Optimal. Leaf size=42 \[ -\frac {2 \sqrt {1+x^3} \tanh ^{-1}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x} \sqrt {1-x+x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {929, 272, 65,
213} \begin {gather*} -\frac {2 \sqrt {x^3+1} \tanh ^{-1}\left (\sqrt {x^3+1}\right )}{3 \sqrt {x+1} \sqrt {x^2-x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 213
Rule 272
Rule 929
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {1+x} \sqrt {1-x+x^2}} \, dx &=\frac {\sqrt {1+x^3} \int \frac {1}{x \sqrt {1+x^3}} \, dx}{\sqrt {1+x} \sqrt {1-x+x^2}}\\ &=\frac {\sqrt {1+x^3} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )}{3 \sqrt {1+x} \sqrt {1-x+x^2}}\\ &=\frac {\left (2 \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )}{3 \sqrt {1+x} \sqrt {1-x+x^2}}\\ &=-\frac {2 \sqrt {1+x^3} \tanh ^{-1}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x} \sqrt {1-x+x^2}}\\ \end {align*}
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Mathematica [A]
time = 5.31, size = 29, normalized size = 0.69 \begin {gather*} -\frac {2}{3} \tanh ^{-1}\left (\sqrt {1+x} \sqrt {3-3 (1+x)+(1+x)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 33, normalized size = 0.79
method | result | size |
default | \(-\frac {2 \arctanh \left (\sqrt {x^{3}+1}\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{3 \sqrt {x^{3}+1}}\) | \(33\) |
elliptic | \(-\frac {2 \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \arctanh \left (\sqrt {x^{3}+1}\right )}{3 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(40\) |
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 = 1.76, size = 43, normalized size = 1.02 \begin {gather*} -\frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} - 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 {1}{x \sqrt {x + 1} \sqrt {x^{2} - x + 1}}\, 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 = -0.02 \begin {gather*} \int \frac {1}{x\,\sqrt {x+1}\,\sqrt {x^2-x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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