 a^{4} m^{4} x^{4}-4 a^{3} m^{6} x^{3}+4076 a^{5} m \,x^{5}-925 a^{4} m^{3} x^{4}-96 a^{3} m^{5} x^{3}+1680 a^{5} x^{5}-2144 a^{4} m^{2} x^{4}-904 a^{3} m^{4} x^{3}-a^{2} m^{6} x^{2}-2412 a^{4} m \,x^{4}-4224 a^{3} m^{3} x^{3}-25 a^{2} m^{5} x^{2}-1008 a^{4} x^{4}-10180 a^{3} m^{2} x^{3}-247 a^{2} m^{4} x^{2}+2 a \,m^{6} x -11808 a^{3} m \,x^{3}-1219 a^{2} m^{3} x^{2}+52 a \,m^{5} x -5040 a^{3} x^{3}-3112 a^{2} m^{2} x^{2}+540 a \,m^{4} x +m^{6}-3796 a^{2} m \,x^{2}+2840 a \,m^{3} x +27 m^{5}-1680 a^{2} x^{2}+7858 a \,m^{2} x +295 m^{4}+10548 a m x +1665 m^{3}+5040 a x +5104 m^{2}+8028 m +5040\right )}{m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^3,x)
Output:
(x**m*c**3*x*(a**6*m**6*x**6 + 21*a**6*m**5*x**6 + 175*a**6*m**4*x**6 + 73 5*a**6*m**3*x**6 + 1624*a**6*m**2*x**6 + 1764*a**6*m*x**6 + 720*a**6*x**6 + 2*a**5*m**6*x**5 + 44*a**5*m**5*x**5 + 380*a**5*m**4*x**5 + 1640*a**5*m* *3*x**5 + 3698*a**5*m**2*x**5 + 4076*a**5*m*x**5 + 1680*a**5*x**5 - a**4*m **6*x**4 - 23*a**4*m**5*x**4 - 207*a**4*m**4*x**4 - 925*a**4*m**3*x**4 - 2 144*a**4*m**2*x**4 - 2412*a**4*m*x**4 - 1008*a**4*x**4 - 4*a**3*m**6*x**3 - 96*a**3*m**5*x**3 - 904*a**3*m**4*x**3 - 4224*a**3*m**3*x**3 - 10180*a** 3*m**2*x**3 - 11808*a**3*m*x**3 - 5040*a**3*x**3 - a**2*m**6*x**2 - 25*a** 2*m**5*x**2 - 247*a**2*m**4*x**2 - 1219*a**2*m**3*x**2 - 3112*a**2*m**2*x* *2 - 3796*a**2*m*x**2 - 1680*a**2*x**2 + 2*a*m**6*x + 52*a*m**5*x + 540*a* m**4*x + 2840*a*m**3*x + 7858*a*m**2*x + 10548*a*m*x + 5040*a*x + m**6 + 2 7*m**5 + 295*m**4 + 1665*m**3 + 5104*m**2 + 8028*m + 5040))/(m**7 + 28*m** 6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040)