Integrand size = 16, antiderivative size = 22 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=-2 \text {arctanh}\left (\sqrt {1+\log (x)}\right )+2 \sqrt {1+\log (x)} \] Output:
-2*arctanh((1+ln(x))^(1/2))+2*(1+ln(x))^(1/2)
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=-2 \text {arctanh}\left (\sqrt {1+\log (x)}\right )+2 \sqrt {1+\log (x)} \] Input:
Integrate[Sqrt[1 + Log[x]]/(x*Log[x]),x]
Output:
-2*ArcTanh[Sqrt[1 + Log[x]]] + 2*Sqrt[1 + Log[x]]
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2812, 60, 73, 220}
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.
\(\displaystyle \int \frac {\sqrt {\log (x)+1}}{x \log (x)} \, dx\) |
\(\Big \downarrow \) 2812 |
\(\displaystyle \int \frac {\sqrt {\log (x)+1}}{\log (x)}d\log (x)\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \int \frac {1}{\log (x) \sqrt {\log (x)+1}}d\log (x)+2 \sqrt {\log (x)+1}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 2 \int \frac {1}{\log (x)}d\sqrt {\log (x)+1}+2 \sqrt {\log (x)+1}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle 2 \sqrt {\log (x)+1}-2 \text {arctanh}\left (\sqrt {\log (x)+1}\right )\) |
Input:
Int[Sqrt[1 + Log[x]]/(x*Log[x]),x]
Output:
-2*ArcTanh[Sqrt[1 + Log[x]]] + 2*Sqrt[1 + Log[x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(c_.)*(x_)^(n _.)]*(e_.))^(q_.))/(x_), x_Symbol] :> Simp[1/n Subst[Int[(a + b*x)^p*(d + e*x)^q, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x]
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36
method | result | size |
derivativedivides | \(2 \sqrt {1+\ln \left (x \right )}+\ln \left (\sqrt {1+\ln \left (x \right )}-1\right )-\ln \left (\sqrt {1+\ln \left (x \right )}+1\right )\) | \(30\) |
default | \(2 \sqrt {1+\ln \left (x \right )}+\ln \left (\sqrt {1+\ln \left (x \right )}-1\right )-\ln \left (\sqrt {1+\ln \left (x \right )}+1\right )\) | \(30\) |
Input:
int((1+ln(x))^(1/2)/x/ln(x),x,method=_RETURNVERBOSE)
Output:
2*(1+ln(x))^(1/2)+ln((1+ln(x))^(1/2)-1)-ln((1+ln(x))^(1/2)+1)
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2 \, \sqrt {\log \left (x\right ) + 1} - \log \left (\sqrt {\log \left (x\right ) + 1} + 1\right ) + \log \left (\sqrt {\log \left (x\right ) + 1} - 1\right ) \] Input:
integrate((1+log(x))^(1/2)/x/log(x),x, algorithm="fricas")
Output:
2*sqrt(log(x) + 1) - log(sqrt(log(x) + 1) + 1) + log(sqrt(log(x) + 1) - 1)
Time = 0.52 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2 \sqrt {\log {\left (x \right )} + 1} + \log {\left (\sqrt {\log {\left (x \right )} + 1} - 1 \right )} - \log {\left (\sqrt {\log {\left (x \right )} + 1} + 1 \right )} \] Input:
integrate((1+ln(x))**(1/2)/x/ln(x),x)
Output:
2*sqrt(log(x) + 1) + log(sqrt(log(x) + 1) - 1) - log(sqrt(log(x) + 1) + 1)
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2 \, \sqrt {\log \left (x\right ) + 1} - \log \left (\sqrt {\log \left (x\right ) + 1} + 1\right ) + \log \left (\sqrt {\log \left (x\right ) + 1} - 1\right ) \] Input:
integrate((1+log(x))^(1/2)/x/log(x),x, algorithm="maxima")
Output:
2*sqrt(log(x) + 1) - log(sqrt(log(x) + 1) + 1) + log(sqrt(log(x) + 1) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (18) = 36\).
Time = 0.31 (sec) , antiderivative size = 268, normalized size of antiderivative = 12.18 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2 \, \sqrt {\log \left (x\right ) + 1} - \log \left (\sqrt {\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \sqrt {-\pi ^{2} \mathrm {sgn}\left (x\right ) + \pi ^{2} + 2 \, \log \left ({\left | x \right |}\right )^{2} + 4 \, \log \left ({\left | x \right |}\right ) + 2}\right )} + {\left (-8 \, \pi ^{2} \mathrm {sgn}\left (x\right ) + 8 \, \pi ^{2} + 16 \, \log \left ({\left | x \right |}\right )^{2} + 32 \, \log \left ({\left | x \right |}\right ) + 16\right )}^{\frac {1}{4}} \cos \left (-\frac {1}{4} \, \pi \mathrm {sgn}\left (\frac {1}{2} \, \pi - \frac {1}{2} \, \pi \mathrm {sgn}\left (x\right )\right ) \mathrm {sgn}\left (\log \left ({\left | x \right |}\right ) + 1\right ) + \frac {1}{4} \, \pi \mathrm {sgn}\left (\frac {1}{2} \, \pi - \frac {1}{2} \, \pi \mathrm {sgn}\left (x\right )\right ) + \frac {1}{2} \, \arctan \left (-\frac {\pi \mathrm {sgn}\left (x\right )}{2 \, {\left (\log \left ({\left | x \right |}\right ) + 1\right )}} + \frac {\pi }{2 \, {\left (\log \left ({\left | x \right |}\right ) + 1\right )}}\right )\right )}\right ) + \log \left (\sqrt {\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \sqrt {-\pi ^{2} \mathrm {sgn}\left (x\right ) + \pi ^{2} + 2 \, \log \left ({\left | x \right |}\right )^{2} + 4 \, \log \left ({\left | x \right |}\right ) + 2}\right )} - {\left (-8 \, \pi ^{2} \mathrm {sgn}\left (x\right ) + 8 \, \pi ^{2} + 16 \, \log \left ({\left | x \right |}\right )^{2} + 32 \, \log \left ({\left | x \right |}\right ) + 16\right )}^{\frac {1}{4}} \cos \left (-\frac {1}{4} \, \pi \mathrm {sgn}\left (\frac {1}{2} \, \pi - \frac {1}{2} \, \pi \mathrm {sgn}\left (x\right )\right ) \mathrm {sgn}\left (\log \left ({\left | x \right |}\right ) + 1\right ) + \frac {1}{4} \, \pi \mathrm {sgn}\left (\frac {1}{2} \, \pi - \frac {1}{2} \, \pi \mathrm {sgn}\left (x\right )\right ) + \frac {1}{2} \, \arctan \left (-\frac {\pi \mathrm {sgn}\left (x\right )}{2 \, {\left (\log \left ({\left | x \right |}\right ) + 1\right )}} + \frac {\pi }{2 \, {\left (\log \left ({\left | x \right |}\right ) + 1\right )}}\right )\right )}\right ) \] Input:
integrate((1+log(x))^(1/2)/x/log(x),x, algorithm="giac")
Output:
2*sqrt(log(x) + 1) - log(sqrt(1/2*sqrt(2)*(sqrt(2) + sqrt(-pi^2*sgn(x) + p i^2 + 2*log(abs(x))^2 + 4*log(abs(x)) + 2)) + (-8*pi^2*sgn(x) + 8*pi^2 + 1 6*log(abs(x))^2 + 32*log(abs(x)) + 16)^(1/4)*cos(-1/4*pi*sgn(1/2*pi - 1/2* pi*sgn(x))*sgn(log(abs(x)) + 1) + 1/4*pi*sgn(1/2*pi - 1/2*pi*sgn(x)) + 1/2 *arctan(-1/2*pi*sgn(x)/(log(abs(x)) + 1) + 1/2*pi/(log(abs(x)) + 1))))) + log(sqrt(1/2*sqrt(2)*(sqrt(2) + sqrt(-pi^2*sgn(x) + pi^2 + 2*log(abs(x))^2 + 4*log(abs(x)) + 2)) - (-8*pi^2*sgn(x) + 8*pi^2 + 16*log(abs(x))^2 + 32* log(abs(x)) + 16)^(1/4)*cos(-1/4*pi*sgn(1/2*pi - 1/2*pi*sgn(x))*sgn(log(ab s(x)) + 1) + 1/4*pi*sgn(1/2*pi - 1/2*pi*sgn(x)) + 1/2*arctan(-1/2*pi*sgn(x )/(log(abs(x)) + 1) + 1/2*pi/(log(abs(x)) + 1)))))
Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2\,\sqrt {\ln \left (x\right )+1}-2\,\mathrm {atanh}\left (\sqrt {\ln \left (x\right )+1}\right ) \] Input:
int((log(x) + 1)^(1/2)/(x*log(x)),x)
Output:
2*(log(x) + 1)^(1/2) - 2*atanh((log(x) + 1)^(1/2))
\[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2 \sqrt {\mathrm {log}\left (x \right )+1}+\int \frac {\sqrt {\mathrm {log}\left (x \right )+1}}{\mathrm {log}\left (x \right )^{2} x +\mathrm {log}\left (x \right ) x}d x \] Input:
int((1+log(x))^(1/2)/x/log(x),x)
Output:
2*sqrt(log(x) + 1) + int(sqrt(log(x) + 1)/(log(x)**2*x + log(x)*x),x)