Integrand size = 13, antiderivative size = 19 \[ \int \frac {-1+x^5}{-1+x^2} \, dx=\frac {x^2}{2}+\frac {x^4}{4}+\log (1+x) \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1824, 641, 31} \[ \int \frac {-1+x^5}{-1+x^2} \, dx=\frac {x^4}{4}+\frac {x^2}{2}+\log (x+1) \]
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Rule 31
Rule 641
Rule 1824
Rubi steps \begin{align*} \text {integral}& = \int \left (x+x^3-\frac {1-x}{-1+x^2}\right ) \, dx \\ & = \frac {x^2}{2}+\frac {x^4}{4}-\int \frac {1-x}{-1+x^2} \, dx \\ & = \frac {x^2}{2}+\frac {x^4}{4}-\int \frac {1}{-1-x} \, dx \\ & = \frac {x^2}{2}+\frac {x^4}{4}+\log (1+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^5}{-1+x^2} \, dx=\frac {x^2}{2}+\frac {x^4}{4}+\log (1+x) \]
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Time = 0.80 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {x^{2}}{2}+\frac {x^{4}}{4}+\ln \left (x +1\right )\) | \(16\) |
norman | \(\frac {x^{2}}{2}+\frac {x^{4}}{4}+\ln \left (x +1\right )\) | \(16\) |
parallelrisch | \(\frac {x^{2}}{2}+\frac {x^{4}}{4}+\ln \left (x +1\right )\) | \(16\) |
risch | \(\frac {x^{4}}{4}+\frac {x^{2}}{2}+\frac {1}{4}+\ln \left (x +1\right )\) | \(17\) |
meijerg | \(\operatorname {arctanh}\left (x \right )+\frac {x^{2} \left (3 x^{2}+6\right )}{12}+\frac {\ln \left (-x^{2}+1\right )}{2}\) | \(26\) |
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Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-1+x^5}{-1+x^2} \, dx=\frac {1}{4} \, x^{4} + \frac {1}{2} \, x^{2} + \log \left (x + 1\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {-1+x^5}{-1+x^2} \, dx=\frac {x^{4}}{4} + \frac {x^{2}}{2} + \log {\left (x + 1 \right )} \]
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Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-1+x^5}{-1+x^2} \, dx=\frac {1}{4} \, x^{4} + \frac {1}{2} \, x^{2} + \log \left (x + 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {-1+x^5}{-1+x^2} \, dx=\frac {1}{4} \, x^{4} + \frac {1}{2} \, x^{2} + \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 9.67 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-1+x^5}{-1+x^2} \, dx=\ln \left (x+1\right )+\frac {x^2}{2}+\frac {x^4}{4} \]
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