Integrand size = 22, antiderivative size = 52 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {4 c^4 (1+a x)^7}{7 a}-\frac {c^4 (1+a x)^8}{2 a}+\frac {c^4 (1+a x)^9}{9 a} \] Output:
4/7*c^4*(a*x+1)^7/a-1/2*c^4*(a*x+1)^8/a+1/9*c^4*(a*x+1)^9/a
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.60 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^4 (1+a x)^7 \left (23-35 a x+14 a^2 x^2\right )}{126 a} \] Input:
Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^4,x]
Output:
(c^4*(1 + a*x)^7*(23 - 35*a*x + 14*a^2*x^2))/(126*a)
Time = 0.56 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6717, 27, 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.
\(\displaystyle \int \left (c-a^2 c x^2\right )^4 e^{4 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle \int c^4 \left (1-a^2 x^2\right )^4 e^{4 \text {arctanh}(a x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^4 \int e^{4 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^4dx\) |
\(\Big \downarrow \) 6690 |
\(\displaystyle c^4 \int (1-a x)^2 (a x+1)^6dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle c^4 \int \left ((a x+1)^8-4 (a x+1)^7+4 (a x+1)^6\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle c^4 \left (\frac {(a x+1)^9}{9 a}-\frac {(a x+1)^8}{2 a}+\frac {4 (a x+1)^7}{7 a}\right )\) |
Input:
Int[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^4,x]
Output:
c^4*((4*(1 + a*x)^7)/(7*a) - (1 + a*x)^8/(2*a) + (1 + a*x)^9/(9*a))
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_))^(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])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.33
method | result | size |
gosper | \(\frac {c^{4} x \left (14 a^{8} x^{8}+63 a^{7} x^{7}+72 x^{6} a^{6}-84 a^{5} x^{5}-252 a^{4} x^{4}-126 a^{3} x^{3}+168 a^{2} x^{2}+252 a x +126\right )}{126}\) | \(69\) |
default | \(c^{4} \left (\frac {1}{9} a^{8} x^{9}+\frac {1}{2} a^{7} x^{8}+\frac {4}{7} a^{6} x^{7}-\frac {2}{3} a^{5} x^{6}-2 x^{5} a^{4}-a^{3} x^{4}+\frac {4}{3} a^{2} x^{3}+2 a \,x^{2}+x \right )\) | \(69\) |
risch | \(\frac {1}{9} a^{8} c^{4} x^{9}+\frac {1}{2} a^{7} c^{4} x^{8}+\frac {4}{7} a^{6} c^{4} x^{7}-\frac {2}{3} a^{5} c^{4} x^{6}-2 a^{4} c^{4} x^{5}-a^{3} c^{4} x^{4}+\frac {4}{3} a^{2} c^{4} x^{3}+2 a \,c^{4} x^{2}+c^{4} x\) | \(93\) |
parallelrisch | \(\frac {1}{9} a^{8} c^{4} x^{9}+\frac {1}{2} a^{7} c^{4} x^{8}+\frac {4}{7} a^{6} c^{4} x^{7}-\frac {2}{3} a^{5} c^{4} x^{6}-2 a^{4} c^{4} x^{5}-a^{3} c^{4} x^{4}+\frac {4}{3} a^{2} c^{4} x^{3}+2 a \,c^{4} x^{2}+c^{4} x\) | \(93\) |
orering | \(\frac {x \left (14 a^{8} x^{8}+63 a^{7} x^{7}+72 x^{6} a^{6}-84 a^{5} x^{5}-252 a^{4} x^{4}-126 a^{3} x^{3}+168 a^{2} x^{2}+252 a x +126\right ) \left (-a^{2} c \,x^{2}+c \right )^{4}}{126 \left (a x +1\right )^{4} \left (a x -1\right )^{4}}\) | \(93\) |
norman | \(\frac {-c^{4} x +a^{4} c^{4} x^{5}-a \,c^{4} x^{2}+\frac {2}{3} a^{2} c^{4} x^{3}+\frac {7}{3} a^{3} c^{4} x^{4}-\frac {4}{3} a^{5} c^{4} x^{6}-\frac {26}{21} a^{6} c^{4} x^{7}+\frac {1}{14} a^{7} c^{4} x^{8}+\frac {7}{18} a^{8} c^{4} x^{9}+\frac {1}{9} a^{9} c^{4} x^{10}}{a x -1}\) | \(112\) |
meijerg | \(-\frac {c^{4} \left (-\frac {x a \left (-308 a^{9} x^{9}-385 a^{8} x^{8}-495 a^{7} x^{7}-660 x^{6} a^{6}-924 a^{5} x^{5}-1386 a^{4} x^{4}-2310 a^{3} x^{3}-4620 a^{2} x^{2}-13860 a x +27720\right )}{2772 \left (-a x +1\right )}-10 \ln \left (-a x +1\right )\right )}{a}+\frac {3 c^{4} \left (-\frac {a x \left (-45 a^{7} x^{7}-60 x^{6} a^{6}-84 a^{5} x^{5}-126 a^{4} x^{4}-210 a^{3} x^{3}-420 a^{2} x^{2}-1260 a x +2520\right )}{315 \left (-a x +1\right )}-8 \ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{4} \left (-\frac {a x \left (-14 a^{5} x^{5}-21 a^{4} x^{4}-35 a^{3} x^{3}-70 a^{2} x^{2}-210 a x +420\right )}{70 \left (-a x +1\right )}-6 \ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{4} \left (-\frac {a x \left (-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{15 \left (-a x +1\right )}-4 \ln \left (-a x +1\right )\right )}{a}+\frac {3 c^{4} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{4} \left (\frac {a x \left (-35 a^{8} x^{8}-45 a^{7} x^{7}-60 x^{6} a^{6}-84 a^{5} x^{5}-126 a^{4} x^{4}-210 a^{3} x^{3}-420 a^{2} x^{2}-1260 a x +2520\right )}{-280 a x +280}+9 \ln \left (-a x +1\right )\right )}{a}-\frac {8 c^{4} \left (\frac {a x \left (-20 x^{6} a^{6}-28 a^{5} x^{5}-42 a^{4} x^{4}-70 a^{3} x^{3}-140 a^{2} x^{2}-420 a x +840\right )}{-120 a x +120}+7 \ln \left (-a x +1\right )\right )}{a}+\frac {12 c^{4} \left (\frac {x a \left (-3 a^{4} x^{4}-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{-12 a x +12}+5 \ln \left (-a x +1\right )\right )}{a}-\frac {8 c^{4} \left (\frac {a x \left (-2 a^{2} x^{2}-6 a x +12\right )}{-4 a x +4}+3 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{4} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{4} x}{-a x +1}\) | \(654\) |
Input:
int(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)
Output:
1/126*c^4*x*(14*a^8*x^8+63*a^7*x^7+72*a^6*x^6-84*a^5*x^5-252*a^4*x^4-126*a ^3*x^3+168*a^2*x^2+252*a*x+126)
Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.77 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{2} \, a^{7} c^{4} x^{8} + \frac {4}{7} \, a^{6} c^{4} x^{7} - \frac {2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4}{3} \, a^{2} c^{4} x^{3} + 2 \, a c^{4} x^{2} + c^{4} x \] Input:
integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^4,x, algorithm="fricas")
Output:
1/9*a^8*c^4*x^9 + 1/2*a^7*c^4*x^8 + 4/7*a^6*c^4*x^7 - 2/3*a^5*c^4*x^6 - 2* a^4*c^4*x^5 - a^3*c^4*x^4 + 4/3*a^2*c^4*x^3 + 2*a*c^4*x^2 + c^4*x
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (41) = 82\).
Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.92 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {a^{8} c^{4} x^{9}}{9} + \frac {a^{7} c^{4} x^{8}}{2} + \frac {4 a^{6} c^{4} x^{7}}{7} - \frac {2 a^{5} c^{4} x^{6}}{3} - 2 a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4 a^{2} c^{4} x^{3}}{3} + 2 a c^{4} x^{2} + c^{4} x \] Input:
integrate(1/(a*x-1)**2*(a*x+1)**2*(-a**2*c*x**2+c)**4,x)
Output:
a**8*c**4*x**9/9 + a**7*c**4*x**8/2 + 4*a**6*c**4*x**7/7 - 2*a**5*c**4*x** 6/3 - 2*a**4*c**4*x**5 - a**3*c**4*x**4 + 4*a**2*c**4*x**3/3 + 2*a*c**4*x* *2 + c**4*x
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.77 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{2} \, a^{7} c^{4} x^{8} + \frac {4}{7} \, a^{6} c^{4} x^{7} - \frac {2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4}{3} \, a^{2} c^{4} x^{3} + 2 \, a c^{4} x^{2} + c^{4} x \] Input:
integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^4,x, algorithm="maxima")
Output:
1/9*a^8*c^4*x^9 + 1/2*a^7*c^4*x^8 + 4/7*a^6*c^4*x^7 - 2/3*a^5*c^4*x^6 - 2* a^4*c^4*x^5 - a^3*c^4*x^4 + 4/3*a^2*c^4*x^3 + 2*a*c^4*x^2 + c^4*x
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.73 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {{\left (14 \, c^{4} + \frac {189 \, c^{4}}{a x - 1} + \frac {1080 \, c^{4}}{{\left (a x - 1\right )}^{2}} + \frac {3360 \, c^{4}}{{\left (a x - 1\right )}^{3}} + \frac {6048 \, c^{4}}{{\left (a x - 1\right )}^{4}} + \frac {6048 \, c^{4}}{{\left (a x - 1\right )}^{5}} + \frac {2688 \, c^{4}}{{\left (a x - 1\right )}^{6}}\right )} {\left (a x - 1\right )}^{9}}{126 \, a} \] Input:
integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^4,x, algorithm="giac")
Output:
1/126*(14*c^4 + 189*c^4/(a*x - 1) + 1080*c^4/(a*x - 1)^2 + 3360*c^4/(a*x - 1)^3 + 6048*c^4/(a*x - 1)^4 + 6048*c^4/(a*x - 1)^5 + 2688*c^4/(a*x - 1)^6 )*(a*x - 1)^9/a
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.77 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {a^8\,c^4\,x^9}{9}+\frac {a^7\,c^4\,x^8}{2}+\frac {4\,a^6\,c^4\,x^7}{7}-\frac {2\,a^5\,c^4\,x^6}{3}-2\,a^4\,c^4\,x^5-a^3\,c^4\,x^4+\frac {4\,a^2\,c^4\,x^3}{3}+2\,a\,c^4\,x^2+c^4\,x \] Input:
int(((c - a^2*c*x^2)^4*(a*x + 1)^2)/(a*x - 1)^2,x)
Output:
c^4*x + 2*a*c^4*x^2 + (4*a^2*c^4*x^3)/3 - a^3*c^4*x^4 - 2*a^4*c^4*x^5 - (2 *a^5*c^4*x^6)/3 + (4*a^6*c^4*x^7)/7 + (a^7*c^4*x^8)/2 + (a^8*c^4*x^9)/9
Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.31 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^{4} x \left (14 a^{8} x^{8}+63 a^{7} x^{7}+72 a^{6} x^{6}-84 a^{5} x^{5}-252 a^{4} x^{4}-126 a^{3} x^{3}+168 a^{2} x^{2}+252 a x +126\right )}{126} \] Input:
int(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^4,x)
Output:
(c**4*x*(14*a**8*x**8 + 63*a**7*x**7 + 72*a**6*x**6 - 84*a**5*x**5 - 252*a **4*x**4 - 126*a**3*x**3 + 168*a**2*x**2 + 252*a*x + 126))/126