Integrand size = 6, antiderivative size = 52 \[ \int e^{\text {sech}^{-1}(a x)} \, dx=\frac {(1+a x) \sqrt {-1+\frac {2}{1+a x}}}{a}-\frac {2 \text {arctanh}\left (\sqrt {-1+\frac {2}{1+a x}}\right )}{a}+\frac {\log (x)}{a} \] Output:
(a*x+1)*(-1+2/(a*x+1))^(1/2)/a-2*arctanh((-1+2/(a*x+1))^(1/2))/a+ln(x)/a
Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.52 \[ \int e^{\text {sech}^{-1}(a x)} \, dx=\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)+2 \log (a x)-\log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{a} \] Input:
Integrate[E^ArcSech[a*x],x]
Output:
(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) + 2*Log[a*x] - Log[1 + Sqrt[(1 - a*x) /(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/a
Time = 0.47 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6883, 2056, 221}
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.
\(\displaystyle \int e^{\text {sech}^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6883 |
\(\displaystyle \frac {\int \frac {\sqrt {\frac {1-a x}{a x+1}}}{x (1-a x)}dx}{a}+\frac {\log (x)}{a}+x e^{\text {sech}^{-1}(a x)}\) |
\(\Big \downarrow \) 2056 |
\(\displaystyle -4 \int \frac {1}{2 a-\frac {2 a (1-a x)}{a x+1}}d\sqrt {\frac {1-a x}{a x+1}}+\frac {\log (x)}{a}+x e^{\text {sech}^{-1}(a x)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\sqrt {\frac {1-a x}{a x+1}}\right )}{a}+\frac {\log (x)}{a}+x e^{\text {sech}^{-1}(a x)}\) |
Input:
Int[E^ArcSech[a*x],x]
Output:
E^ArcSech[a*x]*x - (2*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]])/a + Log[x]/a
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(u_)^(r_.)*(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.) *(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Denominator[p]}, Simp[q*e*((b*c - a*d)/n) Subst[Int[SimplifyIntegrand[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q) ^((m + 1)/n - 1)/(b*e - d*x^q)^((m + 1)/n + 1))*(u /. x -> ((-a)*e + c*x^q) ^(1/n)/(b*e - d*x^q)^(1/n))^r, x], x], x, (e*((a + b*x^n)/(c + d*x^n)))^(1/ q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && FractionQ[p] && IntegerQ[1/n] && IntegersQ[m, r]
Int[E^ArcSech[(a_.)*(x_)], x_Symbol] :> Simp[x*E^ArcSech[a*x], x] + (Simp[L og[x]/a, x] + Simp[1/a Int[(1/(x*(1 - a*x)))*Sqrt[(1 - a*x)/(1 + a*x)], x ], x]) /; FreeQ[a, x]
Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.54
method | result | size |
default | \(\frac {\ln \left (x \right )}{a}-\frac {\sqrt {-\frac {a x -1}{a x}}\, x \sqrt {\frac {a x +1}{a x}}\, \left (-\sqrt {-a^{2} x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{\sqrt {-a^{2} x^{2}+1}}\) | \(80\) |
Input:
int(1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2),x,method=_RETURNVERBOSE)
Output:
ln(x)/a-(-(a*x-1)/a/x)^(1/2)*x*((a*x+1)/a/x)^(1/2)*(-(-a^2*x^2+1)^(1/2)+ar ctanh(1/(-a^2*x^2+1)^(1/2)))/(-a^2*x^2+1)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (48) = 96\).
Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.21 \[ \int e^{\text {sech}^{-1}(a x)} \, dx=\frac {2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) + \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, \log \left (x\right )}{2 \, a} \] Input:
integrate(1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2),x, algorithm="fricas")
Output:
1/2*(2*a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - log(a*x*sqrt((a* x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) + log(a*x*sqrt((a*x + 1)/(a*x))* sqrt(-(a*x - 1)/(a*x)) - 1) + 2*log(x))/a
\[ \int e^{\text {sech}^{-1}(a x)} \, dx=\frac {\int \frac {1}{x}\, dx + \int a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a} \] Input:
integrate(1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2),x)
Output:
(Integral(1/x, x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)), x))/a
\[ \int e^{\text {sech}^{-1}(a x)} \, dx=\int { \sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x} \,d x } \] Input:
integrate(1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x), x)
\[ \int e^{\text {sech}^{-1}(a x)} \, dx=\int { \sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x} \,d x } \] Input:
integrate(1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x), x)
Time = 26.12 (sec) , antiderivative size = 182, normalized size of antiderivative = 3.50 \[ \int e^{\text {sech}^{-1}(a x)} \, dx=\frac {\ln \left (x\right )}{a}-\frac {4\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )}{a}+\frac {\frac {5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1}{\frac {4\,a\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}+\frac {4\,a\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}}+\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{4\,a\,\left (\sqrt {\frac {1}{a\,x}+1}-1\right )} \] Input:
int((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x),x)
Output:
log(x)/a - (4*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1))) /a + ((5*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^2 + 1)/(( 4*a*((1/(a*x) - 1)^(1/2) - 1i))/((1/(a*x) + 1)^(1/2) - 1) + (4*a*((1/(a*x) - 1)^(1/2) - 1i)^3)/((1/(a*x) + 1)^(1/2) - 1)^3) + ((1/(a*x) - 1)^(1/2) - 1i)/(4*a*((1/(a*x) + 1)^(1/2) - 1))
Time = 0.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.17 \[ \int e^{\text {sech}^{-1}(a x)} \, dx=\frac {\sqrt {a x +1}\, \sqrt {-a x +1}-\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right )+\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right )-\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right )+\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right )+\mathrm {log}\left (x \right )}{a} \] Input:
int(1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2),x)
Output:
(sqrt(a*x + 1)*sqrt( - a*x + 1) - log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) - 1) + log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/ 2) + 1) - log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) - 1) + log(s qrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1) + log(x))/a