Integrand size = 18, antiderivative size = 28 \[ \int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (3+4 x^6+23 x^{12}\right )}{45 x^{15}} \]
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Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1501, 464, 277, 270} \[ \int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {x^6-1}}{15 x^{15}}+\frac {4 \sqrt {x^6-1}}{45 x^9}+\frac {23 \sqrt {x^6-1}}{45 x^3} \]
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Rule 270
Rule 277
Rule 464
Rule 1501
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {-1+x^6}}{6 x^9}-\frac {1}{6} \int \frac {-6-9 x^6}{x^{16} \sqrt {-1+x^6}} \, dx \\ & = \frac {\sqrt {-1+x^6}}{15 x^{15}}-\frac {\sqrt {-1+x^6}}{6 x^9}+\frac {23}{10} \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx \\ & = \frac {\sqrt {-1+x^6}}{15 x^{15}}+\frac {4 \sqrt {-1+x^6}}{45 x^9}+\frac {23}{15} \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx \\ & = \frac {\sqrt {-1+x^6}}{15 x^{15}}+\frac {4 \sqrt {-1+x^6}}{45 x^9}+\frac {23 \sqrt {-1+x^6}}{45 x^3} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (3+4 x^6+23 x^{12}\right )}{45 x^{15}} \]
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Time = 0.86 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (23 x^{12}+4 x^{6}+3\right )}{45 x^{15}}\) | \(25\) |
pseudoelliptic | \(\frac {\sqrt {x^{6}-1}\, \left (23 x^{12}+4 x^{6}+3\right )}{45 x^{15}}\) | \(25\) |
risch | \(\frac {23 x^{18}-19 x^{12}-x^{6}-3}{45 x^{15} \sqrt {x^{6}-1}}\) | \(30\) |
gosper | \(\frac {\left (23 x^{12}+4 x^{6}+3\right ) \left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{45 \sqrt {x^{6}-1}\, x^{15}}\) | \(45\) |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \sqrt {-x^{6}+1}}{3 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, x^{3}}-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (\frac {8}{3} x^{12}+\frac {4}{3} x^{6}+1\right ) \sqrt {-x^{6}+1}}{15 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, x^{15}}\) | \(78\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx=\frac {23 \, x^{15} + {\left (23 \, x^{12} + 4 \, x^{6} + 3\right )} \sqrt {x^{6} - 1}}{45 \, x^{15}} \]
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Time = 1.54 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx=\frac {\begin {cases} \frac {\sqrt {x^{6} - 1}}{x^{3}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} + \frac {\begin {cases} \frac {\sqrt {x^{6} - 1}}{x^{3}} - \frac {2 \left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} + \frac {\left (x^{6} - 1\right )^{\frac {5}{2}}}{5 x^{15}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} \]
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Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx=\frac {2 \, \sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {2 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} + \frac {{\left (x^{6} - 1\right )}^{\frac {5}{2}}}{15 \, x^{15}} \]
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Exception generated. \[ \int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx=\text {Exception raised: TypeError} \]
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Time = 5.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {x^6-1}\,\left (23\,x^{12}+4\,x^6+3\right )}{45\,x^{15}} \]
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