Integrand size = 12, antiderivative size = 183 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^4} \, dx=-\frac {4 a^3}{5 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^5}+\frac {2 a^3}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}-\frac {7 a^3}{3 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}+\frac {3 a^3}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}-\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^3}{4 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )} \] Output:
-4/5*a^3/(1-((-a*x+1)/(a*x+1))^(1/2))^5+2*a^3/(1-((-a*x+1)/(a*x+1))^(1/2)) ^4-7/3*a^3/(1-((-a*x+1)/(a*x+1))^(1/2))^3+3/2*a^3/(1-((-a*x+1)/(a*x+1))^(1 /2))^2-a^3/(4-4*((-a*x+1)/(a*x+1))^(1/2))-a^3/(4+4*((-a*x+1)/(a*x+1))^(1/2 ))
Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.38 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^4} \, dx=\frac {-6+5 a^2 x^2+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (-3+3 a x-2 a^2 x^2+2 a^3 x^3\right )}{15 a^2 x^5} \] Input:
Integrate[E^(2*ArcSech[a*x])/x^4,x]
Output:
(-6 + 5*a^2*x^2 + 2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2*(-3 + 3*a*x - 2* a^2*x^2 + 2*a^3*x^3))/(15*a^2*x^5)
Time = 1.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6891, 7268, 27, 2115, 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.
\(\displaystyle \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^4} \, dx\) |
\(\Big \downarrow \) 6891 |
\(\displaystyle \int \frac {\left (\frac {\sqrt {\frac {1-a x}{a x+1}}}{a x}+\sqrt {\frac {1-a x}{a x+1}}+\frac {1}{a x}\right )^2}{x^4}dx\) |
\(\Big \downarrow \) 7268 |
\(\displaystyle -4 a \int \frac {a^2 \sqrt {\frac {1-a x}{a x+1}} \left (\frac {1-a x}{a x+1}+1\right )^2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^6 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}d\sqrt {\frac {1-a x}{a x+1}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -4 a^3 \int \frac {\sqrt {\frac {1-a x}{a x+1}} \left (\frac {1-a x}{a x+1}+1\right )^2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^6 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}d\sqrt {\frac {1-a x}{a x+1}}\) |
\(\Big \downarrow \) 2115 |
\(\displaystyle -4 a^3 \int \left (-\frac {1}{16 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}+\frac {1}{16 \left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^2}+\frac {3}{4 \left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^3}+\frac {7}{4 \left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^4}+\frac {2}{\left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^5}+\frac {1}{\left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^6}\right )d\sqrt {\frac {1-a x}{a x+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 a^3 \left (\frac {1}{16 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}+\frac {1}{16 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {3}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}+\frac {7}{12 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}-\frac {1}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}+\frac {1}{5 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^5}\right )\) |
Input:
Int[E^(2*ArcSech[a*x])/x^4,x]
Output:
-4*a^3*(1/(5*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5) - 1/(2*(1 - Sqrt[(1 - a*x) /(1 + a*x)])^4) + 7/(12*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) - 3/(8*(1 - Sqr t[(1 - a*x)/(1 + a*x)])^2) + 1/(16*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) + 1/(1 6*(1 + Sqrt[(1 - a*x)/(1 + a*x)])))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^ n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[m, n]
Int[E^(ArcSech[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + (1/u)*Sqrt[(1 - u)/(1 + u)])^n, x] /; FreeQ[m, x] && Integer Q[n]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears [u, x]}, Simp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls t[[2]])], x] /; !FalseQ[lst]]
Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.46
method | result | size |
default | \(\frac {\frac {a^{2}}{3 x^{3}}-\frac {1}{5 x^{5}}}{a^{2}}+\frac {2 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (2 a^{2} x^{2}+3\right )}{15 a \,x^{4}}-\frac {1}{5 a^{2} x^{5}}\) | \(84\) |
orering | \(\frac {\left (-\frac {8}{15} x^{5} a^{4}+\frac {28}{15} a^{2} x^{3}-\frac {7}{5} x \right ) \left (\frac {1}{a x}+\sqrt {-1+\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )^{2}}{x^{4}}-\frac {\left (2 a^{2} x^{2}-3\right ) x^{2} \left (a x -1\right ) \left (a x +1\right ) \left (\frac {2 \left (\frac {1}{a x}+\sqrt {-1+\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right ) \left (-\frac {1}{a \,x^{2}}-\frac {\sqrt {1+\frac {1}{a x}}}{2 \sqrt {-1+\frac {1}{a x}}\, a \,x^{2}}-\frac {\sqrt {-1+\frac {1}{a x}}}{2 \sqrt {1+\frac {1}{a x}}\, a \,x^{2}}\right )}{x^{4}}-\frac {4 \left (\frac {1}{a x}+\sqrt {-1+\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )^{2}}{x^{5}}\right )}{15}\) | \(228\) |
Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x^4,x,method=_RETURNVERBOSE )
Output:
1/a^2*(1/3*a^2/x^3-1/5/x^5)+2/15/a*(-(a*x-1)/a/x)^(1/2)/x^4*((a*x+1)/a/x)^ (1/2)*(a^2*x^2-1)*(2*a^2*x^2+3)-1/5/a^2/x^5
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.38 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^4} \, dx=\frac {5 \, a^{2} x^{2} + 2 \, {\left (2 \, a^{5} x^{5} + a^{3} x^{3} - 3 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 6}{15 \, a^{2} x^{5}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x^4,x, algorithm="fri cas")
Output:
1/15*(5*a^2*x^2 + 2*(2*a^5*x^5 + a^3*x^3 - 3*a*x)*sqrt((a*x + 1)/(a*x))*sq rt(-(a*x - 1)/(a*x)) - 6)/(a^2*x^5)
\[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^4} \, dx=\frac {\int \frac {2}{x^{6}}\, dx + \int \left (- \frac {a^{2}}{x^{4}}\right )\, dx + \int \frac {2 a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{5}}\, dx}{a^{2}} \] Input:
integrate((1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2))**2/x**4,x)
Output:
(Integral(2/x**6, x) + Integral(-a**2/x**4, x) + Integral(2*a*sqrt(-1 + 1/ (a*x))*sqrt(1 + 1/(a*x))/x**5, x))/a**2
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.31 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^4} \, dx=\frac {1}{3 \, x^{3}} + \frac {2 \, {\left (2 \, a^{4} x^{5} + a^{2} x^{3} - 3 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{15 \, a^{2} x^{6}} - \frac {2}{5 \, a^{2} x^{5}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x^4,x, algorithm="max ima")
Output:
1/3/x^3 + 2/15*(2*a^4*x^5 + a^2*x^3 - 3*x)*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a ^2*x^6) - 2/5/(a^2*x^5)
\[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^4} \, dx=\int { \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x^4,x, algorithm="gia c")
Output:
integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))^2/x^4, x)
Time = 23.97 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.47 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^4} \, dx=\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {2\,a\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{15}-\frac {2\,\sqrt {\frac {1}{a\,x}+1}}{5\,a}+\frac {4\,a^3\,x^4\,\sqrt {\frac {1}{a\,x}+1}}{15}\right )}{x^4}+\frac {\frac {a^2\,x^2}{3}-\frac {2}{5}}{a^2\,x^5} \] Input:
int(((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))^2/x^4,x)
Output:
((1/(a*x) - 1)^(1/2)*((2*a*x^2*(1/(a*x) + 1)^(1/2))/15 - (2*(1/(a*x) + 1)^ (1/2))/(5*a) + (4*a^3*x^4*(1/(a*x) + 1)^(1/2))/15))/x^4 + ((a^2*x^2)/3 - 2 /5)/(a^2*x^5)
Time = 0.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.41 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^4} \, dx=\frac {4 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{4} x^{4}+2 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{2} x^{2}-6 \sqrt {a x +1}\, \sqrt {-a x +1}+5 a^{2} x^{2}-6}{15 a^{2} x^{5}} \] Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x^4,x)
Output:
(4*sqrt(a*x + 1)*sqrt( - a*x + 1)*a**4*x**4 + 2*sqrt(a*x + 1)*sqrt( - a*x + 1)*a**2*x**2 - 6*sqrt(a*x + 1)*sqrt( - a*x + 1) + 5*a**2*x**2 - 6)/(15*a **2*x**5)