Integrand size = 10, antiderivative size = 46 \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\text {sech}^{-1}\left (\sqrt {x}\right )^2-2 \text {sech}^{-1}\left (\sqrt {x}\right ) \log \left (1+e^{2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right )-\operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right ) \]
arcsech(x^(1/2))^2-2*arcsech(x^(1/2))*ln(1+(1/x^(1/2)+(-1+1/x^(1/2))^(1/2) *(1+1/x^(1/2))^(1/2))^2)-polylog(2,-(1/x^(1/2)+(-1+1/x^(1/2))^(1/2)*(1+1/x ^(1/2))^(1/2))^2)
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=-\text {sech}^{-1}\left (\sqrt {x}\right ) \left (\text {sech}^{-1}\left (\sqrt {x}\right )+2 \log \left (1+e^{-2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right ) \]
-(ArcSech[Sqrt[x]]*(ArcSech[Sqrt[x]] + 2*Log[1 + E^(-2*ArcSech[Sqrt[x]])]) ) + PolyLog[2, -E^(-2*ArcSech[Sqrt[x]])]
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {7267, 6835, 6297, 3042, 26, 4201, 2620, 2715, 2838}
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.
\(\displaystyle \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}d\sqrt {x}\) |
\(\Big \downarrow \) 6835 |
\(\displaystyle -2 \int \frac {\text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}{\sqrt {x}}d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 6297 |
\(\displaystyle -2 \int \sqrt {\frac {\frac {1}{\sqrt {x}}-1}{1+\frac {1}{\sqrt {x}}}} \left (1+\frac {1}{\sqrt {x}}\right ) \sqrt {x} \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )d\text {arccosh}\left (\frac {1}{\sqrt {x}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -2 \int -i \text {arccosh}\left (\frac {1}{\sqrt {x}}\right ) \tan \left (i \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )\right )d\text {arccosh}\left (\frac {1}{\sqrt {x}}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 i \int \text {arccosh}\left (\frac {1}{\sqrt {x}}\right ) \tan \left (i \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )\right )d\text {arccosh}\left (\frac {1}{\sqrt {x}}\right )\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle 2 i \left (2 i \int \frac {e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )} \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}{1+e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}}d\text {arccosh}\left (\frac {1}{\sqrt {x}}\right )-\frac {i x}{2}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle 2 i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (\frac {1}{\sqrt {x}}\right ) \log \left (e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}\right )d\text {arccosh}\left (\frac {1}{\sqrt {x}}\right )\right )-\frac {i x}{2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle 2 i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (\frac {1}{\sqrt {x}}\right ) \log \left (e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}+1\right )-\frac {1}{4} \int e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )} \log \left (1+e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}\right )de^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}\right )-\frac {i x}{2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle 2 i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}\right )+\frac {1}{2} \text {arccosh}\left (\frac {1}{\sqrt {x}}\right ) \log \left (e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}+1\right )\right )-\frac {i x}{2}\right )\) |
(2*I)*((-1/2*I)*x + (2*I)*((ArcCosh[1/Sqrt[x]]*Log[1 + E^(2*ArcCosh[1/Sqrt [x]])])/2 + PolyLog[2, -E^(2*ArcCosh[1/Sqrt[x]])]/4))
3.1.24.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + b*ArcCosh[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.47 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.41
method | result | size |
derivativedivides | \(\operatorname {arcsech}\left (\sqrt {x}\right )^{2}-2 \,\operatorname {arcsech}\left (\sqrt {x}\right ) \ln \left (1+\left (\frac {1}{\sqrt {x}}+\sqrt {-1+\frac {1}{\sqrt {x}}}\, \sqrt {1+\frac {1}{\sqrt {x}}}\right )^{2}\right )-\operatorname {polylog}\left (2, -\left (\frac {1}{\sqrt {x}}+\sqrt {-1+\frac {1}{\sqrt {x}}}\, \sqrt {1+\frac {1}{\sqrt {x}}}\right )^{2}\right )\) | \(65\) |
default | \(\operatorname {arcsech}\left (\sqrt {x}\right )^{2}-2 \,\operatorname {arcsech}\left (\sqrt {x}\right ) \ln \left (1+\left (\frac {1}{\sqrt {x}}+\sqrt {-1+\frac {1}{\sqrt {x}}}\, \sqrt {1+\frac {1}{\sqrt {x}}}\right )^{2}\right )-\operatorname {polylog}\left (2, -\left (\frac {1}{\sqrt {x}}+\sqrt {-1+\frac {1}{\sqrt {x}}}\, \sqrt {1+\frac {1}{\sqrt {x}}}\right )^{2}\right )\) | \(65\) |
arcsech(x^(1/2))^2-2*arcsech(x^(1/2))*ln(1+(1/x^(1/2)+(-1+1/x^(1/2))^(1/2) *(1+1/x^(1/2))^(1/2))^2)-polylog(2,-(1/x^(1/2)+(-1+1/x^(1/2))^(1/2)*(1+1/x ^(1/2))^(1/2))^2)
\[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{x} \,d x } \]
\[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int \frac {\operatorname {asech}{\left (\sqrt {x} \right )}}{x}\, dx \]
\[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{x} \,d x } \]
-1/4*log(x)^2 + log(x)*log(sqrt(sqrt(x) + 1)*sqrt(-sqrt(x) + 1) + 1) - log (sqrt(x) + 1)*log(sqrt(x)) - log(sqrt(x))*log(-sqrt(x) + 1) - dilog(-sqrt( x)) - dilog(sqrt(x)) + integrate(1/2*log(x)/((x - 1)*e^(1/2*log(sqrt(x) + 1) + 1/2*log(-sqrt(x) + 1)) + x - 1), x)
\[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int \frac {\mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right )}{x} \,d x \]