Integrand size = 18, antiderivative size = 53 \[ \int e^{4 \text {arctanh}(a x)} (c-a c x)^5 \, dx=-\frac {c^5 (1-a x)^4}{a}+\frac {4 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \] Output:
-c^5*(-a*x+1)^4/a+4/5*c^5*(-a*x+1)^5/a-1/6*c^5*(-a*x+1)^6/a
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.58 \[ \int e^{4 \text {arctanh}(a x)} (c-a c x)^5 \, dx=-\frac {c^5 (-1+a x)^4 \left (11+14 a x+5 a^2 x^2\right )}{30 a} \] Input:
Integrate[E^(4*ArcTanh[a*x])*(c - a*c*x)^5,x]
Output:
-1/30*(c^5*(-1 + a*x)^4*(11 + 14*a*x + 5*a^2*x^2))/a
Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6679, 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.
\(\displaystyle \int e^{4 \text {arctanh}(a x)} (c-a c x)^5 \, dx\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle c^5 \int (1-a x)^3 (a x+1)^2dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle c^5 \int \left ((1-a x)^5-4 (1-a x)^4+4 (1-a x)^3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle c^5 \left (-\frac {(1-a x)^6}{6 a}+\frac {4 (1-a x)^5}{5 a}-\frac {(1-a x)^4}{a}\right )\) |
Input:
Int[E^(4*ArcTanh[a*x])*(c - a*c*x)^5,x]
Output:
c^5*(-((1 - a*x)^4/a) + (4*(1 - a*x)^5)/(5*a) - (1 - a*x)^6/(6*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_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[c^p Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] , x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(-\frac {\left (5 a^{5} x^{5}-6 a^{4} x^{4}-15 a^{3} x^{3}+20 a^{2} x^{2}+15 a x -30\right ) c^{5} x}{30}\) | \(45\) |
default | \(c^{5} \left (-\frac {1}{6} a^{5} x^{6}+\frac {1}{5} x^{5} a^{4}+\frac {1}{2} a^{3} x^{4}-\frac {2}{3} a^{2} x^{3}-\frac {1}{2} a \,x^{2}+x \right )\) | \(45\) |
risch | \(-\frac {1}{6} a^{5} c^{5} x^{6}+\frac {1}{5} a^{4} c^{5} x^{5}+\frac {1}{2} c^{5} a^{3} x^{4}-\frac {2}{3} c^{5} a^{2} x^{3}-\frac {1}{2} c^{5} a \,x^{2}+c^{5} x\) | \(60\) |
parallelrisch | \(-\frac {1}{6} a^{5} c^{5} x^{6}+\frac {1}{5} a^{4} c^{5} x^{5}+\frac {1}{2} c^{5} a^{3} x^{4}-\frac {2}{3} c^{5} a^{2} x^{3}-\frac {1}{2} c^{5} a \,x^{2}+c^{5} x\) | \(60\) |
orering | \(\frac {x \left (5 a^{5} x^{5}-6 a^{4} x^{4}-15 a^{3} x^{3}+20 a^{2} x^{2}+15 a x -30\right ) \left (a x +1\right )^{2} \left (-a c x +c \right )^{5}}{30 \left (a x -1\right )^{3} \left (-a^{2} x^{2}+1\right )^{2}}\) | \(77\) |
norman | \(\frac {\frac {1}{2} c^{5} a \,x^{2}-c^{5} x -\frac {13}{15} a^{4} c^{5} x^{5}+\frac {2}{3} a^{5} c^{5} x^{6}+\frac {1}{5} a^{6} c^{5} x^{7}-\frac {1}{6} a^{7} c^{5} x^{8}+\frac {5}{3} c^{5} a^{2} x^{3}-c^{5} a^{3} x^{4}}{a^{2} x^{2}-1}\) | \(95\) |
meijerg | \(\frac {2 c^{5} \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \operatorname {arctanh}\left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}-\frac {a \,c^{5} x^{2}}{2 \left (-a^{2} x^{2}+1\right )}+\frac {2 c^{5} \left (\frac {x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-70 a^{2} x^{2}+105\right )}{21 a^{6} \left (-a^{2} x^{2}+1\right )}-\frac {5 \left (-a^{2}\right )^{\frac {7}{2}} \operatorname {arctanh}\left (a x \right )}{a^{7}}\right )}{\sqrt {-a^{2}}}+\frac {3 c^{5} \left (-\frac {x^{2} a^{2} \left (-3 a^{2} x^{2}+6\right )}{3 \left (-a^{2} x^{2}+1\right )}-2 \ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {3 c^{5} \left (\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \operatorname {arctanh}\left (a x \right )}{a^{5}}\right )}{\sqrt {-a^{2}}}+\frac {2 c^{5} \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {c^{5} \left (-\frac {a^{2} x^{2} \left (-5 x^{6} a^{6}-10 a^{4} x^{4}-30 a^{2} x^{2}+60\right )}{15 \left (-a^{2} x^{2}+1\right )}-4 \ln \left (-a^{2} x^{2}+1\right )\right )}{2 a}+\frac {c^{5} \left (\frac {x \left (-a^{2}\right )^{\frac {9}{2}} \left (-18 x^{6} a^{6}-42 a^{4} x^{4}-210 a^{2} x^{2}+315\right )}{45 a^{8} \left (-a^{2} x^{2}+1\right )}-\frac {7 \left (-a^{2}\right )^{\frac {9}{2}} \operatorname {arctanh}\left (a x \right )}{a^{9}}\right )}{2 \sqrt {-a^{2}}}+\frac {2 c^{5} \left (\frac {a^{2} x^{2} \left (-2 a^{4} x^{4}-6 a^{2} x^{2}+12\right )}{-4 a^{2} x^{2}+4}+3 \ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {c^{5} \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \operatorname {arctanh}\left (a x \right )}{a}\right )}{2 \sqrt {-a^{2}}}\) | \(561\) |
Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^5,x,method=_RETURNVERBOSE)
Output:
-1/30*(5*a^5*x^5-6*a^4*x^4-15*a^3*x^3+20*a^2*x^2+15*a*x-30)*c^5*x
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int e^{4 \text {arctanh}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {1}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {1}{2} \, a c^{5} x^{2} + c^{5} x \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^5,x, algorithm="fricas")
Output:
-1/6*a^5*c^5*x^6 + 1/5*a^4*c^5*x^5 + 1/2*a^3*c^5*x^4 - 2/3*a^2*c^5*x^3 - 1 /2*a*c^5*x^2 + c^5*x
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.19 \[ \int e^{4 \text {arctanh}(a x)} (c-a c x)^5 \, dx=- \frac {a^{5} c^{5} x^{6}}{6} + \frac {a^{4} c^{5} x^{5}}{5} + \frac {a^{3} c^{5} x^{4}}{2} - \frac {2 a^{2} c^{5} x^{3}}{3} - \frac {a c^{5} x^{2}}{2} + c^{5} x \] Input:
integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a*c*x+c)**5,x)
Output:
-a**5*c**5*x**6/6 + a**4*c**5*x**5/5 + a**3*c**5*x**4/2 - 2*a**2*c**5*x**3 /3 - a*c**5*x**2/2 + c**5*x
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int e^{4 \text {arctanh}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {1}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {1}{2} \, a c^{5} x^{2} + c^{5} x \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^5,x, algorithm="maxima")
Output:
-1/6*a^5*c^5*x^6 + 1/5*a^4*c^5*x^5 + 1/2*a^3*c^5*x^4 - 2/3*a^2*c^5*x^3 - 1 /2*a*c^5*x^2 + c^5*x
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int e^{4 \text {arctanh}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {1}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {1}{2} \, a c^{5} x^{2} + c^{5} x \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^5,x, algorithm="giac")
Output:
-1/6*a^5*c^5*x^6 + 1/5*a^4*c^5*x^5 + 1/2*a^3*c^5*x^4 - 2/3*a^2*c^5*x^3 - 1 /2*a*c^5*x^2 + c^5*x
Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int e^{4 \text {arctanh}(a x)} (c-a c x)^5 \, dx=-\frac {a^5\,c^5\,x^6}{6}+\frac {a^4\,c^5\,x^5}{5}+\frac {a^3\,c^5\,x^4}{2}-\frac {2\,a^2\,c^5\,x^3}{3}-\frac {a\,c^5\,x^2}{2}+c^5\,x \] Input:
int(((c - a*c*x)^5*(a*x + 1)^4)/(a^2*x^2 - 1)^2,x)
Output:
c^5*x - (a*c^5*x^2)/2 - (2*a^2*c^5*x^3)/3 + (a^3*c^5*x^4)/2 + (a^4*c^5*x^5 )/5 - (a^5*c^5*x^6)/6
Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int e^{4 \text {arctanh}(a x)} (c-a c x)^5 \, dx=\frac {c^{5} x \left (-5 a^{5} x^{5}+6 a^{4} x^{4}+15 a^{3} x^{3}-20 a^{2} x^{2}-15 a x +30\right )}{30} \] Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^5,x)
Output:
(c**5*x*( - 5*a**5*x**5 + 6*a**4*x**4 + 15*a**3*x**3 - 20*a**2*x**2 - 15*a *x + 30))/30