Integrand size = 23, antiderivative size = 42 \[ \int \frac {1}{x \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=-\frac {2 \sqrt {1+x^3} \text {arctanh}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x} \sqrt {1-x+x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {929, 272, 65, 213} \[ \int \frac {1}{x \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=-\frac {2 \sqrt {x^3+1} \text {arctanh}\left (\sqrt {x^3+1}\right )}{3 \sqrt {x+1} \sqrt {x^2-x+1}} \]
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Rule 65
Rule 213
Rule 272
Rule 929
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 4.92 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=-\frac {2}{3} \text {arctanh}\left (\sqrt {1+x} \sqrt {3-3 (1+x)+(1+x)^2}\right ) \]
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Time = 0.58 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{3 \sqrt {x^{3}+1}}\) | \(33\) |
elliptic | \(-\frac {2 \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{3 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(40\) |
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Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=-\frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} - 1\right ) \]
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\[ \int \frac {1}{x \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {1}{x \sqrt {x + 1} \sqrt {x^{2} - x + 1}}\, dx \]
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\[ \int \frac {1}{x \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - x + 1} \sqrt {x + 1} x} \,d x } \]
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\[ \int \frac {1}{x \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - x + 1} \sqrt {x + 1} x} \,d x } \]
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Timed out. \[ \int \frac {1}{x \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {1}{x\,\sqrt {x+1}\,\sqrt {x^2-x+1}} \,d x \]
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