Integrand size = 24, antiderivative size = 75 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}} \]
-2/5*(-a*x+1)/a/(-a^2*c*x^2+c)^(5/2)+1/5*x/c/(-a^2*c*x^2+c)^(3/2)+2/5*x/c^ 2/(-a^2*c*x^2+c)^(1/2)
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (-2+a x+4 a^2 x^2+2 a^3 x^3\right )}{5 a c^2 \sqrt {1-a x} (1+a x)^{5/2} \sqrt {c-a^2 c x^2}} \]
(Sqrt[1 - a^2*x^2]*(-2 + a*x + 4*a^2*x^2 + 2*a^3*x^3))/(5*a*c^2*Sqrt[1 - a *x]*(1 + a*x)^(5/2)*Sqrt[c - a^2*c*x^2])
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6692, 457, 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.
\(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6692 |
\(\displaystyle c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{7/2}}dx\) |
\(\Big \downarrow \) 457 |
\(\displaystyle c \left (\frac {3 \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}-\frac {2 (1-a x)}{5 a c \left (c-a^2 c x^2\right )^{5/2}}\right )\) |
\(\Big \downarrow \) 209 |
\(\displaystyle c \left (\frac {3 \left (\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {x}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}-\frac {2 (1-a x)}{5 a c \left (c-a^2 c x^2\right )^{5/2}}\right )\) |
\(\Big \downarrow \) 208 |
\(\displaystyle c \left (\frac {3 \left (\frac {2 x}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}-\frac {2 (1-a x)}{5 a c \left (c-a^2 c x^2\right )^{5/2}}\right )\) |
c*((-2*(1 - a*x))/(5*a*c*(c - a^2*c*x^2)^(5/2)) + (3*(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))
3.13.50.3.1 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[1/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]) && ILtQ[ n/2, 0]
Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(\frac {\left (a x -1\right )^{2} \left (2 a^{3} x^{3}+4 a^{2} x^{2}+a x -2\right )}{5 \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} a}\) | \(47\) |
trager | \(-\frac {\left (2 a^{3} x^{3}+4 a^{2} x^{2}+a x -2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{5 c^{3} \left (a x +1\right )^{3} a \left (a x -1\right )}\) | \(57\) |
default | \(-\frac {x}{3 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 x}{3 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}+\frac {-\frac {2}{5 a c \left (x +\frac {1}{a}\right ) \left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}+\frac {8 a \left (-\frac {-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c}{6 a^{2} c^{2} \left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}-\frac {-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c}{3 a^{2} c^{3} \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{5}}{a}\) | \(188\) |
Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {{\left (2 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x - 2\right )} \sqrt {-a^{2} c x^{2} + c}}{5 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
-1/5*(2*a^3*x^3 + 4*a^2*x^2 + a*x - 2)*sqrt(-a^2*c*x^2 + c)/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x - a*c^3)
\[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=- \int \frac {a x}{a^{5} c^{2} x^{5} \sqrt {- a^{2} c x^{2} + c} + a^{4} c^{2} x^{4} \sqrt {- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt {- a^{2} c x^{2} + c} - 2 a^{2} c^{2} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{2} x \sqrt {- a^{2} c x^{2} + c} + c^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {1}{a^{5} c^{2} x^{5} \sqrt {- a^{2} c x^{2} + c} + a^{4} c^{2} x^{4} \sqrt {- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt {- a^{2} c x^{2} + c} - 2 a^{2} c^{2} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{2} x \sqrt {- a^{2} c x^{2} + c} + c^{2} \sqrt {- a^{2} c x^{2} + c}}\right )\, dx \]
-Integral(a*x/(a**5*c**2*x**5*sqrt(-a**2*c*x**2 + c) + a**4*c**2*x**4*sqrt (-a**2*c*x**2 + c) - 2*a**3*c**2*x**3*sqrt(-a**2*c*x**2 + c) - 2*a**2*c**2 *x**2*sqrt(-a**2*c*x**2 + c) + a*c**2*x*sqrt(-a**2*c*x**2 + c) + c**2*sqrt (-a**2*c*x**2 + c)), x) - Integral(-1/(a**5*c**2*x**5*sqrt(-a**2*c*x**2 + c) + a**4*c**2*x**4*sqrt(-a**2*c*x**2 + c) - 2*a**3*c**2*x**3*sqrt(-a**2*c *x**2 + c) - 2*a**2*c**2*x**2*sqrt(-a**2*c*x**2 + c) + a*c**2*x*sqrt(-a**2 *c*x**2 + c) + c**2*sqrt(-a**2*c*x**2 + c)), x)
Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {2}{5 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a c\right )}} + \frac {2 \, x}{5 \, \sqrt {-a^{2} c x^{2} + c} c^{2}} + \frac {x}{5 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c} \]
-2/5/((-a^2*c*x^2 + c)^(3/2)*a^2*c*x + (-a^2*c*x^2 + c)^(3/2)*a*c) + 2/5*x /(sqrt(-a^2*c*x^2 + c)*c^2) + 1/5*x/((-a^2*c*x^2 + c)^(3/2)*c)
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (62) = 124\).
Time = 0.31 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.93 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {a^{3} {\left (\frac {5}{a^{3} c^{2} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )} - \frac {a^{12} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{20} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{4} \mathrm {sgn}\left (a\right )^{4} + 15 \, a^{12} c^{22} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{4} \mathrm {sgn}\left (a\right )^{4} + 5 \, a^{12} c^{21} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{4} \mathrm {sgn}\left (a\right )^{4}}{a^{15} c^{25} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{5} \mathrm {sgn}\left (a\right )^{5}}\right )} - \frac {16 \, \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}{\sqrt {-c} c^{2}}}{40 \, {\left | a \right |}} \]
1/40*(a^3*(5/(a^3*c^2*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a)) - (a^12*(c - 2*c/(a*x + 1))^2*c^20*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1)) ^4*sgn(a)^4 + 15*a^12*c^22*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))^4*sgn (a)^4 + 5*a^12*c^21*(-c + 2*c/(a*x + 1))^(3/2)*sgn(1/(a*x + 1))^4*sgn(a)^4 )/(a^15*c^25*sgn(1/(a*x + 1))^5*sgn(a)^5)) - 16*sgn(1/(a*x + 1))*sgn(a)/(s qrt(-c)*c^2))/abs(a)
Time = 3.90 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c-a^2\,c\,x^2}\,\left (2\,a^3\,x^3+4\,a^2\,x^2+a\,x-2\right )}{5\,a\,c^3\,\left (a\,x-1\right )\,{\left (a\,x+1\right )}^3} \]