Integrand size = 25, antiderivative size = 67 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^2 x^{1+m}}{1+m}+\frac {2 a c^2 x^{2+m}}{2+m}-\frac {2 a^3 c^2 x^{4+m}}{4+m}-\frac {a^4 c^2 x^{5+m}}{5+m} \] Output:
c^2*x^(1+m)/(1+m)+2*a*c^2*x^(2+m)/(2+m)-2*a^3*c^2*x^(4+m)/(4+m)-a^4*c^2*x^ (5+m)/(5+m)
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.03 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^2 x^{1+m} \left (-(1+a x)^4+2 (3+m) \left (\frac {1}{1+m}+\frac {3 a x}{2+m}+\frac {3 a^2 x^2}{3+m}+\frac {a^3 x^3}{4+m}\right )\right )}{5+m} \] Input:
Integrate[E^(2*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^2,x]
Output:
(c^2*x^(1 + m)*(-(1 + a*x)^4 + 2*(3 + m)*((1 + m)^(-1) + (3*a*x)/(2 + m) + (3*a^2*x^2)/(3 + m) + (a^3*x^3)/(4 + m))))/(5 + m)
Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6700, 84, 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.
| \(\displaystyle \int x^m e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 6700 |
| \(\displaystyle c^2 \int x^m (1-a x) (a x+1)^3dx\) |
| \(\Big \downarrow \) 84 |
| \(\displaystyle c^2 \int \left (x^m+2 a x^{m+1}-2 a^3 x^{m+3}-a^4 x^{m+4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c^2 \left (-\frac {a^4 x^{m+5}}{m+5}-\frac {2 a^3 x^{m+4}}{m+4}+\frac {2 a x^{m+2}}{m+2}+\frac {x^{m+1}}{m+1}\right )\) |
Input:
Int[E^(2*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^2,x]
Output:
c^2*(x^(1 + m)/(1 + m) + (2*a*x^(2 + m))/(2 + m) - (2*a^3*x^(4 + m))/(4 + m) - (a^4*x^(5 + m))/(5 + m))
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && !(ILtQ[n + p + 2, 0 ] && GtQ[n + 2*p, 0])
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])
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.16
| method | result | size |
| norman | \(\frac {c^{2} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {2 a \,c^{2} x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}-\frac {2 a^{3} c^{2} x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}-\frac {a^{4} c^{2} x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}\) | \(78\) |
| risch | \(-\frac {c^{2} x^{m} \left (a^{4} x^{4} m^{3}+7 a^{4} x^{4} m^{2}+14 a^{4} x^{4} m +2 a^{3} m^{3} x^{3}+8 a^{4} x^{4}+16 a^{3} m^{2} x^{3}+34 a^{3} m \,x^{3}+20 a^{3} x^{3}-2 a \,m^{3} x -20 a x \,m^{2}-58 a m x -m^{3}-40 a x -11 m^{2}-38 m -40\right ) x}{\left (5+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(145\) |
| gosper | \(-\frac {x^{1+m} c^{2} \left (a^{4} x^{4} m^{3}+7 a^{4} x^{4} m^{2}+14 a^{4} x^{4} m +2 a^{3} m^{3} x^{3}+8 a^{4} x^{4}+16 a^{3} m^{2} x^{3}+34 a^{3} m \,x^{3}+20 a^{3} x^{3}-2 a \,m^{3} x -20 a x \,m^{2}-58 a m x -m^{3}-40 a x -11 m^{2}-38 m -40\right )}{\left (1+m \right ) \left (2+m \right ) \left (4+m \right ) \left (5+m \right )}\) | \(146\) |
| orering | \(\frac {\left (a^{4} x^{4} m^{3}+7 a^{4} x^{4} m^{2}+14 a^{4} x^{4} m +2 a^{3} m^{3} x^{3}+8 a^{4} x^{4}+16 a^{3} m^{2} x^{3}+34 a^{3} m \,x^{3}+20 a^{3} x^{3}-2 a \,m^{3} x -20 a x \,m^{2}-58 a m x -m^{3}-40 a x -11 m^{2}-38 m -40\right ) x \,x^{m} \left (-a^{2} c \,x^{2}+c \right )^{2}}{\left (5+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) \left (a x +1\right ) \left (a x -1\right ) \left (-a^{2} x^{2}+1\right )}\) | \(180\) |
| parallelrisch | \(-\frac {x^{5} x^{m} a^{4} c^{2} m^{3}+7 x^{5} x^{m} a^{4} c^{2} m^{2}+14 x^{5} x^{m} a^{4} c^{2} m +2 x^{4} x^{m} a^{3} c^{2} m^{3}+8 x^{5} x^{m} a^{4} c^{2}+16 x^{4} x^{m} a^{3} c^{2} m^{2}+34 x^{4} x^{m} a^{3} c^{2} m +20 x^{4} x^{m} a^{3} c^{2}-2 x^{2} x^{m} a \,c^{2} m^{3}-20 x^{2} x^{m} a \,c^{2} m^{2}-58 x^{2} x^{m} a \,c^{2} m -x \,x^{m} c^{2} m^{3}-40 x^{2} x^{m} a \,c^{2}-11 x \,x^{m} c^{2} m^{2}-38 x \,x^{m} c^{2} m -40 c^{2} x^{m} x}{\left (5+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(247\) |
| meijerg | \(-\frac {\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} c^{2} \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 )}{2}-\frac {\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} c^{2} \left (-\frac {2 x^{1+m} \left (-a^{2}\right )^{\frac {5}{2}+\frac {m}{2}} \left (a^{2} m \,x^{2}+a^{2} x^{2}+m +3\right )}{a^{4} \left (3+m \right ) \left (1+m \right )}+\frac {x^{1+m} \left (-a^{2}\right )^{\frac {5}{2}+\frac {m}{2}} \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{a^{4}}\right )}{2}+\frac {\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} c^{2} \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 )}{2}-\frac {\left (-a^{2}\right )^{-\frac {m}{2}} c^{2} \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 {2 \left (-a^{2}\right )^{-\frac {m}{2}} c^{2} \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^{2} \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^{2} 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}\) | \(635\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
c^2/(1+m)*x*exp(m*ln(x))+2*a*c^2/(2+m)*x^2*exp(m*ln(x))-2*a^3*c^2/(4+m)*x^ 4*exp(m*ln(x))-a^4*c^2/(5+m)*x^5*exp(m*ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (67) = 134\).
Time = 0.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.67 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx=-\frac {{\left ({\left (a^{4} c^{2} m^{3} + 7 \, a^{4} c^{2} m^{2} + 14 \, a^{4} c^{2} m + 8 \, a^{4} c^{2}\right )} x^{5} + 2 \, {\left (a^{3} c^{2} m^{3} + 8 \, a^{3} c^{2} m^{2} + 17 \, a^{3} c^{2} m + 10 \, a^{3} c^{2}\right )} x^{4} - 2 \, {\left (a c^{2} m^{3} + 10 \, a c^{2} m^{2} + 29 \, a c^{2} m + 20 \, a c^{2}\right )} x^{2} - {\left (c^{2} m^{3} + 11 \, c^{2} m^{2} + 38 \, c^{2} m + 40 \, c^{2}\right )} x\right )} x^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^2,x, algorithm="fricas ")
Output:
-((a^4*c^2*m^3 + 7*a^4*c^2*m^2 + 14*a^4*c^2*m + 8*a^4*c^2)*x^5 + 2*(a^3*c^ 2*m^3 + 8*a^3*c^2*m^2 + 17*a^3*c^2*m + 10*a^3*c^2)*x^4 - 2*(a*c^2*m^3 + 10 *a*c^2*m^2 + 29*a*c^2*m + 20*a*c^2)*x^2 - (c^2*m^3 + 11*c^2*m^2 + 38*c^2*m + 40*c^2)*x)*x^m/(m^4 + 12*m^3 + 49*m^2 + 78*m + 40)
Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (58) = 116\).
Time = 0.51 (sec) , antiderivative size = 706, normalized size of antiderivative = 10.54 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx=\begin {cases} - a^{4} c^{2} \log {\left (x \right )} + \frac {2 a^{3} c^{2}}{x} - \frac {2 a c^{2}}{3 x^{3}} - \frac {c^{2}}{4 x^{4}} & \text {for}\: m = -5 \\- a^{4} c^{2} x - 2 a^{3} c^{2} \log {\left (x \right )} - \frac {a c^{2}}{x^{2}} - \frac {c^{2}}{3 x^{3}} & \text {for}\: m = -4 \\- \frac {a^{4} c^{2} x^{3}}{3} - a^{3} c^{2} x^{2} + 2 a c^{2} \log {\left (x \right )} - \frac {c^{2}}{x} & \text {for}\: m = -2 \\- \frac {a^{4} c^{2} x^{4}}{4} - \frac {2 a^{3} c^{2} x^{3}}{3} + 2 a c^{2} x + c^{2} \log {\left (x \right )} & \text {for}\: m = -1 \\- \frac {a^{4} c^{2} m^{3} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {7 a^{4} c^{2} m^{2} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {14 a^{4} c^{2} m x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {8 a^{4} c^{2} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {2 a^{3} c^{2} m^{3} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {16 a^{3} c^{2} m^{2} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {34 a^{3} c^{2} m x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {20 a^{3} c^{2} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {2 a c^{2} m^{3} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {20 a c^{2} m^{2} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {58 a c^{2} m x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {40 a c^{2} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {c^{2} m^{3} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {11 c^{2} m^{2} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {38 c^{2} m x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {40 c^{2} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} & \text {otherwise} \end {cases} \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*x**m*(-a**2*c*x**2+c)**2,x)
Output:
Piecewise((-a**4*c**2*log(x) + 2*a**3*c**2/x - 2*a*c**2/(3*x**3) - c**2/(4 *x**4), Eq(m, -5)), (-a**4*c**2*x - 2*a**3*c**2*log(x) - a*c**2/x**2 - c** 2/(3*x**3), Eq(m, -4)), (-a**4*c**2*x**3/3 - a**3*c**2*x**2 + 2*a*c**2*log (x) - c**2/x, Eq(m, -2)), (-a**4*c**2*x**4/4 - 2*a**3*c**2*x**3/3 + 2*a*c* *2*x + c**2*log(x), Eq(m, -1)), (-a**4*c**2*m**3*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 7*a**4*c**2*m**2*x**5*x**m/(m**4 + 12*m**3 + 49* m**2 + 78*m + 40) - 14*a**4*c**2*m*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 7 8*m + 40) - 8*a**4*c**2*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 2*a**3*c**2*m**3*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 16*a* *3*c**2*m**2*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 34*a**3*c* *2*m*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 20*a**3*c**2*x**4* x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 2*a*c**2*m**3*x**2*x**m/(m** 4 + 12*m**3 + 49*m**2 + 78*m + 40) + 20*a*c**2*m**2*x**2*x**m/(m**4 + 12*m **3 + 49*m**2 + 78*m + 40) + 58*a*c**2*m*x**2*x**m/(m**4 + 12*m**3 + 49*m* *2 + 78*m + 40) + 40*a*c**2*x**2*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 4 0) + c**2*m**3*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 11*c**2*m** 2*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 38*c**2*m*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 40*c**2*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40), True))
Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.70 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx=-\frac {{\left ({\left (m^{3} + 7 \, m^{2} + 14 \, m + 8\right )} a^{4} c^{2} x^{5} + 2 \, {\left (m^{3} + 8 \, m^{2} + 17 \, m + 10\right )} a^{3} c^{2} x^{4} - 2 \, {\left (m^{3} + 10 \, m^{2} + 29 \, m + 20\right )} a c^{2} x^{2} - {\left (m^{3} + 11 \, m^{2} + 38 \, m + 40\right )} c^{2} x\right )} x^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^2,x, algorithm="maxima ")
Output:
-((m^3 + 7*m^2 + 14*m + 8)*a^4*c^2*x^5 + 2*(m^3 + 8*m^2 + 17*m + 10)*a^3*c ^2*x^4 - 2*(m^3 + 10*m^2 + 29*m + 20)*a*c^2*x^2 - (m^3 + 11*m^2 + 38*m + 4 0)*c^2*x)*x^m/(m^4 + 12*m^3 + 49*m^2 + 78*m + 40)
\[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{2} {\left (a x + 1\right )}^{2} x^{m}}{a^{2} x^{2} - 1} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^2,x, algorithm="giac")
Output:
integrate(-(a^2*c*x^2 - c)^2*(a*x + 1)^2*x^m/(a^2*x^2 - 1), x)
Time = 25.88 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.58 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx=x^m\,\left (\frac {c^2\,x\,\left (m^3+11\,m^2+38\,m+40\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}+\frac {2\,a\,c^2\,x^2\,\left (m^3+10\,m^2+29\,m+20\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}-\frac {a^4\,c^2\,x^5\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}-\frac {2\,a^3\,c^2\,x^4\,\left (m^3+8\,m^2+17\,m+10\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}\right ) \] Input:
int(-(x^m*(c - a^2*c*x^2)^2*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
x^m*((c^2*x*(38*m + 11*m^2 + m^3 + 40))/(78*m + 49*m^2 + 12*m^3 + m^4 + 40 ) + (2*a*c^2*x^2*(29*m + 10*m^2 + m^3 + 20))/(78*m + 49*m^2 + 12*m^3 + m^4 + 40) - (a^4*c^2*x^5*(14*m + 7*m^2 + m^3 + 8))/(78*m + 49*m^2 + 12*m^3 + m^4 + 40) - (2*a^3*c^2*x^4*(17*m + 8*m^2 + m^3 + 10))/(78*m + 49*m^2 + 12* m^3 + m^4 + 40))
Time = 0.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.12 \[ \int e^{2 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx=\frac {x^{m} c^{2} x \left (-a^{4} m^{3} x^{4}-7 a^{4} m^{2} x^{4}-14 a^{4} m \,x^{4}-2 a^{3} m^{3} x^{3}-8 a^{4} x^{4}-16 a^{3} m^{2} x^{3}-34 a^{3} m \,x^{3}-20 a^{3} x^{3}+2 a \,m^{3} x +20 a \,m^{2} x +58 a m x +m^{3}+40 a x +11 m^{2}+38 m +40\right )}{m^{4}+12 m^{3}+49 m^{2}+78 m +40} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^2,x)
Output:
(x**m*c**2*x*( - a**4*m**3*x**4 - 7*a**4*m**2*x**4 - 14*a**4*m*x**4 - 8*a* *4*x**4 - 2*a**3*m**3*x**3 - 16*a**3*m**2*x**3 - 34*a**3*m*x**3 - 20*a**3* x**3 + 2*a*m**3*x + 20*a*m**2*x + 58*a*m*x + 40*a*x + m**3 + 11*m**2 + 38* m + 40))/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40)