Integrand size = 22, antiderivative size = 66 \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=\frac {c^5 (1+a x)^8}{a}-\frac {4 c^5 (1+a x)^9}{3 a}+\frac {3 c^5 (1+a x)^{10}}{5 a}-\frac {c^5 (1+a x)^{11}}{11 a} \] Output:
c^5*(a*x+1)^8/a-4/3*c^5*(a*x+1)^9/a+3/5*c^5*(a*x+1)^10/a-1/11*c^5*(a*x+1)^ 11/a
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.59 \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {c^5 (1+a x)^8 \left (-29+67 a x-54 a^2 x^2+15 a^3 x^3\right )}{165 a} \] Input:
Integrate[E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^5,x]
Output:
-1/165*(c^5*(1 + a*x)^8*(-29 + 67*a*x - 54*a^2*x^2 + 15*a^3*x^3))/a
Time = 0.47 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.
\(\displaystyle \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx\) |
\(\Big \downarrow \) 6690 |
\(\displaystyle c^5 \int (1-a x)^3 (a x+1)^7dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle c^5 \int \left (-(a x+1)^{10}+6 (a x+1)^9-12 (a x+1)^8+8 (a x+1)^7\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle c^5 \left (-\frac {(a x+1)^{11}}{11 a}+\frac {3 (a x+1)^{10}}{5 a}-\frac {4 (a x+1)^9}{3 a}+\frac {(a x+1)^8}{a}\right )\) |
Input:
Int[E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^5,x]
Output:
c^5*((1 + a*x)^8/a - (4*(1 + a*x)^9)/(3*a) + (3*(1 + a*x)^10)/(5*a) - (1 + a*x)^11/(11*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 = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14
method | result | size |
default | \(c^{5} \left (-\frac {1}{11} a^{10} x^{11}-\frac {2}{5} a^{9} x^{10}-\frac {1}{3} a^{8} x^{9}+a^{7} x^{8}+2 a^{6} x^{7}-\frac {14}{5} x^{5} a^{4}-2 a^{3} x^{4}+a^{2} x^{3}+2 a \,x^{2}+x \right )\) | \(75\) |
gosper | \(-\frac {c^{5} x \left (15 a^{10} x^{10}+66 a^{9} x^{9}+55 a^{8} x^{8}-165 a^{7} x^{7}-330 x^{6} a^{6}+462 a^{4} x^{4}+330 a^{3} x^{3}-165 a^{2} x^{2}-330 a x -165\right )}{165}\) | \(77\) |
risch | \(-\frac {1}{11} a^{10} c^{5} x^{11}-\frac {2}{5} a^{9} c^{5} x^{10}-\frac {1}{3} a^{8} c^{5} x^{9}+a^{7} c^{5} x^{8}+2 a^{6} c^{5} x^{7}-\frac {14}{5} a^{4} c^{5} x^{5}-2 a^{3} c^{5} x^{4}+c^{5} a^{2} x^{3}+2 a \,c^{5} x^{2}+c^{5} x\) | \(102\) |
parallelrisch | \(-\frac {1}{11} a^{10} c^{5} x^{11}-\frac {2}{5} a^{9} c^{5} x^{10}-\frac {1}{3} a^{8} c^{5} x^{9}+a^{7} c^{5} x^{8}+2 a^{6} c^{5} x^{7}-\frac {14}{5} a^{4} c^{5} x^{5}-2 a^{3} c^{5} x^{4}+c^{5} a^{2} x^{3}+2 a \,c^{5} x^{2}+c^{5} x\) | \(102\) |
orering | \(\frac {x \left (15 a^{10} x^{10}+66 a^{9} x^{9}+55 a^{8} x^{8}-165 a^{7} x^{7}-330 x^{6} a^{6}+462 a^{4} x^{4}+330 a^{3} x^{3}-165 a^{2} x^{2}-330 a x -165\right ) \left (-a^{2} c \,x^{2}+c \right )^{5}}{165 \left (a x -1\right )^{3} \left (a x +1\right )^{3} \left (-a^{2} x^{2}+1\right )^{2}}\) | \(113\) |
norman | \(\frac {-2 a \,c^{5} x^{2}-c^{5} x +4 a^{3} c^{5} x^{4}+\frac {19}{5} a^{4} c^{5} x^{5}-2 a^{5} c^{5} x^{6}-\frac {24}{5} a^{6} c^{5} x^{7}-a^{7} c^{5} x^{8}+\frac {7}{3} a^{8} c^{5} x^{9}+\frac {7}{5} a^{9} c^{5} x^{10}-\frac {8}{33} a^{10} c^{5} x^{11}-\frac {2}{5} a^{11} c^{5} x^{12}-\frac {1}{11} a^{12} c^{5} x^{13}}{a^{2} x^{2}-1}\) | \(139\) |
meijerg | \(-\frac {45 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 )}{2 \sqrt {-a^{2}}}+\frac {8 c^{5} \left (\frac {a^{2} x^{2} \left (-3 a^{8} x^{8}-5 x^{6} a^{6}-10 a^{4} x^{4}-30 a^{2} x^{2}+60\right )}{-12 a^{2} x^{2}+12}+5 \ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {10 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 )}{a}+\frac {2 a \,c^{5} x^{2}}{-a^{2} x^{2}+1}-\frac {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 )}{2 \sqrt {-a^{2}}}-\frac {19 c^{5} \left (\frac {x \left (-a^{2}\right )^{\frac {11}{2}} \left (-110 a^{8} x^{8}-198 x^{6} a^{6}-462 a^{4} x^{4}-2310 a^{2} x^{2}+3465\right )}{385 a^{10} \left (-a^{2} x^{2}+1\right )}-\frac {9 \left (-a^{2}\right )^{\frac {11}{2}} \operatorname {arctanh}\left (a x \right )}{a^{11}}\right )}{2 \sqrt {-a^{2}}}-\frac {c^{5} \left (\frac {x \left (-a^{2}\right )^{\frac {13}{2}} \left (-910 a^{10} x^{10}-1430 a^{8} x^{8}-2574 x^{6} a^{6}-6006 a^{4} x^{4}-30030 a^{2} x^{2}+45045\right )}{4095 a^{12} \left (-a^{2} x^{2}+1\right )}-\frac {11 \left (-a^{2}\right )^{\frac {13}{2}} \operatorname {arctanh}\left (a x \right )}{a^{13}}\right )}{2 \sqrt {-a^{2}}}-\frac {19 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 )}{2 \sqrt {-a^{2}}}-\frac {10 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 {8 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 {x \left (-a^{2}\right )^{\frac {15}{2}} \left (-630 a^{12} x^{12}-910 a^{10} x^{10}-1430 a^{8} x^{8}-2574 x^{6} a^{6}-6006 a^{4} x^{4}-30030 a^{2} x^{2}+45045\right )}{3465 a^{14} \left (-a^{2} x^{2}+1\right )}-\frac {13 \left (-a^{2}\right )^{\frac {15}{2}} \operatorname {arctanh}\left (a x \right )}{a^{15}}\right )}{2 \sqrt {-a^{2}}}-\frac {45 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 (-14 a^{10} x^{10}-21 a^{8} x^{8}-35 x^{6} a^{6}-70 a^{4} x^{4}-210 a^{2} x^{2}+420\right )}{70 \left (-a^{2} x^{2}+1\right )}-6 \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}}}\) | \(949\) |
Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^5,x,method=_RETURNVERBOSE)
Output:
c^5*(-1/11*a^10*x^11-2/5*a^9*x^10-1/3*a^8*x^9+a^7*x^8+2*a^6*x^7-14/5*x^5*a ^4-2*a^3*x^4+a^2*x^3+2*a*x^2+x)
Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.53 \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {2}{5} \, a^{9} c^{5} x^{10} - \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac {14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^5,x, algorithm="fricas")
Output:
-1/11*a^10*c^5*x^11 - 2/5*a^9*c^5*x^10 - 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 + 2 *a^6*c^5*x^7 - 14/5*a^4*c^5*x^5 - 2*a^3*c^5*x^4 + a^2*c^5*x^3 + 2*a*c^5*x^ 2 + c^5*x
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (54) = 108\).
Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.65 \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=- \frac {a^{10} c^{5} x^{11}}{11} - \frac {2 a^{9} c^{5} x^{10}}{5} - \frac {a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} + 2 a^{6} c^{5} x^{7} - \frac {14 a^{4} c^{5} x^{5}}{5} - 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 a c^{5} x^{2} + c^{5} x \] Input:
integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a**2*c*x**2+c)**5,x)
Output:
-a**10*c**5*x**11/11 - 2*a**9*c**5*x**10/5 - a**8*c**5*x**9/3 + a**7*c**5* x**8 + 2*a**6*c**5*x**7 - 14*a**4*c**5*x**5/5 - 2*a**3*c**5*x**4 + a**2*c* *5*x**3 + 2*a*c**5*x**2 + c**5*x
Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.53 \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {2}{5} \, a^{9} c^{5} x^{10} - \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac {14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^5,x, algorithm="maxima")
Output:
-1/11*a^10*c^5*x^11 - 2/5*a^9*c^5*x^10 - 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 + 2 *a^6*c^5*x^7 - 14/5*a^4*c^5*x^5 - 2*a^3*c^5*x^4 + a^2*c^5*x^3 + 2*a*c^5*x^ 2 + c^5*x
Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.53 \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {2}{5} \, a^{9} c^{5} x^{10} - \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac {14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^5,x, algorithm="giac")
Output:
-1/11*a^10*c^5*x^11 - 2/5*a^9*c^5*x^10 - 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 + 2 *a^6*c^5*x^7 - 14/5*a^4*c^5*x^5 - 2*a^3*c^5*x^4 + a^2*c^5*x^3 + 2*a*c^5*x^ 2 + c^5*x
Time = 23.25 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.53 \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {a^{10}\,c^5\,x^{11}}{11}-\frac {2\,a^9\,c^5\,x^{10}}{5}-\frac {a^8\,c^5\,x^9}{3}+a^7\,c^5\,x^8+2\,a^6\,c^5\,x^7-\frac {14\,a^4\,c^5\,x^5}{5}-2\,a^3\,c^5\,x^4+a^2\,c^5\,x^3+2\,a\,c^5\,x^2+c^5\,x \] Input:
int(((c - a^2*c*x^2)^5*(a*x + 1)^4)/(a^2*x^2 - 1)^2,x)
Output:
c^5*x + 2*a*c^5*x^2 + a^2*c^5*x^3 - 2*a^3*c^5*x^4 - (14*a^4*c^5*x^5)/5 + 2 *a^6*c^5*x^7 + a^7*c^5*x^8 - (a^8*c^5*x^9)/3 - (2*a^9*c^5*x^10)/5 - (a^10* c^5*x^11)/11
Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15 \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=\frac {c^{5} x \left (-15 a^{10} x^{10}-66 a^{9} x^{9}-55 a^{8} x^{8}+165 a^{7} x^{7}+330 a^{6} x^{6}-462 a^{4} x^{4}-330 a^{3} x^{3}+165 a^{2} x^{2}+330 a x +165\right )}{165} \] Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^5,x)
Output:
(c**5*x*( - 15*a**10*x**10 - 66*a**9*x**9 - 55*a**8*x**8 + 165*a**7*x**7 + 330*a**6*x**6 - 462*a**4*x**4 - 330*a**3*x**3 + 165*a**2*x**2 + 330*a*x + 165))/165