Integrand size = 15, antiderivative size = 24 \[ \int \frac {1}{\sqrt {3-5 x+x^2+x^3}} \, dx=-\text {arctanh}\left (\frac {-2+2 x}{\sqrt {3-5 x+x^2+x^3}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2092, 2089, 65, 212} \[ \int \frac {1}{\sqrt {3-5 x+x^2+x^3}} \, dx=\frac {(1-x) \sqrt {x+3} \text {arctanh}\left (\frac {\sqrt {x+3}}{2}\right )}{\sqrt {x^3+x^2-5 x+3}} \]
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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 {1}{3}+x\right ) \\ & = \frac {\left (128 (1-x) \sqrt {3+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 {1}{3}+x\right )}{3 \sqrt {3} \sqrt {3-5 x+x^2+x^3}} \\ & = \frac {\left (16 (1-x) \sqrt {3+x}\right ) \text {Subst}\left (\int \frac {1}{\frac {128}{3}-2 x^2} \, dx,x,\frac {4 \sqrt {3+x}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {3-5 x+x^2+x^3}} \\ & = \frac {(1-x) \sqrt {3+x} \text {arctanh}\left (\frac {\sqrt {3+x}}{2}\right )}{\sqrt {3-5 x+x^2+x^3}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\sqrt {3-5 x+x^2+x^3}} \, dx=-\frac {(-1+x) \sqrt {3+x} \text {arctanh}\left (\frac {\sqrt {3+x}}{2}\right )}{\sqrt {(-1+x)^2 (3+x)}} \]
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Time = 0.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46
method | result | size |
trager | \(\frac {\ln \left (\frac {-x^{2}+4 \sqrt {x^{3}+x^{2}-5 x +3}-6 x +7}{\left (x -1\right )^{2}}\right )}{2}\) | \(35\) |
default | \(-\frac {\left (x -1\right ) \sqrt {3+x}\, \left (\ln \left (\sqrt {3+x}+2\right )-\ln \left (\sqrt {3+x}-2\right )\right )}{2 \sqrt {x^{3}+x^{2}-5 x +3}}\) | \(43\) |
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.42 \[ \int \frac {1}{\sqrt {3-5 x+x^2+x^3}} \, dx=-\frac {1}{2} \, \log \left (\frac {2 \, x + \sqrt {x^{3} + x^{2} - 5 \, x + 3} - 2}{x - 1}\right ) + \frac {1}{2} \, \log \left (-\frac {2 \, x - \sqrt {x^{3} + x^{2} - 5 \, x + 3} - 2}{x - 1}\right ) \]
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\[ \int \frac {1}{\sqrt {3-5 x+x^2+x^3}} \, dx=\int \frac {1}{\sqrt {x^{3} + x^{2} - 5 x + 3}}\, dx \]
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\[ \int \frac {1}{\sqrt {3-5 x+x^2+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} + x^{2} - 5 \, x + 3}} \,d x } \]
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none
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\sqrt {3-5 x+x^2+x^3}} \, dx=-\frac {\log \left (\sqrt {x + 3} + 2\right )}{2 \, \mathrm {sgn}\left (x - 1\right )} + \frac {\log \left ({\left | \sqrt {x + 3} - 2 \right |}\right )}{2 \, \mathrm {sgn}\left (x - 1\right )} \]
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Timed out. \[ \int \frac {1}{\sqrt {3-5 x+x^2+x^3}} \, dx=\int \frac {1}{\sqrt {x^3+x^2-5\,x+3}} \,d x \]
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