Optimal. Leaf size=43 \[ -\frac {1}{2} \sqrt {x} \sqrt {1+x}+\frac {1}{2} \sinh ^{-1}\left (\sqrt {x}\right )+x \log \left (\sqrt {x}+\sqrt {1+x}\right ) \]
[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 = {2628, 12, 1978,
52, 56, 221} \begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 52
Rule 56
Rule 221
Rule 1978
Rule 2628
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} \text {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 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 52, normalized size = 1.21
method | result | size |
default | \(x \ln \left (\sqrt {x}+\sqrt {1+x}\right )-\frac {\sqrt {x}\, \sqrt {1+x}}{2}+\frac {\sqrt {x \left (1+x \right )}\, \ln \left (\frac {1}{2}+x +\sqrt {x^{2}+x}\right )}{4 \sqrt {x}\, \sqrt {1+x}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 28, normalized size = 0.65 \begin {gather*} \frac {1}{2} \, {\left (2 \, x + 1\right )} \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x + 1} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \log {\left (\sqrt {x} + \sqrt {x + 1} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 5.46, size = 40, normalized size = 0.93 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.08, size = 37, normalized size = 0.86 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________