Optimal. Leaf size=25 \[ x+4 \sqrt {x+1}+4 \log \left (1-\sqrt {x+1}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {431, 376, 77} \[ x+4 \sqrt {x+1}+4 \log \left (1-\sqrt {x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 77
Rule 376
Rule 431
Rubi steps
\begin {align*} \int \frac {1+\sqrt {1+x}}{-1+\sqrt {1+x}} \, dx &=\operatorname {Subst}\left (\int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx,x,1+x\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x (1+x)}{-1+x} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (2+\frac {2}{-1+x}+x\right ) \, dx,x,\sqrt {1+x}\right )\\ &=x+4 \sqrt {1+x}+4 \log \left (1-\sqrt {1+x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.96 \[ x+4 \left (\sqrt {x+1}+\log \left (1-\sqrt {x+1}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 23, normalized size = 0.92 \[ x+4 \sqrt {x+1}+4 \log \left (\sqrt {x+1}-1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 19, normalized size = 0.76 \[ x + 4 \, \sqrt {x + 1} + 4 \, \log \left (\sqrt {x + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 21, normalized size = 0.84 \[ x + 4 \, \sqrt {x + 1} + 4 \, \log \left ({\left | \sqrt {x + 1} - 1 \right |}\right ) + 1 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 21, normalized size = 0.84
method | result | size |
derivativedivides | \(1+x +4 \sqrt {1+x}+4 \ln \left (-1+\sqrt {1+x}\right )\) | \(21\) |
default | \(1+x +4 \sqrt {1+x}+4 \ln \left (-1+\sqrt {1+x}\right )\) | \(21\) |
trager | \(-1+x +4 \sqrt {1+x}+2 \ln \left (2 \sqrt {1+x}-2-x \right )\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 20, normalized size = 0.80 \[ x + 4 \, \sqrt {x + 1} + 4 \, \log \left (\sqrt {x + 1} - 1\right ) + 1 \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 19, normalized size = 0.76 \[ x+4\,\ln \left (\sqrt {x+1}-1\right )+4\,\sqrt {x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 20, normalized size = 0.80 \[ x + 4 \sqrt {x + 1} + 4 \log {\left (\sqrt {x + 1} - 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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