Optimal. Leaf size=25 \[ x+4 \sqrt {1+x}+4 \log \left (1-\sqrt {1+x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, 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 = {442, 383, 78}
\begin {gather*} x+4 \sqrt {x+1}+4 \log \left (1-\sqrt {x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 383
Rule 442
Rubi steps
\begin {align*} \int \frac {1+\sqrt {1+x}}{-1+\sqrt {1+x}} \, dx &=\text {Subst}\left (\int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx,x,1+x\right )\\ &=2 \text {Subst}\left (\int \frac {x (1+x)}{-1+x} \, dx,x,\sqrt {1+x}\right )\\ &=2 \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 24, normalized size = 0.96 \begin {gather*} 1+x+4 \sqrt {1+x}+4 \log \left (-1+\sqrt {1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.05, 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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 2.49, size = 20, normalized size = 0.80 \begin {gather*} x + 4 \, \sqrt {x + 1} + 4 \, \log \left (\sqrt {x + 1} - 1\right ) + 1 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 19, normalized size = 0.76 \begin {gather*} x + 4 \, \sqrt {x + 1} + 4 \, \log \left (\sqrt {x + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 20, normalized size = 0.80 \begin {gather*} x + 4 \sqrt {x + 1} + 4 \log {\left (\sqrt {x + 1} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.70, size = 21, normalized size = 0.84 \begin {gather*} x + 4 \, \sqrt {x + 1} + 4 \, \log \left ({\left | \sqrt {x + 1} - 1 \right |}\right ) + 1 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.22, size = 19, normalized size = 0.76 \begin {gather*} x+4\,\ln \left (\sqrt {x+1}-1\right )+4\,\sqrt {x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________