Integrand size = 22, antiderivative size = 54 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {2 x}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {1}{3 a c^2 (1+a x) \sqrt {1-a^2 x^2}} \] Output:
2/3*x/c^2/(-a^2*x^2+1)^(1/2)-1/3/a/c^2/(a*x+1)/(-a^2*x^2+1)^(1/2)
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {-1+2 a x+2 a^2 x^2}{3 a c^2 \sqrt {1-a x} (1+a x)^{3/2}} \] Input:
Integrate[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)^2),x]
Output:
(-1 + 2*a*x + 2*a^2*x^2)/(3*a*c^2*Sqrt[1 - a*x]*(1 + a*x)^(3/2))
Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6689, 454, 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.
\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6689 |
\(\displaystyle \frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\) |
\(\Big \downarrow \) 454 |
\(\displaystyle \frac {\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {1-a x}{3 a \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {\frac {2 x}{3 \sqrt {1-a^2 x^2}}-\frac {1-a x}{3 a \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
Input:
Int[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)^2),x]
Output:
(-1/3*(1 - a*x)/(a*(1 - a^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 - a^2*x^2]))/c^2
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*d - b*c*x)/(2*a*b*(p + 1)))*(a + b*x^2)^(p + 1), x] + Simp[c*((2*p + 3)/(2*a *(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && L tQ[p, -1] && NeQ[p, -3/2]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - a^2*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] && ILtQ[(n - 1)/2, 0] && !In tegerQ[p - n/2]
Time = 0.53 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(\frac {2 a^{2} x^{2}+2 a x -1}{3 a \,c^{2} \left (a x +1\right ) \sqrt {-a^{2} x^{2}+1}}\) | \(42\) |
trager | \(-\frac {\left (2 a^{2} x^{2}+2 a x -1\right ) \sqrt {-a^{2} x^{2}+1}}{3 c^{2} \left (a x +1\right )^{2} a \left (a x -1\right )}\) | \(49\) |
orering | \(-\frac {\left (2 a^{2} x^{2}+2 a x -1\right ) \left (a x -1\right ) \sqrt {-a^{2} x^{2}+1}}{3 a \left (-a^{2} c \,x^{2}+c \right )^{2}}\) | \(50\) |
default | \(\frac {\frac {\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}+a \left (\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{8 a^{2}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{12 a^{4} \left (x +\frac {1}{a}\right )^{3}}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}-a \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{4 a^{2}}+\frac {\frac {3 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16}+\frac {3 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{16 \sqrt {a^{2}}}}{a}-\frac {3 \left (\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{16 a}}{c^{2}}\) | \(409\) |
Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
1/3/a/c^2/(a*x+1)/(-a^2*x^2+1)^(1/2)*(2*a^2*x^2+2*a*x-1)
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.61 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {a^{3} x^{3} + a^{2} x^{2} - a x + {\left (2 \, a^{2} x^{2} + 2 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} - 1}{3 \, {\left (a^{4} c^{2} x^{3} + a^{3} c^{2} x^{2} - a^{2} c^{2} x - a c^{2}\right )}} \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^2,x, algorithm="fric as")
Output:
-1/3*(a^3*x^3 + a^2*x^2 - a*x + (2*a^2*x^2 + 2*a*x - 1)*sqrt(-a^2*x^2 + 1) - 1)/(a^4*c^2*x^3 + a^3*c^2*x^2 - a^2*c^2*x - a*c^2)
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {1}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \] Input:
integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**2,x)
Output:
Integral(1/(-a**3*x**3*sqrt(-a**2*x**2 + 1) - a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**2
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{2} {\left (a x + 1\right )}} \,d x } \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^2,x, algorithm="maxi ma")
Output:
integrate(sqrt(-a^2*x^2 + 1)/((a^2*c*x^2 - c)^2*(a*x + 1)), x)
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{2} {\left (a x + 1\right )}} \,d x } \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^2,x, algorithm="giac ")
Output:
integrate(sqrt(-a^2*x^2 + 1)/((a^2*c*x^2 - c)^2*(a*x + 1)), x)
Time = 26.91 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {\sqrt {1-a^2\,x^2}\,\left (2\,a^2\,x^2+2\,a\,x-1\right )}{3\,a\,c^2\,\left (a\,x-1\right )\,{\left (a\,x+1\right )}^2} \] Input:
int((1 - a^2*x^2)^(1/2)/((c - a^2*c*x^2)^2*(a*x + 1)),x)
Output:
-((1 - a^2*x^2)^(1/2)*(2*a*x + 2*a^2*x^2 - 1))/(3*a*c^2*(a*x - 1)*(a*x + 1 )^2)
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {2 \sqrt {-a^{2} x^{2}+1}\, a x +2 \sqrt {-a^{2} x^{2}+1}+2 a^{2} x^{2}+2 a x -1}{3 \sqrt {-a^{2} x^{2}+1}\, a \,c^{2} \left (a x +1\right )} \] Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^2,x)
Output:
(2*sqrt( - a**2*x**2 + 1)*a*x + 2*sqrt( - a**2*x**2 + 1) + 2*a**2*x**2 + 2 *a*x - 1)/(3*sqrt( - a**2*x**2 + 1)*a*c**2*(a*x + 1))