Integrand size = 25, antiderivative size = 86 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {1}{12 a^4 c^3 (1-a x)^3}-\frac {1}{4 a^4 c^3 (1-a x)^2}+\frac {3}{16 a^4 c^3 (1-a x)}+\frac {1}{16 a^4 c^3 (1+a x)}+\frac {\text {arctanh}(a x)}{8 a^4 c^3} \] Output:
1/12/a^4/c^3/(-a*x+1)^3-1/4/a^4/c^3/(-a*x+1)^2+3/16/a^4/c^3/(-a*x+1)+1/16/ a^4/c^3/(a*x+1)+1/8*arctanh(a*x)/a^4/c^3
Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {-2+7 a x-6 a^2 x^2-3 a^3 x^3+3 (-1+a x)^3 (1+a x) \text {arctanh}(a x)}{24 a^4 c^3 (-1+a x)^3 (1+a x)} \] Input:
Integrate[(E^(2*ArcTanh[a*x])*x^3)/(c - a^2*c*x^2)^3,x]
Output:
(-2 + 7*a*x - 6*a^2*x^2 - 3*a^3*x^3 + 3*(-1 + a*x)^3*(1 + a*x)*ArcTanh[a*x ])/(24*a^4*c^3*(-1 + a*x)^3*(1 + a*x))
Time = 0.54 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87, 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.
\(\displaystyle \int \frac {x^3 e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\int \frac {x^3}{(1-a x)^4 (a x+1)^2}dx}{c^3}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\int \left (\frac {3}{16 a^3 (a x-1)^2}-\frac {1}{16 a^3 (a x+1)^2}+\frac {1}{2 a^3 (a x-1)^3}+\frac {1}{4 a^3 (a x-1)^4}-\frac {1}{8 a^3 \left (a^2 x^2-1\right )}\right )dx}{c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\text {arctanh}(a x)}{8 a^4}+\frac {3}{16 a^4 (1-a x)}+\frac {1}{16 a^4 (a x+1)}-\frac {1}{4 a^4 (1-a x)^2}+\frac {1}{12 a^4 (1-a x)^3}}{c^3}\) |
Input:
Int[(E^(2*ArcTanh[a*x])*x^3)/(c - a^2*c*x^2)^3,x]
Output:
(1/(12*a^4*(1 - a*x)^3) - 1/(4*a^4*(1 - a*x)^2) + 3/(16*a^4*(1 - a*x)) + 1 /(16*a^4*(1 + a*x)) + ArcTanh[a*x]/(8*a^4))/c^3
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])
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\frac {1}{16 a^{4} \left (a x +1\right )}+\frac {\ln \left (a x +1\right )}{16 a^{4}}-\frac {1}{12 a^{4} \left (a x -1\right )^{3}}-\frac {1}{4 a^{4} \left (a x -1\right )^{2}}-\frac {3}{16 a^{4} \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{16 a^{4}}}{c^{3}}\) | \(76\) |
risch | \(\frac {-\frac {x^{3}}{8 a}-\frac {x^{2}}{4 a^{2}}+\frac {7 x}{24 a^{3}}-\frac {1}{12 a^{4}}}{c^{3} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )}-\frac {\ln \left (a x -1\right )}{16 c^{3} a^{4}}+\frac {\ln \left (-a x -1\right )}{16 c^{3} a^{4}}\) | \(81\) |
norman | \(\frac {-\frac {x^{3}}{3 a c}-\frac {a \,x^{5}}{8 c}+\frac {x}{8 a^{3} c}-\frac {x^{4}}{4 c}-\frac {a^{2} x^{6}}{12 c}}{\left (a^{2} x^{2}-1\right )^{3} c^{2}}-\frac {\ln \left (a x -1\right )}{16 c^{3} a^{4}}+\frac {\ln \left (a x +1\right )}{16 c^{3} a^{4}}\) | \(94\) |
parallelrisch | \(\frac {-3 \ln \left (a x -1\right ) x^{4} a^{4}+3 \ln \left (a x +1\right ) x^{4} a^{4}-4 a^{4} x^{4}+6 a^{3} \ln \left (a x -1\right ) x^{3}-6 \ln \left (a x +1\right ) x^{3} a^{3}+2 a^{3} x^{3}-12 a^{2} x^{2}-6 a \ln \left (a x -1\right ) x +6 \ln \left (a x +1\right ) x a +6 a x +3 \ln \left (a x -1\right )-3 \ln \left (a x +1\right )}{48 a^{4} c^{3} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )}\) | \(148\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
1/c^3*(1/16/a^4/(a*x+1)+1/16*ln(a*x+1)/a^4-1/12/a^4/(a*x-1)^3-1/4/a^4/(a*x -1)^2-3/16/a^4/(a*x-1)-1/16/a^4*ln(a*x-1))
Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.43 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 14 \, a x - 3 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x - 1\right ) + 4}{48 \, {\left (a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{5} c^{3} x - a^{4} c^{3}\right )}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c)^3,x, algorithm="fricas ")
Output:
-1/48*(6*a^3*x^3 + 12*a^2*x^2 - 14*a*x - 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x - 1) + 4)/(a ^8*c^3*x^4 - 2*a^7*c^3*x^3 + 2*a^5*c^3*x - a^4*c^3)
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {- 3 a^{3} x^{3} - 6 a^{2} x^{2} + 7 a x - 2}{24 a^{8} c^{3} x^{4} - 48 a^{7} c^{3} x^{3} + 48 a^{5} c^{3} x - 24 a^{4} c^{3}} + \frac {- \frac {\log {\left (x - \frac {1}{a} \right )}}{16} + \frac {\log {\left (x + \frac {1}{a} \right )}}{16}}{a^{4} c^{3}} \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*x**3/(-a**2*c*x**2+c)**3,x)
Output:
(-3*a**3*x**3 - 6*a**2*x**2 + 7*a*x - 2)/(24*a**8*c**3*x**4 - 48*a**7*c**3 *x**3 + 48*a**5*c**3*x - 24*a**4*c**3) + (-log(x - 1/a)/16 + log(x + 1/a)/ 16)/(a**4*c**3)
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.09 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 7 \, a x + 2}{24 \, {\left (a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{5} c^{3} x - a^{4} c^{3}\right )}} + \frac {\log \left (a x + 1\right )}{16 \, a^{4} c^{3}} - \frac {\log \left (a x - 1\right )}{16 \, a^{4} c^{3}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c)^3,x, algorithm="maxima ")
Output:
-1/24*(3*a^3*x^3 + 6*a^2*x^2 - 7*a*x + 2)/(a^8*c^3*x^4 - 2*a^7*c^3*x^3 + 2 *a^5*c^3*x - a^4*c^3) + 1/16*log(a*x + 1)/(a^4*c^3) - 1/16*log(a*x - 1)/(a ^4*c^3)
Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{4} c^{3}} - \frac {\log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{4} c^{3}} - \frac {3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 7 \, a x + 2}{24 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{3} a^{4} c^{3}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c)^3,x, algorithm="giac")
Output:
1/16*log(abs(a*x + 1))/(a^4*c^3) - 1/16*log(abs(a*x - 1))/(a^4*c^3) - 1/24 *(3*a^3*x^3 + 6*a^2*x^2 - 7*a*x + 2)/((a*x + 1)*(a*x - 1)^3*a^4*c^3)
Time = 23.85 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\frac {1}{12\,a^4}-\frac {7\,x}{24\,a^3}+\frac {x^3}{8\,a}+\frac {x^2}{4\,a^2}}{-a^4\,c^3\,x^4+2\,a^3\,c^3\,x^3-2\,a\,c^3\,x+c^3}+\frac {\mathrm {atanh}\left (a\,x\right )}{8\,a^4\,c^3} \] Input:
int(-(x^3*(a*x + 1)^2)/((c - a^2*c*x^2)^3*(a^2*x^2 - 1)),x)
Output:
(1/(12*a^4) - (7*x)/(24*a^3) + x^3/(8*a) + x^2/(4*a^2))/(c^3 + 2*a^3*c^3*x ^3 - a^4*c^3*x^4 - 2*a*c^3*x) + atanh(a*x)/(8*a^4*c^3)
Time = 0.16 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.69 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {-3 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}+6 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}-6 \,\mathrm {log}\left (a x -1\right ) a x +3 \,\mathrm {log}\left (a x -1\right )+3 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}-6 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}+6 \,\mathrm {log}\left (a x +1\right ) a x -3 \,\mathrm {log}\left (a x +1\right )-3 a^{4} x^{4}-12 a^{2} x^{2}+8 a x -1}{48 a^{4} c^{3} \left (a^{4} x^{4}-2 a^{3} x^{3}+2 a x -1\right )} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c)^3,x)
Output:
( - 3*log(a*x - 1)*a**4*x**4 + 6*log(a*x - 1)*a**3*x**3 - 6*log(a*x - 1)*a *x + 3*log(a*x - 1) + 3*log(a*x + 1)*a**4*x**4 - 6*log(a*x + 1)*a**3*x**3 + 6*log(a*x + 1)*a*x - 3*log(a*x + 1) - 3*a**4*x**4 - 12*a**2*x**2 + 8*a*x - 1)/(48*a**4*c**3*(a**4*x**4 - 2*a**3*x**3 + 2*a*x - 1))