Optimal. Leaf size=79 \[ -\frac {5 \sqrt {2 x-x^2}}{6 (1+x)^2}-\frac {2 \sqrt {2 x-x^2}}{3 (1+x)}+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3} \sqrt {2 x-x^2}}\right )}{2 \sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {848, 820, 738,
210} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 x}{\sqrt {3} \sqrt {2 x-x^2}}\right )}{2 \sqrt {3}}-\frac {2 \sqrt {2 x-x^2}}{3 (x+1)}-\frac {5 \sqrt {2 x-x^2}}{6 (x+1)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 738
Rule 820
Rule 848
Rubi steps
\begin {align*} \int \frac {-2+3 x}{(1+x)^3 \sqrt {2 x-x^2}} \, dx &=-\frac {5 \sqrt {2 x-x^2}}{6 (1+x)^2}+\frac {1}{6} \int \frac {-7+5 x}{(1+x)^2 \sqrt {2 x-x^2}} \, dx\\ &=-\frac {5 \sqrt {2 x-x^2}}{6 (1+x)^2}-\frac {2 \sqrt {2 x-x^2}}{3 (1+x)}-\frac {1}{2} \int \frac {1}{(1+x) \sqrt {2 x-x^2}} \, dx\\ &=-\frac {5 \sqrt {2 x-x^2}}{6 (1+x)^2}-\frac {2 \sqrt {2 x-x^2}}{3 (1+x)}+\text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,\frac {-2+4 x}{\sqrt {2 x-x^2}}\right )\\ &=-\frac {5 \sqrt {2 x-x^2}}{6 (1+x)^2}-\frac {2 \sqrt {2 x-x^2}}{3 (1+x)}-\frac {\tan ^{-1}\left (\frac {-2+4 x}{2 \sqrt {3} \sqrt {2 x-x^2}}\right )}{2 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 78, normalized size = 0.99 \begin {gather*} \frac {x \left (-18+x+4 x^2\right )-2 \sqrt {3} \sqrt {-2+x} \sqrt {x} (1+x)^2 \tanh ^{-1}\left (\frac {1-\sqrt {-2+x} \sqrt {x}+x}{\sqrt {3}}\right )}{6 \sqrt {-((-2+x) x)} (1+x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 74, normalized size = 0.94
method | result | size |
risch | \(\frac {x \left (-2+x \right ) \left (4 x +9\right )}{6 \left (1+x \right )^{2} \sqrt {-x \left (-2+x \right )}}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2+4 x \right ) \sqrt {3}}{6 \sqrt {-\left (1+x \right )^{2}+1+4 x}}\right )}{6}\) | \(56\) |
trager | \(-\frac {\left (4 x +9\right ) \sqrt {-x^{2}+2 x}}{6 \left (1+x \right )^{2}}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x +3 \sqrt {-x^{2}+2 x}+\RootOf \left (\textit {\_Z}^{2}+3\right )}{1+x}\right )}{6}\) | \(69\) |
default | \(-\frac {5 \sqrt {-\left (1+x \right )^{2}+1+4 x}}{6 \left (1+x \right )^{2}}-\frac {2 \sqrt {-\left (1+x \right )^{2}+1+4 x}}{3 \left (1+x \right )}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2+4 x \right ) \sqrt {3}}{6 \sqrt {-\left (1+x \right )^{2}+1+4 x}}\right )}{6}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 3.73, size = 66, normalized size = 0.84 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arcsin \left (\frac {2 \, x}{{\left | x + 1 \right |}} - \frac {1}{{\left | x + 1 \right |}}\right ) - \frac {5 \, \sqrt {-x^{2} + 2 \, x}}{6 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 2 \, x}}{3 \, {\left (x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.43, size = 64, normalized size = 0.81 \begin {gather*} \frac {2 \, \sqrt {3} {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {3} \sqrt {-x^{2} + 2 \, x}}{3 \, x}\right ) - \sqrt {-x^{2} + 2 \, x} {\left (4 \, x + 9\right )}}{6 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x - 2}{\sqrt {- x \left (x - 2\right )} \left (x + 1\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 147 vs.
\(2 (64) = 128\).
time = 1.00, size = 147, normalized size = 1.86 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}}{x - 1} - 1\right )}\right ) + \frac {\frac {34 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}}{x - 1} - \frac {39 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}^{2}}{{\left (x - 1\right )}^{2}} + \frac {18 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}^{3}}{{\left (x - 1\right )}^{3}} - 26}{24 \, {\left (\frac {\sqrt {-x^{2} + 2 \, x} - 1}{x - 1} - \frac {{\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}^{2}}{{\left (x - 1\right )}^{2}} - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,x-2}{\sqrt {2\,x-x^2}\,{\left (x+1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________