Optimal. Leaf size=29 \[ -\frac {\sqrt {x^2+2 x+2}}{x+1}+\frac {1}{x+1}+\sinh ^{-1}(x+1) \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6742, 684, 619, 215} \[ -\frac {\sqrt {x^2+2 x+2}}{x+1}+\frac {1}{x+1}+\sinh ^{-1}(x+1) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 619
Rule 684
Rule 6742
Rubi steps
\begin {align*} \int \frac {1}{1+\sqrt {2+2 x+x^2}} \, dx &=\int \left (-\frac {1}{(1+x)^2}+\frac {\sqrt {2+2 x+x^2}}{(1+x)^2}\right ) \, dx\\ &=\frac {1}{1+x}+\int \frac {\sqrt {2+2 x+x^2}}{(1+x)^2} \, dx\\ &=\frac {1}{1+x}-\frac {\sqrt {2+2 x+x^2}}{1+x}+\int \frac {1}{\sqrt {2+2 x+x^2}} \, dx\\ &=\frac {1}{1+x}-\frac {\sqrt {2+2 x+x^2}}{1+x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4}}} \, dx,x,2+2 x\right )\\ &=\frac {1}{1+x}-\frac {\sqrt {2+2 x+x^2}}{1+x}+\sinh ^{-1}(1+x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 30, normalized size = 1.03 \[ \frac {-\sqrt {x^2+2 x+2}+(x+1) \sinh ^{-1}(x+1)+1}{x+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.29, size = 46, normalized size = 1.59 \[ \frac {\sqrt {x^2+2 x+2}}{-x-1}-\log \left (\sqrt {x^2+2 x+2}-x-1\right )+\frac {1}{x+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.68, size = 39, normalized size = 1.34 \[ -\frac {{\left (x + 1\right )} \log \left (-x + \sqrt {x^{2} + 2 \, x + 2} - 1\right ) + x + \sqrt {x^{2} + 2 \, x + 2}}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.66, size = 60, normalized size = 2.07 \[ \frac {2}{{\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )}^{2} + 2 \, x - 2 \, \sqrt {x^{2} + 2 \, x + 2}} + \frac {1}{x + 1} - \log \left (-x + \sqrt {x^{2} + 2 \, x + 2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.14, size = 40, normalized size = 1.38
method | result | size |
default | \(-\frac {\left (\left (1+x \right )^{2}+1\right )^{\frac {3}{2}}}{1+x}+\left (1+x \right ) \sqrt {\left (1+x \right )^{2}+1}+\arcsinh \left (1+x \right )+\frac {1}{1+x}\) | \(40\) |
trager | \(-\frac {x}{1+x}-\frac {\sqrt {x^{2}+2 x +2}}{1+x}-\ln \left (\sqrt {x^{2}+2 x +2}-1-x \right )\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} + 2 \, x + 2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \frac {1}{x+1}+\int \frac {\sqrt {x^2+2\,x+2}}{{\left (x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} + 2 x + 2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________