Integrand size = 22, antiderivative size = 84 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {16 c^5 (1+a x)^7}{7 a}+\frac {4 c^5 (1+a x)^8}{a}-\frac {8 c^5 (1+a x)^9}{3 a}+\frac {4 c^5 (1+a x)^{10}}{5 a}-\frac {c^5 (1+a x)^{11}}{11 a} \]
-16/7*c^5*(a*x+1)^7/a+4*c^5*(a*x+1)^8/a-8/3*c^5*(a*x+1)^9/a+4/5*c^5*(a*x+1 )^10/a-1/11*c^5*(a*x+1)^11/a
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.56 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {c^5 (1+a x)^7 \left (281-812 a x+938 a^2 x^2-504 a^3 x^3+105 a^4 x^4\right )}{1155 a} \]
Time = 0.37 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.88, 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.
\(\displaystyle \int \left (c-a^2 c x^2\right )^5 e^{2 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int c^5 e^{2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^5dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c^5 \int e^{2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^5dx\) |
\(\Big \downarrow \) 6690 |
\(\displaystyle -c^5 \int (1-a x)^4 (a x+1)^6dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -c^5 \int \left ((a x+1)^{10}-8 (a x+1)^9+24 (a x+1)^8-32 (a x+1)^7+16 (a x+1)^6\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -c^5 \left (\frac {(a x+1)^{11}}{11 a}-\frac {4 (a x+1)^{10}}{5 a}+\frac {8 (a x+1)^9}{3 a}-\frac {4 (a x+1)^8}{a}+\frac {16 (a x+1)^7}{7 a}\right )\) |
-(c^5*((16*(1 + a*x)^7)/(7*a) - (4*(1 + a*x)^8)/a + (8*(1 + a*x)^9)/(3*a) - (4*(1 + a*x)^10)/(5*a) + (1 + a*x)^11/(11*a)))
3.6.64.3.1 Defintions of rubi rules used
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 = 0.65 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01
method | result | size |
gosper | \(-\frac {c^{5} x \left (105 a^{10} x^{10}+231 a^{9} x^{9}-385 a^{8} x^{8}-1155 a^{7} x^{7}+330 a^{6} x^{6}+2310 a^{5} x^{5}+462 a^{4} x^{4}-2310 a^{3} x^{3}-1155 a^{2} x^{2}+1155 a x +1155\right )}{1155}\) | \(85\) |
default | \(c^{5} \left (-\frac {1}{11} a^{10} x^{11}-\frac {1}{5} a^{9} x^{10}+\frac {1}{3} a^{8} x^{9}+a^{7} x^{8}-\frac {2}{7} a^{6} x^{7}-2 a^{5} x^{6}-\frac {2}{5} a^{4} x^{5}+2 a^{3} x^{4}+a^{2} x^{3}-a \,x^{2}-x \right )\) | \(85\) |
norman | \(a^{7} c^{5} x^{8}+c^{5} a^{2} x^{3}-c^{5} x -a \,c^{5} x^{2}+2 a^{3} c^{5} x^{4}-\frac {2}{5} a^{4} c^{5} x^{5}-2 a^{5} c^{5} x^{6}-\frac {2}{7} a^{6} c^{5} x^{7}+\frac {1}{3} a^{8} c^{5} x^{9}-\frac {1}{5} a^{9} c^{5} x^{10}-\frac {1}{11} a^{10} c^{5} x^{11}\) | \(114\) |
risch | \(a^{7} c^{5} x^{8}+c^{5} a^{2} x^{3}-c^{5} x -a \,c^{5} x^{2}+2 a^{3} c^{5} x^{4}-\frac {2}{5} a^{4} c^{5} x^{5}-2 a^{5} c^{5} x^{6}-\frac {2}{7} a^{6} c^{5} x^{7}+\frac {1}{3} a^{8} c^{5} x^{9}-\frac {1}{5} a^{9} c^{5} x^{10}-\frac {1}{11} a^{10} c^{5} x^{11}\) | \(114\) |
parallelrisch | \(a^{7} c^{5} x^{8}+c^{5} a^{2} x^{3}-c^{5} x -a \,c^{5} x^{2}+2 a^{3} c^{5} x^{4}-\frac {2}{5} a^{4} c^{5} x^{5}-2 a^{5} c^{5} x^{6}-\frac {2}{7} a^{6} c^{5} x^{7}+\frac {1}{3} a^{8} c^{5} x^{9}-\frac {1}{5} a^{9} c^{5} x^{10}-\frac {1}{11} a^{10} c^{5} x^{11}\) | \(114\) |
meijerg | \(\frac {c^{5} \left (-\frac {x a \left (2520 a^{10} x^{10}+2772 a^{9} x^{9}+3080 a^{8} x^{8}+3465 a^{7} x^{7}+3960 a^{6} x^{6}+4620 a^{5} x^{5}+5544 a^{4} x^{4}+6930 a^{3} x^{3}+9240 a^{2} x^{2}+13860 a x +27720\right )}{27720}-\ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (-\frac {a x \left (280 a^{8} x^{8}+315 a^{7} x^{7}+360 a^{6} x^{6}+420 a^{5} x^{5}+504 a^{4} x^{4}+630 a^{3} x^{3}+840 a^{2} x^{2}+1260 a x +2520\right )}{2520}-\ln \left (-a x +1\right )\right )}{a}+\frac {10 c^{5} \left (-\frac {a x \left (120 a^{6} x^{6}+140 a^{5} x^{5}+168 a^{4} x^{4}+210 a^{3} x^{3}+280 a^{2} x^{2}+420 a x +840\right )}{840}-\ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (-\frac {a x \left (12 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}-\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (-a x -\ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (\frac {x a \left (2772 a^{9} x^{9}+3080 a^{8} x^{8}+3465 a^{7} x^{7}+3960 a^{6} x^{6}+4620 a^{5} x^{5}+5544 a^{4} x^{4}+6930 a^{3} x^{3}+9240 a^{2} x^{2}+13860 a x +27720\right )}{27720}+\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (\frac {a x \left (315 a^{7} x^{7}+360 a^{6} x^{6}+420 a^{5} x^{5}+504 a^{4} x^{4}+630 a^{3} x^{3}+840 a^{2} x^{2}+1260 a x +2520\right )}{2520}+\ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (\frac {a x \left (70 a^{5} x^{5}+84 a^{4} x^{4}+105 a^{3} x^{3}+140 a^{2} x^{2}+210 a x +420\right )}{420}+\ln \left (-a x +1\right )\right )}{a}+\frac {10 c^{5} \left (\frac {a x \left (15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}+\ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} \ln \left (-a x +1\right )}{a}\) | \(667\) |
-1/1155*c^5*x*(105*a^10*x^10+231*a^9*x^9-385*a^8*x^8-1155*a^7*x^7+330*a^6* x^6+2310*a^5*x^5+462*a^4*x^4-2310*a^3*x^3-1155*a^2*x^2+1155*a*x+1155)
Time = 0.24 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.35 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
-1/11*a^10*c^5*x^11 - 1/5*a^9*c^5*x^10 + 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 - 2 /7*a^6*c^5*x^7 - 2*a^5*c^5*x^6 - 2/5*a^4*c^5*x^5 + 2*a^3*c^5*x^4 + a^2*c^5 *x^3 - a*c^5*x^2 - c^5*x
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.42 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=- \frac {a^{10} c^{5} x^{11}}{11} - \frac {a^{9} c^{5} x^{10}}{5} + \frac {a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} - \frac {2 a^{6} c^{5} x^{7}}{7} - 2 a^{5} c^{5} x^{6} - \frac {2 a^{4} c^{5} x^{5}}{5} + 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
-a**10*c**5*x**11/11 - 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**7/7 - 2*a**5*c**5*x**6 - 2*a**4*c**5*x**5/5 + 2*a**3*c **5*x**4 + a**2*c**5*x**3 - a*c**5*x**2 - c**5*x
Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.35 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
-1/11*a^10*c^5*x^11 - 1/5*a^9*c^5*x^10 + 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 - 2 /7*a^6*c^5*x^7 - 2*a^5*c^5*x^6 - 2/5*a^4*c^5*x^5 + 2*a^3*c^5*x^4 + a^2*c^5 *x^3 - a*c^5*x^2 - c^5*x
Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.35 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
-1/11*a^10*c^5*x^11 - 1/5*a^9*c^5*x^10 + 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 - 2 /7*a^6*c^5*x^7 - 2*a^5*c^5*x^6 - 2/5*a^4*c^5*x^5 + 2*a^3*c^5*x^4 + a^2*c^5 *x^3 - a*c^5*x^2 - c^5*x
Time = 3.90 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.35 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {a^{10}\,c^5\,x^{11}}{11}-\frac {a^9\,c^5\,x^{10}}{5}+\frac {a^8\,c^5\,x^9}{3}+a^7\,c^5\,x^8-\frac {2\,a^6\,c^5\,x^7}{7}-2\,a^5\,c^5\,x^6-\frac {2\,a^4\,c^5\,x^5}{5}+2\,a^3\,c^5\,x^4+a^2\,c^5\,x^3-a\,c^5\,x^2-c^5\,x \]