Integrand size = 13, antiderivative size = 49 \[ \int \frac {1}{x^7 \left (1+x^4\right )^{3/2}} \, dx=-\frac {1}{6 x^6 \sqrt {1+x^4}}+\frac {2}{3 x^2 \sqrt {1+x^4}}+\frac {4 x^2}{3 \sqrt {1+x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {277, 270} \[ \int \frac {1}{x^7 \left (1+x^4\right )^{3/2}} \, dx=-\frac {1}{6 \sqrt {x^4+1} x^6}+\frac {4 x^2}{3 \sqrt {x^4+1}}+\frac {2}{3 \sqrt {x^4+1} x^2} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{6 x^6 \sqrt {1+x^4}}-\frac {4}{3} \int \frac {1}{x^3 \left (1+x^4\right )^{3/2}} \, dx \\ & = -\frac {1}{6 x^6 \sqrt {1+x^4}}+\frac {2}{3 x^2 \sqrt {1+x^4}}+\frac {8}{3} \int \frac {x}{\left (1+x^4\right )^{3/2}} \, dx \\ & = -\frac {1}{6 x^6 \sqrt {1+x^4}}+\frac {2}{3 x^2 \sqrt {1+x^4}}+\frac {4 x^2}{3 \sqrt {1+x^4}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^7 \left (1+x^4\right )^{3/2}} \, dx=\frac {-1+4 x^4+8 x^8}{6 x^6 \sqrt {1+x^4}} \]
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Time = 4.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\) | \(25\) |
default | \(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\) | \(25\) |
trager | \(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\) | \(25\) |
meijerg | \(-\frac {-8 x^{8}-4 x^{4}+1}{6 x^{6} \sqrt {x^{4}+1}}\) | \(25\) |
risch | \(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\) | \(25\) |
elliptic | \(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\) | \(25\) |
pseudoelliptic | \(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\) | \(25\) |
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none
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^7 \left (1+x^4\right )^{3/2}} \, dx=\frac {8 \, x^{10} + 8 \, x^{6} + {\left (8 \, x^{8} + 4 \, x^{4} - 1\right )} \sqrt {x^{4} + 1}}{6 \, {\left (x^{10} + x^{6}\right )}} \]
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Time = 0.60 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^7 \left (1+x^4\right )^{3/2}} \, dx=\frac {8 x^{8} \sqrt {1 + \frac {1}{x^{4}}}}{6 x^{8} + 6 x^{4}} + \frac {4 x^{4} \sqrt {1 + \frac {1}{x^{4}}}}{6 x^{8} + 6 x^{4}} - \frac {\sqrt {1 + \frac {1}{x^{4}}}}{6 x^{8} + 6 x^{4}} \]
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none
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^7 \left (1+x^4\right )^{3/2}} \, dx=\frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} + \frac {\sqrt {x^{4} + 1}}{x^{2}} - \frac {{\left (x^{4} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]
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none
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^7 \left (1+x^4\right )^{3/2}} \, dx=\frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} - \frac {3 \, {\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{4} - 12 \, {\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} + 5}{3 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1\right )}^{3}} \]
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Time = 5.73 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^7 \left (1+x^4\right )^{3/2}} \, dx=-\frac {12\,x^4-8\,{\left (x^4+1\right )}^2+9}{6\,x^6\,\sqrt {x^4+1}} \]
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