Optimal. Leaf size=43 \[ -\frac {1}{2} \sqrt {x} \sqrt {x+1}+x \log \left (\sqrt {x}+\sqrt {x+1}\right )+\frac {1}{2} \sinh ^{-1}\left (\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2548, 12, 1958, 50, 54, 215} \[ -\frac {1}{2} \sqrt {x} \sqrt {x+1}+x \log \left (\sqrt {x}+\sqrt {x+1}\right )+\frac {1}{2} \sinh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 50
Rule 54
Rule 215
Rule 1958
Rule 2548
Rubi steps
\begin {align*} \int \log \left (\sqrt {x}+\sqrt {1+x}\right ) \, dx &=x \log \left (\sqrt {x}+\sqrt {1+x}\right )-\int \frac {1}{2} \sqrt {\frac {x}{1+x}} \, dx\\ &=x \log \left (\sqrt {x}+\sqrt {1+x}\right )-\frac {1}{2} \int \sqrt {\frac {x}{1+x}} \, dx\\ &=x \log \left (\sqrt {x}+\sqrt {1+x}\right )-\frac {1}{2} \int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx\\ &=-\frac {1}{2} \sqrt {x} \sqrt {1+x}+x \log \left (\sqrt {x}+\sqrt {1+x}\right )+\frac {1}{4} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx\\ &=-\frac {1}{2} \sqrt {x} \sqrt {1+x}+x \log \left (\sqrt {x}+\sqrt {1+x}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{2} \sqrt {x} \sqrt {1+x}+\frac {1}{2} \sinh ^{-1}\left (\sqrt {x}\right )+x \log \left (\sqrt {x}+\sqrt {1+x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 43, normalized size = 1.00 \[ -\frac {1}{2} \sqrt {x} \sqrt {x+1}+x \log \left (\sqrt {x}+\sqrt {x+1}\right )+\frac {1}{2} \sinh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 28, normalized size = 0.65 \[ \frac {1}{2} \, {\left (2 \, x + 1\right )} \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x + 1} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 40, normalized size = 0.93 \[ x \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x^{2} + x} - \frac {1}{4} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 52, normalized size = 1.21 \[ x \ln \left (\sqrt {x}+\sqrt {x +1}\right )-\frac {\sqrt {x +1}\, \sqrt {x}}{2}+\frac {\sqrt {\left (x +1\right ) x}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{4 \sqrt {x +1}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ x \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{2} \, x - \int \frac {x}{2 \, {\left (x^{2} + {\left (x^{\frac {3}{2}} + \sqrt {x}\right )} \sqrt {x + 1} + x\right )}}\,{d x} + \frac {1}{2} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.08, size = 37, normalized size = 0.86 \[ \mathrm {atanh}\left (\frac {\sqrt {x}}{\sqrt {x+1}-1}\right )-\frac {\sqrt {x}\,\sqrt {x+1}}{2}+x\,\ln \left (\sqrt {x+1}+\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log {\left (\sqrt {x} + \sqrt {x + 1} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________