Integrand size = 22, antiderivative size = 73 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {4 c^4 (1-a x)^6}{3 a}+\frac {12 c^4 (1-a x)^7}{7 a}-\frac {3 c^4 (1-a x)^8}{4 a}+\frac {c^4 (1-a x)^9}{9 a} \] Output:
-4/3*c^4*(-a*x+1)^6/a+12/7*c^4*(-a*x+1)^7/a-3/4*c^4*(-a*x+1)^8/a+1/9*c^4*( -a*x+1)^9/a
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.53 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {c^4 (-1+a x)^6 \left (65+138 a x+105 a^2 x^2+28 a^3 x^3\right )}{252 a} \] Input:
Integrate[(c - a^2*c*x^2)^4/E^(2*ArcTanh[a*x]),x]
Output:
-1/252*(c^4*(-1 + a*x)^6*(65 + 138*a*x + 105*a^2*x^2 + 28*a^3*x^3))/a
Time = 0.45 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6690, 49, 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.
\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx\) |
\(\Big \downarrow \) 6690 |
\(\displaystyle c^4 \int (1-a x)^5 (a x+1)^3dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle c^4 \int \left (-(1-a x)^8+6 (1-a x)^7-12 (1-a x)^6+8 (1-a x)^5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle c^4 \left (\frac {(1-a x)^9}{9 a}-\frac {3 (1-a x)^8}{4 a}+\frac {12 (1-a x)^7}{7 a}-\frac {4 (1-a x)^6}{3 a}\right )\) |
Input:
Int[(c - a^2*c*x^2)^4/E^(2*ArcTanh[a*x]),x]
Output:
c^4*((-4*(1 - a*x)^6)/(3*a) + (12*(1 - a*x)^7)/(7*a) - (3*(1 - a*x)^8)/(4* a) + (1 - a*x)^9/(9*a))
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a , c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(-\frac {c^{4} x \left (28 a^{8} x^{8}-63 a^{7} x^{7}-72 x^{6} a^{6}+252 a^{5} x^{5}-378 a^{3} x^{3}+168 a^{2} x^{2}+252 a x -252\right )}{252}\) | \(61\) |
default | \(c^{4} \left (-\frac {1}{9} a^{8} x^{9}+\frac {1}{4} a^{7} x^{8}+\frac {2}{7} a^{6} x^{7}-a^{5} x^{6}+\frac {3}{2} a^{3} x^{4}-\frac {2}{3} a^{2} x^{3}-a \,x^{2}+x \right )\) | \(61\) |
risch | \(-\frac {1}{9} a^{8} c^{4} x^{9}+\frac {1}{4} a^{7} c^{4} x^{8}+\frac {2}{7} c^{4} a^{6} x^{7}-a^{5} c^{4} x^{6}+\frac {3}{2} a^{3} c^{4} x^{4}-\frac {2}{3} a^{2} c^{4} x^{3}-a \,c^{4} x^{2}+c^{4} x\) | \(82\) |
parallelrisch | \(-\frac {1}{9} a^{8} c^{4} x^{9}+\frac {1}{4} a^{7} c^{4} x^{8}+\frac {2}{7} c^{4} a^{6} x^{7}-a^{5} c^{4} x^{6}+\frac {3}{2} a^{3} c^{4} x^{4}-\frac {2}{3} a^{2} c^{4} x^{3}-a \,c^{4} x^{2}+c^{4} x\) | \(82\) |
orering | \(\frac {x \left (28 a^{8} x^{8}-63 a^{7} x^{7}-72 x^{6} a^{6}+252 a^{5} x^{5}-378 a^{3} x^{3}+168 a^{2} x^{2}+252 a x -252\right ) \left (-a^{2} c \,x^{2}+c \right )^{4} \left (-a^{2} x^{2}+1\right )}{252 \left (a x +1\right )^{5} \left (a x -1\right )^{5}}\) | \(95\) |
norman | \(\frac {c^{4} x -\frac {5}{3} a^{2} c^{4} x^{3}+\frac {5}{6} a^{3} c^{4} x^{4}+\frac {3}{2} a^{4} c^{4} x^{5}-a^{5} c^{4} x^{6}+\frac {15}{28} a^{7} c^{4} x^{8}+\frac {5}{36} a^{8} c^{4} x^{9}-\frac {1}{9} a^{9} c^{4} x^{10}-\frac {5}{7} c^{4} a^{6} x^{7}}{a x +1}\) | \(103\) |
meijerg | \(-\frac {c^{4} \left (\frac {a x \left (308 a^{9} x^{9}-385 a^{8} x^{8}+495 a^{7} x^{7}-660 x^{6} a^{6}+924 a^{5} x^{5}-1386 a^{4} x^{4}+2310 a^{3} x^{3}-4620 a^{2} x^{2}+13860 a x +27720\right )}{2772 a x +2772}-10 \ln \left (a x +1\right )\right )}{a}+\frac {5 c^{4} \left (\frac {a x \left (45 a^{7} x^{7}-60 x^{6} a^{6}+84 a^{5} x^{5}-126 a^{4} x^{4}+210 a^{3} x^{3}-420 a^{2} x^{2}+1260 a x +2520\right )}{315 a x +315}-8 \ln \left (a x +1\right )\right )}{a}-\frac {10 c^{4} \left (\frac {a x \left (14 a^{5} x^{5}-21 a^{4} x^{4}+35 a^{3} x^{3}-70 a^{2} x^{2}+210 a x +420\right )}{70 a x +70}-6 \ln \left (a x +1\right )\right )}{a}+\frac {10 c^{4} \left (\frac {a x \left (5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{15 a x +15}-4 \ln \left (a x +1\right )\right )}{a}-\frac {5 c^{4} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}+\frac {c^{4} x}{a x +1}\) | \(344\) |
Input:
int((-a^2*c*x^2+c)^4/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)
Output:
-1/252*c^4*x*(28*a^8*x^8-63*a^7*x^7-72*a^6*x^6+252*a^5*x^5-378*a^3*x^3+168 *a^2*x^2+252*a*x-252)
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.11 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{4} \, a^{7} c^{4} x^{8} + \frac {2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac {3}{2} \, a^{3} c^{4} x^{4} - \frac {2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} + c^{4} x \] Input:
integrate((-a^2*c*x^2+c)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas")
Output:
-1/9*a^8*c^4*x^9 + 1/4*a^7*c^4*x^8 + 2/7*a^6*c^4*x^7 - a^5*c^4*x^6 + 3/2*a ^3*c^4*x^4 - 2/3*a^2*c^4*x^3 - a*c^4*x^2 + c^4*x
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=- \frac {a^{8} c^{4} x^{9}}{9} + \frac {a^{7} c^{4} x^{8}}{4} + \frac {2 a^{6} c^{4} x^{7}}{7} - a^{5} c^{4} x^{6} + \frac {3 a^{3} c^{4} x^{4}}{2} - \frac {2 a^{2} c^{4} x^{3}}{3} - a c^{4} x^{2} + c^{4} x \] Input:
integrate((-a**2*c*x**2+c)**4/(a*x+1)**2*(-a**2*x**2+1),x)
Output:
-a**8*c**4*x**9/9 + a**7*c**4*x**8/4 + 2*a**6*c**4*x**7/7 - a**5*c**4*x**6 + 3*a**3*c**4*x**4/2 - 2*a**2*c**4*x**3/3 - a*c**4*x**2 + c**4*x
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.11 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{4} \, a^{7} c^{4} x^{8} + \frac {2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac {3}{2} \, a^{3} c^{4} x^{4} - \frac {2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} + c^{4} x \] Input:
integrate((-a^2*c*x^2+c)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima")
Output:
-1/9*a^8*c^4*x^9 + 1/4*a^7*c^4*x^8 + 2/7*a^6*c^4*x^7 - a^5*c^4*x^6 + 3/2*a ^3*c^4*x^4 - 2/3*a^2*c^4*x^3 - a*c^4*x^2 + c^4*x
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.07 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {{\left (28 \, c^{4} - \frac {315 \, c^{4}}{a x + 1} + \frac {1440 \, c^{4}}{{\left (a x + 1\right )}^{2}} - \frac {3360 \, c^{4}}{{\left (a x + 1\right )}^{3}} + \frac {4032 \, c^{4}}{{\left (a x + 1\right )}^{4}} - \frac {2016 \, c^{4}}{{\left (a x + 1\right )}^{5}}\right )} {\left (a x + 1\right )}^{9}}{252 \, a} \] Input:
integrate((-a^2*c*x^2+c)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")
Output:
-1/252*(28*c^4 - 315*c^4/(a*x + 1) + 1440*c^4/(a*x + 1)^2 - 3360*c^4/(a*x + 1)^3 + 4032*c^4/(a*x + 1)^4 - 2016*c^4/(a*x + 1)^5)*(a*x + 1)^9/a
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.11 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {a^8\,c^4\,x^9}{9}+\frac {a^7\,c^4\,x^8}{4}+\frac {2\,a^6\,c^4\,x^7}{7}-a^5\,c^4\,x^6+\frac {3\,a^3\,c^4\,x^4}{2}-\frac {2\,a^2\,c^4\,x^3}{3}-a\,c^4\,x^2+c^4\,x \] Input:
int(-((c - a^2*c*x^2)^4*(a^2*x^2 - 1))/(a*x + 1)^2,x)
Output:
c^4*x - a*c^4*x^2 - (2*a^2*c^4*x^3)/3 + (3*a^3*c^4*x^4)/2 - a^5*c^4*x^6 + (2*a^6*c^4*x^7)/7 + (a^7*c^4*x^8)/4 - (a^8*c^4*x^9)/9
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^{4} x \left (-28 a^{8} x^{8}+63 a^{7} x^{7}+72 a^{6} x^{6}-252 a^{5} x^{5}+378 a^{3} x^{3}-168 a^{2} x^{2}-252 a x +252\right )}{252} \] Input:
int((-a^2*c*x^2+c)^4/(a*x+1)^2*(-a^2*x^2+1),x)
Output:
(c**4*x*( - 28*a**8*x**8 + 63*a**7*x**7 + 72*a**6*x**6 - 252*a**5*x**5 + 3 78*a**3*x**3 - 168*a**2*x**2 - 252*a*x + 252))/252