Optimal. Leaf size=56 \[ -\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {-1+x}}{\sqrt {2}}\right )}{8 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {43, 44, 65, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {x-1}}{\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {\sqrt {x-1}}{8 (x+1)}-\frac {\sqrt {x-1}}{2 (x+1)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 44
Rule 65
Rule 209
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx &=-\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {1}{4} \int \frac {1}{\sqrt {-1+x} (1+x)^2} \, dx\\ &=-\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {1}{16} \int \frac {1}{\sqrt {-1+x} (1+x)} \, dx\\ &=-\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt {-1+x}\right )\\ &=-\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {-1+x}}{\sqrt {2}}\right )}{8 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 43, normalized size = 0.77 \begin {gather*} \frac {(-3+x) \sqrt {-1+x}}{8 (1+x)^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {-1+x}}{\sqrt {2}}\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.20, size = 40, normalized size = 0.71
method | result | size |
risch | \(\frac {x^{2}-4 x +3}{8 \left (1+x \right )^{2} \sqrt {-1+x}}+\frac {\arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{16}\) | \(38\) |
derivativedivides | \(\frac {\frac {\left (-1+x \right )^{\frac {3}{2}}}{8}-\frac {\sqrt {-1+x}}{4}}{\left (1+x \right )^{2}}+\frac {\arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{16}\) | \(40\) |
default | \(\frac {\frac {\left (-1+x \right )^{\frac {3}{2}}}{8}-\frac {\sqrt {-1+x}}{4}}{\left (1+x \right )^{2}}+\frac {\arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{16}\) | \(40\) |
trager | \(\frac {\left (-3+x \right ) \sqrt {-1+x}}{8 \left (1+x \right )^{2}}+\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {-1+x}+3 \RootOf \left (\textit {\_Z}^{2}+2\right )}{1+x}\right )}{32}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.53, size = 43, normalized size = 0.77 \begin {gather*} \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) + \frac {{\left (x - 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {x - 1}}{8 \, {\left ({\left (x - 1\right )}^{2} + 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.42, size = 46, normalized size = 0.82 \begin {gather*} \frac {\sqrt {2} {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) + 2 \, \sqrt {x - 1} {\left (x - 3\right )}}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.63, size = 168, normalized size = 3.00 \begin {gather*} \begin {cases} \frac {\sqrt {2} i \operatorname {acosh}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )}}{16} - \frac {i}{8 \sqrt {-1 + \frac {2}{x + 1}} \sqrt {x + 1}} + \frac {3 i}{4 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{\frac {3}{2}}} - \frac {i}{\sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{\frac {5}{2}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {\sqrt {2} \operatorname {asin}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )}}{16} + \frac {1}{8 \sqrt {1 - \frac {2}{x + 1}} \sqrt {x + 1}} - \frac {3}{4 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{\frac {3}{2}}} + \frac {1}{\sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.23, size = 37, normalized size = 0.66 \begin {gather*} \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) + \frac {{\left (x - 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {x - 1}}{8 \, {\left (x + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 45, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x-1}}{2}\right )}{16}-\frac {\frac {\sqrt {x-1}}{4}-\frac {{\left (x-1\right )}^{3/2}}{8}}{4\,x+{\left (x-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________