Integrand size = 24, antiderivative size = 97 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}} \]
2/7*(a*x+1)/a/(-a^2*c*x^2+c)^(7/2)+1/7*x/c/(-a^2*c*x^2+c)^(5/2)+4/21*x/c^2 /(-a^2*c*x^2+c)^(3/2)+8/21*x/c^3/(-a^2*c*x^2+c)^(1/2)
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {\sqrt {1-a^2 x^2} \left (-6-9 a x+24 a^2 x^2-4 a^3 x^3-16 a^4 x^4+8 a^5 x^5\right )}{21 a c^3 (1-a x)^{7/2} (1+a x)^{3/2} \sqrt {c-a^2 c x^2}} \]
-1/21*(Sqrt[1 - a^2*x^2]*(-6 - 9*a*x + 24*a^2*x^2 - 4*a^3*x^3 - 16*a^4*x^4 + 8*a^5*x^5))/(a*c^3*(1 - a*x)^(7/2)*(1 + a*x)^(3/2)*Sqrt[c - a^2*c*x^2])
Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6691, 457, 209, 209, 208}
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.
| \(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6691 |
| \(\displaystyle c \int \frac {(a x+1)^2}{\left (c-a^2 c x^2\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 457 |
| \(\displaystyle c \left (\frac {5 \int \frac {1}{\left (c-a^2 c x^2\right )^{7/2}}dx}{7 c}+\frac {2 (a x+1)}{7 a c \left (c-a^2 c x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle c \left (\frac {5 \left (\frac {4 \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{5/2}}\right )}{7 c}+\frac {2 (a x+1)}{7 a c \left (c-a^2 c x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle c \left (\frac {5 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {x}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{5/2}}\right )}{7 c}+\frac {2 (a x+1)}{7 a c \left (c-a^2 c x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle c \left (\frac {5 \left (\frac {4 \left (\frac {2 x}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{5/2}}\right )}{7 c}+\frac {2 (a x+1)}{7 a c \left (c-a^2 c x^2\right )^{7/2}}\right )\) |
c*((2*(1 + a*x))/(7*a*c*(c - a^2*c*x^2)^(7/2)) + (5*(x/(5*c*(c - a^2*c*x^2 )^(5/2)) + (4*(x/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x)/(3*c^2*Sqrt[c - a^2*c *x^2])))/(5*c)))/(7*c))
3.12.24.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((c_) + (d_.)*(x_))^2*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*( c + d*x)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d^2*((p + 2)/(b*(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[ b*c^2 + a*d^2, 0] && LtQ[p, -1]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^(n/2) Int[(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c , d, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && IGtQ[n/ 2, 0]
Time = 0.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.66
| method | result | size |
| gosper | \(-\frac {\left (8 a^{5} x^{5}-16 a^{4} x^{4}-4 a^{3} x^{3}+24 a^{2} x^{2}-9 a x -6\right ) \left (a x +1\right )^{2}}{21 \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} a}\) | \(64\) |
| trager | \(-\frac {\left (8 a^{5} x^{5}-16 a^{4} x^{4}-4 a^{3} x^{3}+24 a^{2} x^{2}-9 a x -6\right ) \sqrt {-a^{2} c \,x^{2}+c}}{21 c^{4} \left (a x -1\right )^{4} \left (a x +1\right )^{2} a}\) | \(74\) |
| default | \(-\frac {x}{5 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}-\frac {4 \left (\frac {x}{3 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}\right )}{5 c}-\frac {2 \left (\frac {1}{7 a c \left (x -\frac {1}{a}\right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}-\frac {6 a \left (-\frac {-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c}{10 a^{2} c^{2} \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right )}{15 a^{2} c^{2} \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}-\frac {4 \left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right )}{15 a^{2} c^{3} \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}}{c}\right )}{7}\right )}{a}\) | \(292\) |
Time = 0.38 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.28 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {{\left (8 \, a^{5} x^{5} - 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 24 \, a^{2} x^{2} - 9 \, a x - 6\right )} \sqrt {-a^{2} c x^{2} + c}}{21 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
-1/21*(8*a^5*x^5 - 16*a^4*x^4 - 4*a^3*x^3 + 24*a^2*x^2 - 9*a*x - 6)*sqrt(- a^2*c*x^2 + c)/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=- \int \frac {a x}{- a^{7} c^{3} x^{7} \sqrt {- a^{2} c x^{2} + c} + a^{6} c^{3} x^{6} \sqrt {- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt {- a^{2} c x^{2} + c} - 3 a^{4} c^{3} x^{4} \sqrt {- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt {- a^{2} c x^{2} + c} + 3 a^{2} c^{3} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{3} x \sqrt {- a^{2} c x^{2} + c} - c^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{- a^{7} c^{3} x^{7} \sqrt {- a^{2} c x^{2} + c} + a^{6} c^{3} x^{6} \sqrt {- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt {- a^{2} c x^{2} + c} - 3 a^{4} c^{3} x^{4} \sqrt {- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt {- a^{2} c x^{2} + c} + 3 a^{2} c^{3} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{3} x \sqrt {- a^{2} c x^{2} + c} - c^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx \]
-Integral(a*x/(-a**7*c**3*x**7*sqrt(-a**2*c*x**2 + c) + a**6*c**3*x**6*sqr t(-a**2*c*x**2 + c) + 3*a**5*c**3*x**5*sqrt(-a**2*c*x**2 + c) - 3*a**4*c** 3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c**3*x**3*sqrt(-a**2*c*x**2 + c) + 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) - c**3*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(-a**7*c**3*x**7*sqrt(-a** 2*c*x**2 + c) + a**6*c**3*x**6*sqrt(-a**2*c*x**2 + c) + 3*a**5*c**3*x**5*s qrt(-a**2*c*x**2 + c) - 3*a**4*c**3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c **3*x**3*sqrt(-a**2*c*x**2 + c) + 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) - c**3*sqrt(-a**2*c*x**2 + c)), x)
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (81) = 162\).
Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.49 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {1}{21} \, a {\left (\frac {3 \, a}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{4} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{3} c} - \frac {3 \, a}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{4} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{3} c} - \frac {3}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{3} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{2} c} - \frac {3}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{3} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{2} c} + \frac {8 \, x}{\sqrt {-a^{2} c x^{2} + c} a c^{3}} + \frac {4 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a c^{2}} + \frac {3 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a c}\right )} \]
1/21*a*(3*a/((-a^2*c*x^2 + c)^(5/2)*a^4*c*x + (-a^2*c*x^2 + c)^(5/2)*a^3*c ) - 3*a/((-a^2*c*x^2 + c)^(5/2)*a^4*c*x - (-a^2*c*x^2 + c)^(5/2)*a^3*c) - 3/((-a^2*c*x^2 + c)^(5/2)*a^3*c*x + (-a^2*c*x^2 + c)^(5/2)*a^2*c) - 3/((-a ^2*c*x^2 + c)^(5/2)*a^3*c*x - (-a^2*c*x^2 + c)^(5/2)*a^2*c) + 8*x/(sqrt(-a ^2*c*x^2 + c)*a*c^3) + 4*x/((-a^2*c*x^2 + c)^(3/2)*a*c^2) + 3*x/((-a^2*c*x ^2 + c)^(5/2)*a*c))
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} {\left (a^{2} x^{2} - 1\right )}} \,d x } \]
Time = 3.76 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.37 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {c-a^2\,c\,x^2}}{28\,a\,c^4\,{\left (a\,x-1\right )}^4}-\frac {\sqrt {c-a^2\,c\,x^2}}{14\,a\,c^4\,{\left (a\,x-1\right )}^3}+\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {11\,x}{42\,c^4}+\frac {5}{28\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {8\,x\,\sqrt {c-a^2\,c\,x^2}}{21\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \]