Integrand size = 12, antiderivative size = 267 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^5} \, dx=-\frac {2 a^4}{3 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^6}+\frac {2 a^4}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^5}-\frac {3 a^4}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {8 a^4}{3 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {11 a^4}{8 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {3 a^4}{8 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^4}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^4}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}+\frac {1}{4} a^4 \text {arctanh}\left (\sqrt {\frac {1-a x}{1+a x}}\right ) \] Output:
-2/3*a^4/(1-((-a*x+1)/(a*x+1))^(1/2))^6+2*a^4/(1-((-a*x+1)/(a*x+1))^(1/2)) ^5-3*a^4/(1-((-a*x+1)/(a*x+1))^(1/2))^4+8/3*a^4/(1-((-a*x+1)/(a*x+1))^(1/2 ))^3-11/8*a^4/(1-((-a*x+1)/(a*x+1))^(1/2))^2+3*a^4/(8-8*((-a*x+1)/(a*x+1)) ^(1/2))-1/8*a^4/(1+((-a*x+1)/(a*x+1))^(1/2))^2+a^4/(8+8*((-a*x+1)/(a*x+1)) ^(1/2))+1/4*a^4*arctanh(((-a*x+1)/(a*x+1))^(1/2))
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.51 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {-8+6 a^2 x^2+\sqrt {\frac {1-a x}{1+a x}} \left (-8-8 a x+2 a^2 x^2+2 a^3 x^3+3 a^4 x^4+3 a^5 x^5\right )-3 a^6 x^6 \log (x)+3 a^6 x^6 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{24 a^2 x^6} \] Input:
Integrate[E^(2*ArcSech[a*x])/x^5,x]
Output:
(-8 + 6*a^2*x^2 + Sqrt[(1 - a*x)/(1 + a*x)]*(-8 - 8*a*x + 2*a^2*x^2 + 2*a^ 3*x^3 + 3*a^4*x^4 + 3*a^5*x^5) - 3*a^6*x^6*Log[x] + 3*a^6*x^6*Log[1 + Sqrt [(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/(24*a^2*x^6)
Time = 1.25 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6891, 7268, 25, 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.
| \(\displaystyle \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^5} \, dx\) |
| \(\Big \downarrow \) 6891 |
| \(\displaystyle \int \frac {\left (\frac {\sqrt {\frac {1-a x}{a x+1}}}{a x}+\sqrt {\frac {1-a x}{a x+1}}+\frac {1}{a x}\right )^2}{x^5}dx\) |
| \(\Big \downarrow \) 7268 |
| \(\displaystyle 4 a \int -\frac {a^3 \sqrt {\frac {1-a x}{a x+1}} \left (\frac {1-a x}{a x+1}+1\right )^3}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^7 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}d\sqrt {\frac {1-a x}{a x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 a \int \frac {a^3 \sqrt {\frac {1-a x}{a x+1}} \left (\frac {1-a x}{a x+1}+1\right )^3}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^7 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}d\sqrt {\frac {1-a x}{a x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 a^4 \int \frac {\sqrt {\frac {1-a x}{a x+1}} \left (\frac {1-a x}{a x+1}+1\right )^3}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^7 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}d\sqrt {\frac {1-a x}{a x+1}}\) |
| \(\Big \downarrow \) 2115 |
| \(\displaystyle -4 a^4 \int \left (-\frac {3}{32 \left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^2}+\frac {1}{32 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}-\frac {11}{16 \left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^3}-\frac {1}{16 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}-\frac {2}{\left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^4}-\frac {3}{\left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^5}-\frac {5}{2 \left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^6}-\frac {1}{\left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^7}+\frac {1}{16 \left (\frac {1-a x}{a x+1}-1\right )}\right )d\sqrt {\frac {1-a x}{a x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 a^4 \left (-\frac {1}{16} \text {arctanh}\left (\sqrt {\frac {1-a x}{a x+1}}\right )-\frac {3}{32 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {1}{32 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}+\frac {11}{32 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}+\frac {1}{32 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}-\frac {2}{3 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}+\frac {3}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}-\frac {1}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^5}+\frac {1}{6 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^6}\right )\) |
Input:
Int[E^(2*ArcSech[a*x])/x^5,x]
Output:
-4*a^4*(1/(6*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^6) - 1/(2*(1 - Sqrt[(1 - a*x) /(1 + a*x)])^5) + 3/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4) - 2/(3*(1 - Sqrt [(1 - a*x)/(1 + a*x)])^3) + 11/(32*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) - 3/ (32*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) + 1/(32*(1 + Sqrt[(1 - a*x)/(1 + a*x) ])^2) - 1/(32*(1 + Sqrt[(1 - a*x)/(1 + a*x)])) - ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]]/16)
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 = 153, normalized size of antiderivative = 0.57
| method | result | size |
| default | \(\frac {-\frac {1}{6 x^{6}}+\frac {a^{2}}{4 x^{4}}}{a^{2}}+\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{6} x^{6}+3 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}+2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-8 \sqrt {-a^{2} x^{2}+1}\right )}{24 a \,x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{6 a^{2} x^{6}}\) | \(153\) |
Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x^5,x,method=_RETURNVERBOSE )
Output:
1/a^2*(-1/6/x^6+1/4*a^2/x^4)+1/24/a*(-(a*x-1)/a/x)^(1/2)/x^5*((a*x+1)/a/x) ^(1/2)*(3*arctanh(1/(-a^2*x^2+1)^(1/2))*a^6*x^6+3*a^4*x^4*(-a^2*x^2+1)^(1/ 2)+2*(-a^2*x^2+1)^(1/2)*a^2*x^2-8*(-a^2*x^2+1)^(1/2))/(-a^2*x^2+1)^(1/2)-1 /6/a^2/x^6
Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.58 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - 3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 12 \, a^{2} x^{2} + 2 \, {\left (3 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - 8 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 16}{48 \, a^{2} x^{6}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x^5,x, algorithm="fri cas")
Output:
1/48*(3*a^6*x^6*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - 3*a^6*x^6*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 1) + 12 *a^2*x^2 + 2*(3*a^5*x^5 + 2*a^3*x^3 - 8*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-( a*x - 1)/(a*x)) - 16)/(a^2*x^6)
\[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {\int \frac {2}{x^{7}}\, dx + \int \left (- \frac {a^{2}}{x^{5}}\right )\, dx + \int \frac {2 a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{6}}\, dx}{a^{2}} \] Input:
integrate((1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2))**2/x**5,x)
Output:
(Integral(2/x**7, x) + Integral(-a**2/x**5, x) + Integral(2*a*sqrt(-1 + 1/ (a*x))*sqrt(1 + 1/(a*x))/x**6, x))/a**2
\[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^5} \, dx=\int { \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x^{5}} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x^5,x, algorithm="max ima")
Output:
2*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^7, x)/a^2 - 1/3/(a^2*x^6) - int egrate(x^(-5), x)
\[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^5} \, dx=\int { \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x^{5}} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x^5,x, algorithm="gia c")
Output:
integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))^2/x^5, x)
Time = 142.60 (sec) , antiderivative size = 2480, normalized size of antiderivative = 9.29 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^5} \, dx=\text {Too large to display} \] Input:
int(((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))^2/x^5,x)
Output:
((311*a^4*((1/(a*x) - 1)^(1/2) - 1i)^5)/(2*((1/(a*x) + 1)^(1/2) - 1)^5) - (175*a^4*((1/(a*x) - 1)^(1/2) - 1i)^3)/(6*((1/(a*x) + 1)^(1/2) - 1)^3) + ( 8361*a^4*((1/(a*x) - 1)^(1/2) - 1i)^7)/(2*((1/(a*x) + 1)^(1/2) - 1)^7) + ( 42259*a^4*((1/(a*x) - 1)^(1/2) - 1i)^9)/(3*((1/(a*x) + 1)^(1/2) - 1)^9) + (25295*a^4*((1/(a*x) - 1)^(1/2) - 1i)^11)/((1/(a*x) + 1)^(1/2) - 1)^11 + ( 25295*a^4*((1/(a*x) - 1)^(1/2) - 1i)^13)/((1/(a*x) + 1)^(1/2) - 1)^13 + (4 2259*a^4*((1/(a*x) - 1)^(1/2) - 1i)^15)/(3*((1/(a*x) + 1)^(1/2) - 1)^15) + (8361*a^4*((1/(a*x) - 1)^(1/2) - 1i)^17)/(2*((1/(a*x) + 1)^(1/2) - 1)^17) + (311*a^4*((1/(a*x) - 1)^(1/2) - 1i)^19)/(2*((1/(a*x) + 1)^(1/2) - 1)^19 ) - (175*a^4*((1/(a*x) - 1)^(1/2) - 1i)^21)/(6*((1/(a*x) + 1)^(1/2) - 1)^2 1) + (5*a^4*((1/(a*x) - 1)^(1/2) - 1i)^23)/(2*((1/(a*x) + 1)^(1/2) - 1)^23 ) + (5*a^4*((1/(a*x) - 1)^(1/2) - 1i))/(2*((1/(a*x) + 1)^(1/2) - 1)))/((66 *((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (12*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^2 - (220*((1/(a*x) - 1)^(1/ 2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + (495*((1/(a*x) - 1)^(1/2) - 1i)^ 8)/((1/(a*x) + 1)^(1/2) - 1)^8 - (792*((1/(a*x) - 1)^(1/2) - 1i)^10)/((1/( a*x) + 1)^(1/2) - 1)^10 + (924*((1/(a*x) - 1)^(1/2) - 1i)^12)/((1/(a*x) + 1)^(1/2) - 1)^12 - (792*((1/(a*x) - 1)^(1/2) - 1i)^14)/((1/(a*x) + 1)^(1/2 ) - 1)^14 + (495*((1/(a*x) - 1)^(1/2) - 1i)^16)/((1/(a*x) + 1)^(1/2) - 1)^ 16 - (220*((1/(a*x) - 1)^(1/2) - 1i)^18)/((1/(a*x) + 1)^(1/2) - 1)^18 +...
Time = 0.17 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.73 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {3 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{4} x^{4}+2 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{2} x^{2}-8 \sqrt {a x +1}\, \sqrt {-a x +1}+3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{6} x^{6}-3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{6} x^{6}+3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{6} x^{6}-3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{6} x^{6}+6 a^{2} x^{2}-8}{24 a^{2} x^{6}} \] Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x^5,x)
Output:
(3*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 - 8*sqrt(a*x + 1)*sqrt( - a*x + 1) + 3*log( - sqrt(2) + tan (asin(sqrt( - a*x + 1)/sqrt(2))/2) - 1)*a**6*x**6 - 3*log( - sqrt(2) + tan (asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1)*a**6*x**6 + 3*log(sqrt(2) + tan(as in(sqrt( - a*x + 1)/sqrt(2))/2) - 1)*a**6*x**6 - 3*log(sqrt(2) + tan(asin( sqrt( - a*x + 1)/sqrt(2))/2) + 1)*a**6*x**6 + 6*a**2*x**2 - 8)/(24*a**2*x* *6)