Optimal. Leaf size=21 \[ \tan ^{-1}\left (\frac {2 \sqrt {x^4-x}}{2 x+1}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 1.76, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-1+2 x+2 x^2}{\left (1+2 x^2\right ) \sqrt {-x+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {-1+2 x+2 x^2}{\left (1+2 x^2\right ) \sqrt {-x+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {-1+2 x+2 x^2}{\sqrt {x} \left (1+2 x^2\right ) \sqrt {-1+x^3}} \, dx}{\sqrt {-x+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {-1+x^3}}-\frac {2 (1-x)}{\sqrt {x} \left (1+2 x^2\right ) \sqrt {-1+x^3}}\right ) \, dx}{\sqrt {-x+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1+x^3}} \, dx}{\sqrt {-x+x^4}}-\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1-x}{\sqrt {x} \left (1+2 x^2\right ) \sqrt {-1+x^3}} \, dx}{\sqrt {-x+x^4}}\\ &=-\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \int \left (\frac {i+\frac {1}{\sqrt {2}}}{2 \sqrt {x} \left (i-\sqrt {2} x\right ) \sqrt {-1+x^3}}+\frac {i-\frac {1}{\sqrt {2}}}{2 \sqrt {x} \left (i+\sqrt {2} x\right ) \sqrt {-1+x^3}}\right ) \, dx}{\sqrt {-x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}\\ &=\frac {(1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}-\frac {\left (\left (2 i-\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (i+\sqrt {2} x\right ) \sqrt {-1+x^3}} \, dx}{2 \sqrt {-x+x^4}}-\frac {\left (\left (2 i+\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (i-\sqrt {2} x\right ) \sqrt {-1+x^3}} \, dx}{2 \sqrt {-x+x^4}}\\ &=\frac {(1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}-\frac {\left (\left (2 i-\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i+\sqrt {2} x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}-\frac {\left (\left (2 i+\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i-\sqrt {2} x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}\\ &=\frac {(1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}-\frac {\left (\left (2 i-\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-\sqrt [4]{2} x\right ) \sqrt {-1+x^6}}-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+\sqrt [4]{2} x\right ) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}-\frac {\left (\left (2 i+\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-\sqrt [4]{2} x\right ) \sqrt {-1+x^6}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+\sqrt [4]{2} x\right ) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}\\ &=\frac {(1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (\sqrt [4]{-1} \left (2 i-\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{3/4}-\sqrt [4]{2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}+\frac {\left (\sqrt [4]{-1} \left (2 i-\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{3/4}+\sqrt [4]{2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}+\frac {\left ((-1)^{3/4} \left (2 i+\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [4]{-1}-\sqrt [4]{2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}+\frac {\left ((-1)^{3/4} \left (2 i+\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [4]{-1}+\sqrt [4]{2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.28, size = 320, normalized size = 15.24 \begin {gather*} -\frac {2 \sqrt {\frac {1}{x^2}+\frac {1}{x}+1} \sqrt {\frac {x-1}{\left (1+\sqrt [3]{-1}\right ) x}} x^2 \left (\frac {\sqrt {3} \left (i \sqrt {3} x+\sqrt [3]{-1}+1\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {x-(-1)^{2/3}}{\left (1+\sqrt [3]{-1}\right ) x}}\right )|\sqrt [3]{-1}\right )}{(-1)^{2/3}-x}+\frac {6 \left (\left (-2 i+\sqrt [6]{-1}+2 \sqrt {2}+(-1)^{2/3} \sqrt {2}\right ) \Pi \left (\frac {2 \sqrt {3}}{-i-2 \sqrt {2}+\sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {x-(-1)^{2/3}}{\left (1+\sqrt [3]{-1}\right ) x}}\right )|\sqrt [3]{-1}\right )+\frac {3}{2} \left (3 i+\sqrt {2}-\sqrt {3}-i \sqrt {6}\right ) \Pi \left (\frac {2 \sqrt {3}}{-i+2 \sqrt {2}+\sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {x-(-1)^{2/3}}{\left (1+\sqrt [3]{-1}\right ) x}}\right )|\sqrt [3]{-1}\right )\right )}{-3 i \sqrt {2}+2 i \sqrt {3}+\sqrt {6}}\right )}{3 \sqrt {x \left (x^3-1\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.26, size = 21, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {2 \sqrt {-x+x^4}}{1+2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 19, normalized size = 0.90 \begin {gather*} -\arctan \left (\frac {2 \, x + 1}{2 \, \sqrt {x^{4} - x}}\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 {2 \, x^{2} + 2 \, x - 1}{\sqrt {x^{4} - x} {\left (2 \, x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.39, size = 46, normalized size = 2.19
method | result | size |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x^{4}-x}}{2 x^{2}+1}\right )\) | \(46\) |
default | \(\text {Expression too large to display}\) | \(13534\) |
elliptic | \(\text {Expression too large to display}\) | \(14836\) |
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 {2 \, x^{2} + 2 \, x - 1}{\sqrt {x^{4} - x} {\left (2 \, x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {2\,x^2+2\,x-1}{\sqrt {x^4-x}\,\left (2\,x^2+1\right )} \,d x \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 {2 x^{2} + 2 x - 1}{\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (2 x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________