Optimal. Leaf size=22 \[ 2 \sqrt {1+x}-2 \log \left (1+\sqrt {1+x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {253, 196, 45}
\begin {gather*} 2 \sqrt {x+1}-2 \log \left (\sqrt {x+1}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 196
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{1+\sqrt {1+x}} \, dx &=\text {Subst}\left (\int \frac {1}{1+\sqrt {x}} \, dx,x,1+x\right )\\ &=2 \text {Subst}\left (\int \frac {x}{1+x} \, dx,x,\sqrt {1+x}\right )\\ &=2 \text {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {1+x}-2 \log \left (1+\sqrt {1+x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} 2 \sqrt {1+x}-2 \log \left (1+\sqrt {1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 31, normalized size = 1.41
method | result | size |
derivativedivides | \(-2 \ln \left (1+\sqrt {1+x}\right )+2 \sqrt {1+x}\) | \(19\) |
trager | \(2 \sqrt {1+x}-\ln \left (2 \sqrt {1+x}+2+x \right )\) | \(22\) |
default | \(2 \sqrt {1+x}+\ln \left (-1+\sqrt {1+x}\right )-\ln \left (1+\sqrt {1+x}\right )-\ln \left (x \right )\) | \(31\) |
meijerg | \(\frac {-4 \sqrt {\pi }+4 \sqrt {\pi }\, \sqrt {1+x}-4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+x}}{2}\right )}{2 \sqrt {\pi }}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.90, size = 18, normalized size = 0.82 \begin {gather*} 2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 18, normalized size = 0.82 \begin {gather*} 2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 19, normalized size = 0.86 \begin {gather*} 2 \sqrt {x + 1} - 2 \log {\left (\sqrt {x + 1} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 18, normalized size = 0.82 \begin {gather*} 2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 18, normalized size = 0.82 \begin {gather*} 2\,\sqrt {x+1}-2\,\ln \left (\sqrt {x+1}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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