Integrand size = 12, antiderivative size = 86 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x} \, dx=-\frac {2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {2}{1-\sqrt {\frac {1-a x}{1+a x}}}-\log (1+a x)-2 \log \left (1-\sqrt {\frac {1-a x}{1+a x}}\right ) \] Output:
-2/(1-((-a*x+1)/(a*x+1))^(1/2))^2+2/(1-((-a*x+1)/(a*x+1))^(1/2))-ln(a*x+1) -2*ln(1-((-a*x+1)/(a*x+1))^(1/2))
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x} \, dx=-\frac {1}{a^2 x^2}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a^2 x^2}-2 \log (x)+\log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right ) \] Input:
Integrate[E^(2*ArcSech[a*x])/x,x]
Output:
-(1/(a^2*x^2)) - (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(a^2*x^2) - 2*Log[x ] + Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]]
Time = 1.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6891, 7268, 25, 2160, 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.
\(\displaystyle \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x} \, dx\) |
\(\Big \downarrow \) 6891 |
\(\displaystyle \int \frac {\left (\frac {\sqrt {\frac {1-a x}{a x+1}}}{a x}+\sqrt {\frac {1-a x}{a x+1}}+\frac {1}{a x}\right )^2}{x}dx\) |
\(\Big \downarrow \) 7268 |
\(\displaystyle 4 \int -\frac {\sqrt {\frac {1-a x}{a x+1}} \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3 \left (\frac {1-a x}{a x+1}+1\right )}d\sqrt {\frac {1-a x}{a x+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -4 \int \frac {\sqrt {\frac {1-a x}{a x+1}} \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3 \left (\frac {1-a x}{a x+1}+1\right )}d\sqrt {\frac {1-a x}{a x+1}}\) |
\(\Big \downarrow \) 2160 |
\(\displaystyle -4 \int \left (-\frac {\sqrt {\frac {1-a x}{a x+1}}}{2 \left (\frac {1-a x}{a x+1}+1\right )}+\frac {1}{2 \left (\sqrt {\frac {1-a x}{a x+1}}-1\right )}-\frac {1}{2 \left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^2}-\frac {1}{\left (\sqrt {\frac {1-a x}{a x+1}}-1\right )^3}\right )d\sqrt {\frac {1-a x}{a x+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (\frac {1}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {1}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\frac {1}{2} \log \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )+\frac {1}{4} \log \left (\frac {1-a x}{a x+1}+1\right )\right )\) |
Input:
Int[E^(2*ArcSech[a*x])/x,x]
Output:
4*(-1/2*1/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2 + 1/(2*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - Log[1 - Sqrt[(1 - a*x)/(1 + a*x)]]/2 + Log[1 + (1 - a*x)/(1 + a*x)]/4)
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
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 = 0.16 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.28
method | result | size |
default | \(\frac {-\frac {1}{2 x^{2}}-a^{2} \ln \left (x \right )}{a^{2}}-\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (-a^{2} x^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\sqrt {-a^{2} x^{2}+1}\right )}{a x \sqrt {-a^{2} x^{2}+1}}-\frac {1}{2 a^{2} x^{2}}\) | \(110\) |
Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x,x,method=_RETURNVERBOSE)
Output:
1/a^2*(-1/2/x^2-a^2*ln(x))-1/a*(-(a*x-1)/a/x)^(1/2)/x*((a*x+1)/a/x)^(1/2)* (-a^2*x^2*arctanh(1/(-a^2*x^2+1)^(1/2))+(-a^2*x^2+1)^(1/2))/(-a^2*x^2+1)^( 1/2)-1/2/a^2/x^2
Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.60 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x} \, dx=\frac {a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 2}{2 \, a^{2} x^{2}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x,x, algorithm="frica s")
Output:
1/2*(a^2*x^2*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - a ^2*x^2*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 1) - 2*a^2*x ^2*log(x) - 2*a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 2)/(a^2*x ^2)
Time = 2.95 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x} \, dx=\frac {- 2 a^{2} \cdot \left (2 \sqrt {-1 + \frac {1}{a x}} \left (\frac {\left (1 + \frac {1}{a x}\right )^{\frac {3}{2}}}{4} - \frac {\sqrt {1 + \frac {1}{a x}}}{4}\right ) - \log {\left (2 \sqrt {-1 + \frac {1}{a x}} + 2 \sqrt {1 + \frac {1}{a x}} \right )}\right ) - a^{2} \log {\left (x \right )} - \frac {1}{x^{2}}}{a^{2}} \] Input:
integrate((1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2))**2/x,x)
Output:
(-2*a**2*(2*sqrt(-1 + 1/(a*x))*((1 + 1/(a*x))**(3/2)/4 - sqrt(1 + 1/(a*x)) /4) - log(2*sqrt(-1 + 1/(a*x)) + 2*sqrt(1 + 1/(a*x)))) - a**2*log(x) - 1/x **2)/a**2
\[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x} \, dx=\int { \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x,x, algorithm="maxim a")
Output:
2*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^3, x)/a^2 - 1/(a^2*x^2) - integ rate(1/x, x)
\[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x} \, dx=\int { \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x,x, algorithm="giac" )
Output:
integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))^2/x, x)
Time = 36.00 (sec) , antiderivative size = 323, normalized size of antiderivative = 3.76 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x} \, dx=\ln \left (\frac {1}{x}\right )-4\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )+2\,\mathrm {acosh}\left (\frac {1}{a\,x}\right )+\frac {\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {4\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}}{1+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}-\frac {1}{a^2\,x^2} \] Input:
int(((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))^2/x,x)
Output:
log(1/x) - 4*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)) + 2*acosh(1/(a*x)) + ((28*((1/(a*x) - 1)^(1/2) - 1i)^3)/((1/(a*x) + 1)^(1/2 ) - 1)^3 + (28*((1/(a*x) - 1)^(1/2) - 1i)^5)/((1/(a*x) + 1)^(1/2) - 1)^5 + (4*((1/(a*x) - 1)^(1/2) - 1i)^7)/((1/(a*x) + 1)^(1/2) - 1)^7 + (4*((1/(a* x) - 1)^(1/2) - 1i))/((1/(a*x) + 1)^(1/2) - 1))/((6*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (4*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1 /(a*x) + 1)^(1/2) - 1)^2 - (4*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1) ^(1/2) - 1)^6 + ((1/(a*x) - 1)^(1/2) - 1i)^8/((1/(a*x) + 1)^(1/2) - 1)^8 + 1) - 1/(a^2*x^2)
Time = 0.15 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.77 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x} \, dx=\frac {-\sqrt {a x +1}\, \sqrt {-a x +1}+\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{2} x^{2}-\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{2} x^{2}+\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{2} x^{2}-\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{2} x^{2}-\mathrm {log}\left (x \right ) a^{2} x^{2}-1}{a^{2} x^{2}} \] Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2/x,x)
Output:
( - sqrt(a*x + 1)*sqrt( - a*x + 1) + log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) - 1)*a**2*x**2 - log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1)*a**2*x**2 + log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/s qrt(2))/2) - 1)*a**2*x**2 - log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2 ))/2) + 1)*a**2*x**2 - log(x)*a**2*x**2 - 1)/(a**2*x**2)