Integrand size = 17, antiderivative size = 42 \[ \int \frac {1}{\sqrt {9+3 x-5 x^2+x^3}} \, dx=\frac {(3-x) \sqrt {1+x} \text {arctanh}\left (\frac {\sqrt {1+x}}{2}\right )}{\sqrt {9+3 x-5 x^2+x^3}} \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2092, 2089, 65, 212} \[ \int \frac {1}{\sqrt {9+3 x-5 x^2+x^3}} \, dx=\frac {(3-x) \sqrt {x+1} \text {arctanh}\left (\frac {\sqrt {x+1}}{2}\right )}{\sqrt {x^3-5 x^2+3 x+9}} \]
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Rule 65
Rule 212
Rule 2089
Rule 2092
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\sqrt {\frac {128}{27}-\frac {16 x}{3}+x^3}} \, dx,x,-\frac {5}{3}+x\right ) \\ & = \frac {\left (128 (3-x) \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {128}{9}-\frac {32 x}{3}\right ) \sqrt {\frac {128}{9}+\frac {16 x}{3}}} \, dx,x,-\frac {5}{3}+x\right )}{3 \sqrt {3} \sqrt {9+3 x-5 x^2+x^3}} \\ & = \frac {\left (16 (3-x) \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1}{\frac {128}{3}-2 x^2} \, dx,x,\frac {4 \sqrt {1+x}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {9+3 x-5 x^2+x^3}} \\ & = \frac {(3-x) \sqrt {1+x} \text {arctanh}\left (\frac {\sqrt {1+x}}{2}\right )}{\sqrt {9+3 x-5 x^2+x^3}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {9+3 x-5 x^2+x^3}} \, dx=-\frac {(-3+x) \sqrt {1+x} \text {arctanh}\left (\frac {\sqrt {1+x}}{2}\right )}{\sqrt {(-3+x)^2 (1+x)}} \]
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Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83
method | result | size |
trager | \(-\frac {\ln \left (\frac {x^{2}+4 \sqrt {x^{3}-5 x^{2}+3 x +9}+2 x -15}{\left (-3+x \right )^{2}}\right )}{2}\) | \(35\) |
default | \(\frac {\left (-3+x \right ) \sqrt {1+x}\, \left (\ln \left (\sqrt {1+x}-2\right )-\ln \left (\sqrt {1+x}+2\right )\right )}{2 \sqrt {x^{3}-5 x^{2}+3 x +9}}\) | \(45\) |
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Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt {9+3 x-5 x^2+x^3}} \, dx=-\frac {1}{2} \, \log \left (\frac {2 \, x + \sqrt {x^{3} - 5 \, x^{2} + 3 \, x + 9} - 6}{x - 3}\right ) + \frac {1}{2} \, \log \left (-\frac {2 \, x - \sqrt {x^{3} - 5 \, x^{2} + 3 \, x + 9} - 6}{x - 3}\right ) \]
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\[ \int \frac {1}{\sqrt {9+3 x-5 x^2+x^3}} \, dx=\int \frac {1}{\sqrt {x^{3} - 5 x^{2} + 3 x + 9}}\, dx \]
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\[ \int \frac {1}{\sqrt {9+3 x-5 x^2+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} - 5 \, x^{2} + 3 \, x + 9}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {9+3 x-5 x^2+x^3}} \, dx=-\frac {\log \left (\sqrt {x + 1} + 2\right )}{2 \, \mathrm {sgn}\left (x - 3\right )} + \frac {\log \left ({\left | \sqrt {x + 1} - 2 \right |}\right )}{2 \, \mathrm {sgn}\left (x - 3\right )} \]
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Timed out. \[ \int \frac {1}{\sqrt {9+3 x-5 x^2+x^3}} \, dx=\int \frac {1}{\sqrt {x^3-5\,x^2+3\,x+9}} \,d x \]
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