Integrand size = 18, antiderivative size = 28 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{(c-a c x)^2} \, dx=-\frac {1-a x}{a c^2 \sqrt {1-a^2 x^2}} \] Output:
-(-a*x+1)/a/c^2/(-a^2*x^2+1)^(1/2)
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{(c-a c x)^2} \, dx=-\frac {\sqrt {1-a x}}{a c^2 \sqrt {1+a x}} \] Input:
Integrate[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^2),x]
Output:
-(Sqrt[1 - a*x]/(a*c^2*Sqrt[1 + a*x]))
Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6677, 27, 453}
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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{(c-a c x)^2} \, dx\) |
\(\Big \downarrow \) 6677 |
\(\displaystyle \frac {\int \frac {c (1-a x)}{\left (1-a^2 x^2\right )^{3/2}}dx}{c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{3/2}}dx}{c^2}\) |
\(\Big \downarrow \) 453 |
\(\displaystyle -\frac {1-a x}{a c^2 \sqrt {1-a^2 x^2}}\) |
Input:
Int[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^2),x]
Output:
-((1 - a*x)/(a*c^2*Sqrt[1 - a^2*x^2]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-(a* d - b*c*x)/(a*b*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b, c, d}, x]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[c^n Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
method | result | size |
trager | \(-\frac {\sqrt {-a^{2} x^{2}+1}}{a \,c^{2} \left (a x +1\right )}\) | \(28\) |
gosper | \(\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{\left (a x -1\right ) c^{2} a \left (a x +1\right )^{2}}\) | \(34\) |
orering | \(\frac {\left (a x -1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{\left (a x +1\right )^{2} a \left (-a c x +c \right )^{2}}\) | \(38\) |
default | \(\frac {\frac {-\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}-3 a \left (\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )}{8 a^{2}}-\frac {3 \left (\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )}{16 a}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-2 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )}{4 a^{3}}+\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )}{4 a^{2}}+\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{16}+\frac {3 a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )}{16}}{a}}{c^{2}}\) | \(766\) |
Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x,method=_RETURNVERBOSE)
Output:
-(-a^2*x^2+1)^(1/2)/a/c^2/(a*x+1)
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{(c-a c x)^2} \, dx=-\frac {a x + \sqrt {-a^{2} x^{2} + 1} + 1}{a^{2} c^{2} x + a c^{2}} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x, algorithm="fricas ")
Output:
-(a*x + sqrt(-a^2*x^2 + 1) + 1)/(a^2*c^2*x + a*c^2)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{(c-a c x)^2} \, dx=\frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\right )\, dx}{c^{2}} \] Input:
integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**2,x)
Output:
(Integral(sqrt(-a**2*x**2 + 1)/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a* *2*x**2 + a*x + 1), x) + Integral(-a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**5*x* *5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1), x))/c**2
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{(c-a c x)^2} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a c x - c\right )}^{2} {\left (a x + 1\right )}^{3}} \,d x } \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x, algorithm="maxima ")
Output:
integrate((-a^2*x^2 + 1)^(3/2)/((a*c*x - c)^2*(a*x + 1)^3), x)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{(c-a c x)^2} \, dx=c^{2} {\left (\frac {i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}{a^{2} c^{4}} + \frac {\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}{a^{2} c^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1}}\right )} {\left | a \right |} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x, algorithm="giac")
Output:
c^2*(I*sgn(1/(a*c*x - c))*sgn(a)*sgn(c)/(a^2*c^4) + sgn(1/(a*c*x - c))*sgn (a)*sgn(c)/(a^2*c^4*sqrt(-2*c/(a*c*x - c) - 1)))*abs(a)
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{(c-a c x)^2} \, dx=\frac {\sqrt {1-a^2\,x^2}}{c^2\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \] Input:
int((1 - a^2*x^2)^(3/2)/((c - a*c*x)^2*(a*x + 1)^3),x)
Output:
(1 - a^2*x^2)^(1/2)/(c^2*(x*(-a^2)^(1/2) + (-a^2)^(1/2)/a)*(-a^2)^(1/2))
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{(c-a c x)^2} \, dx=\frac {2 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )}{a \,c^{2} \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+1\right )} \] Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x)
Output:
(2*tan(asin(a*x)/2))/(a*c**2*(tan(asin(a*x)/2) + 1))