Optimal. Leaf size=73 \[ \frac {1}{4} \left (\sqrt {2}-2\right ) \tanh ^{-1}\left (\frac {x-1}{\left (\sqrt {2}-1\right ) \sqrt {x^3-x}}\right )+\frac {1}{4} \left (2+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {x-1}{\left (1+\sqrt {2}\right ) \sqrt {x^3-x}}\right ) \]
________________________________________________________________________________________
Rubi [C] time = 0.51, antiderivative size = 103, normalized size of antiderivative = 1.41, number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 6728, 933, 168, 537} \begin {gather*} \frac {\sqrt {x} \sqrt {1-x^2} \Pi \left (-\frac {1}{\sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{2 \sqrt {x^3-x}}-\frac {\sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{\sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{2 \sqrt {x^3-x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 168
Rule 537
Rule 933
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {-1+x}{\sqrt {x} \sqrt {-1+x^2} \left (-1-2 x+x^2\right )} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \left (\frac {1}{\sqrt {x} \left (-2-2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}}+\frac {1}{\sqrt {x} \left (-2+2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}}\right ) \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-2-2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-2+2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x} \left (-2-2 \sqrt {2}+2 x\right )} \, dx}{\sqrt {-x+x^3}}+\frac {\left (\sqrt {x} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x} \left (-2+2 \sqrt {2}+2 x\right )} \, dx}{\sqrt {-x+x^3}}\\ &=-\frac {\left (2 \sqrt {x} \sqrt {1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2 \sqrt {2}-2 x^2\right ) \sqrt {1-x^2} \sqrt {2-x^2}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {-x+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {2}-2 x^2\right ) \sqrt {1-x^2} \sqrt {2-x^2}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {-x+x^3}}\\ &=\frac {\sqrt {x} \sqrt {1-x^2} \Pi \left (-\frac {1}{\sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{2 \sqrt {-x+x^3}}-\frac {\sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{\sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{2 \sqrt {-x+x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.80, size = 89, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {1-\frac {1}{x^2}} x^{3/2} \left (2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )-\left (1+\sqrt {2}\right ) \Pi \left (1-\sqrt {2};\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )+\left (\sqrt {2}-1\right ) \Pi \left (1+\sqrt {2};\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {x \left (x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.74, size = 85, normalized size = 1.16 \begin {gather*} \frac {1}{4} \left (2+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {1-\sqrt {2}+\left (-1+\sqrt {2}\right ) x}{\sqrt {-x+x^3}}\right )+\frac {1}{4} \left (-2+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {-1-\sqrt {2}+\left (1+\sqrt {2}\right ) x}{\sqrt {-x+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.56, size = 126, normalized size = 1.73 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} + 12 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} + 2 \, x^{2} - 12 \, x + 1}{x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} + 4 \, x^{3} + 2 \, x^{2} - 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 1\right )} - 4 \, x + 1}{x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {x^{3} - x} {\left (x^{2} - 2 \, x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.22, size = 116, normalized size = 1.59
method | result | size |
default | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}+\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}\) | \(116\) |
elliptic | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}+\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}\) | \(116\) |
trager | \(\frac {\RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) \ln \left (-\frac {278808 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x +63721 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x^{2}+1672848 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2}-1038514 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-2862728 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x -148735 x^{2}+2926449 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-886429 \sqrt {x^{3}-x}-1427856 x +1279121}{\left (4 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x -8 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )+3 x -7\right )^{2}}\right )}{2}-\frac {\ln \left (-\frac {278808 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x +493895 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x^{2}+1672848 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2}+1038514 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}+74648 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x +66352 x^{2}+419247 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )+152085 \sqrt {x^{3}-x}+40832 x +25520}{\left (4 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x -8 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )+x -1\right )^{2}}\right ) \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )}{2}-\frac {\ln \left (-\frac {278808 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x +493895 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x^{2}+1672848 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2}+1038514 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}+74648 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x +66352 x^{2}+419247 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )+152085 \sqrt {x^{3}-x}+40832 x +25520}{\left (4 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x -8 \RootOf \left (8 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )+x -1\right )^{2}}\right )}{2}\) | \(542\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {x^{3} - x} {\left (x^{2} - 2 \, x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.11, size = 102, normalized size = 1.40 \begin {gather*} \frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\frac {1}{\sqrt {2}+1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}\,\left (\sqrt {2}+1\right )}-\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\frac {1}{\sqrt {2}-1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}\,\left (\sqrt {2}-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 {x - 1}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} - 2 x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________