Optimal. Leaf size=23 \[ \tanh ^{-1}\left (\frac {2 \sqrt {-x+x^4}}{1+2 x^2}\right ) \]
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Rubi [F]
time = 0.55, 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}{(1+2 x) \sqrt {-x+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-1+2 x+2 x^2}{(1+2 x) \sqrt {-x+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {-1+2 x+2 x^2}{\sqrt {x} (1+2 x) \sqrt {-1+x^3}} \, dx}{\sqrt {-x+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {-1+x^6}}+\frac {x^2}{\sqrt {-1+x^6}}-\frac {3}{2 \left (1+2 x^2\right ) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (1+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 )}{2 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^{3/2}\right )}{3 \sqrt {-x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \left (\frac {i}{2 \left (i-\sqrt {2} x\right ) \sqrt {-1+x^6}}+\frac {i}{2 \left (i+\sqrt {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 )}{2 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}-\frac {\left (3 i \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (i-\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}-\frac {\left (3 i \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (i+\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {-x+x^4}}\\ &=\frac {2 \sqrt {x} \sqrt {-1+x^3} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \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 )}{2 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}-\frac {\left (3 i \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (i-\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}-\frac {\left (3 i \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (i+\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}\\ \end {align*}
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Mathematica [A]
time = 6.99, size = 46, normalized size = 2.00 \begin {gather*} \frac {2 \sqrt {x} \sqrt {-1+x^3} \tanh ^{-1}\left (\frac {(-1+x) \sqrt {x}}{\sqrt {-1+x^3}}\right )}{\sqrt {x \left (-1+x^3\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.70, size = 774, normalized size = 33.65
method | result | size |
trager | \(\ln \left (-\frac {2 x^{2}+2 \sqrt {x^{4}-x}+1}{2 x +1}\right )\) | \(29\) |
default | \(\text {Expression too large to display}\) | \(774\) |
elliptic | \(\text {Expression too large to display}\) | \(774\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 28, normalized size = 1.22 \begin {gather*} \log \left (-\frac {2 \, x^{2} + 2 \, \sqrt {x^{4} - x} + 1}{2 \, x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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 + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,x^2+2\,x-1}{\sqrt {x^4-x}\,\left (2\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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