Optimal. Leaf size=34 \[ 2 \sqrt {\sqrt {x}+x}-2 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {\sqrt {x}+x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2035, 2038,
634, 212} \begin {gather*} 2 \sqrt {x+\sqrt {x}}-2 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x+\sqrt {x}}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 634
Rule 2035
Rule 2038
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\sqrt {x}+x}} \, dx &=2 \sqrt {\sqrt {x}+x}-\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {\sqrt {x}+x}} \, dx\\ &=2 \sqrt {\sqrt {x}+x}-\text {Subst}\left (\int \frac {1}{\sqrt {x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {\sqrt {x}+x}-2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {\sqrt {x}+x}}\right )\\ &=2 \sqrt {\sqrt {x}+x}-2 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {\sqrt {x}+x}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 37, normalized size = 1.09 \begin {gather*} 2 \sqrt {\sqrt {x}+x}+\log \left (-1-2 \sqrt {x}+2 \sqrt {\sqrt {x}+x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.38, size = 45, normalized size = 1.32
method | result | size |
derivativedivides | \(2 \sqrt {x +\sqrt {x}}-\ln \left (\frac {1}{2}+\sqrt {x}+\sqrt {x +\sqrt {x}}\right )\) | \(26\) |
meijerg | \(\frac {2 \sqrt {\pi }\, x^{\frac {1}{4}} \sqrt {1+\sqrt {x}}-2 \sqrt {\pi }\, \arcsinh \left (x^{\frac {1}{4}}\right )}{\sqrt {\pi }}\) | \(30\) |
default | \(\frac {\sqrt {x +\sqrt {x}}\, \left (2 \sqrt {x +\sqrt {x}}-\ln \left (\frac {1}{2}+\sqrt {x}+\sqrt {x +\sqrt {x}}\right )\right )}{\sqrt {\sqrt {x}\, \left (1+\sqrt {x}\right )}}\) | \(45\) |
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 = 0.56, size = 39, normalized size = 1.15 \begin {gather*} 2 \, \sqrt {x + \sqrt {x}} + \frac {1}{2} \, \log \left (4 \, \sqrt {x + \sqrt {x}} {\left (2 \, \sqrt {x} + 1\right )} - 8 \, x - 8 \, \sqrt {x} - 1\right ) \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 \frac {1}{\sqrt {\sqrt {x} + x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.08, size = 27, normalized size = 0.79 \begin {gather*} 2 \, \sqrt {x + \sqrt {x}} + \log \left (-2 \, \sqrt {x + \sqrt {x}} + 2 \, \sqrt {x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.33, size = 39, normalized size = 1.15 \begin {gather*} \frac {2\,\sqrt {x}\,\left (\sqrt {x}+1\right )+x^{1/4}\,\mathrm {asin}\left (x^{1/4}\,1{}\mathrm {i}\right )\,\sqrt {\sqrt {x}+1}\,2{}\mathrm {i}}{\sqrt {x+\sqrt {x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________