Optimal. Leaf size=78 \[ \frac {1}{6} \sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {1}{12} (-3+8 x) \sqrt {x+\sqrt {1+x}}-\frac {5}{8} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.01, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {654, 626, 635,
212} \begin {gather*} \frac {2}{3} \left (x+\sqrt {x+1}\right )^{3/2}-\frac {1}{4} \left (2 \sqrt {x+1}+1\right ) \sqrt {x+\sqrt {x+1}}+\frac {5}{8} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rubi steps
\begin {align*} \int \sqrt {x+\sqrt {1+x}} \, dx &=2 \text {Subst}\left (\int x \sqrt {-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {2}{3} \left (x+\sqrt {1+x}\right )^{3/2}-\text {Subst}\left (\int \sqrt {-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {2}{3} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {1}{4} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )+\frac {5}{8} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {2}{3} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {1}{4} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )+\frac {5}{4} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=\frac {2}{3} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {1}{4} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )+\frac {5}{8} \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 65, normalized size = 0.83 \begin {gather*} \frac {1}{12} \sqrt {x+\sqrt {1+x}} \left (-11+2 \sqrt {1+x}+8 (1+x)\right )-\frac {5}{8} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 52, normalized size = 0.67
method | result | size |
derivativedivides | \(\frac {2 \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{4}+\frac {5 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{8}\) | \(52\) |
default | \(\frac {2 \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{4}+\frac {5 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{8}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.57, size = 59, normalized size = 0.76 \begin {gather*} \frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x + 1} - 3\right )} \sqrt {x + \sqrt {x + 1}} + \frac {5}{16} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x + 1} + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x + \sqrt {x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 53, normalized size = 0.68 \begin {gather*} \frac {1}{12} \, {\left (2 \, \sqrt {x + 1} {\left (4 \, \sqrt {x + 1} + 1\right )} - 11\right )} \sqrt {x + \sqrt {x + 1}} - \frac {5}{8} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {x+\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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