Integrand size = 13, antiderivative size = 23 \[ \int \frac {1}{x^4 \left (1+x^4\right )^{5/4}} \, dx=\frac {-1-4 x^4}{3 x^3 \sqrt [4]{1+x^4}} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {277, 197} \[ \int \frac {1}{x^4 \left (1+x^4\right )^{5/4}} \, dx=-\frac {4 x}{3 \sqrt [4]{x^4+1}}-\frac {1}{3 \sqrt [4]{x^4+1} x^3} \]
[In]
[Out]
Rule 197
Rule 277
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3 x^3 \sqrt [4]{1+x^4}}-\frac {4}{3} \int \frac {1}{\left (1+x^4\right )^{5/4}} \, dx \\ & = -\frac {1}{3 x^3 \sqrt [4]{1+x^4}}-\frac {4 x}{3 \sqrt [4]{1+x^4}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \left (1+x^4\right )^{5/4}} \, dx=\frac {-1-4 x^4}{3 x^3 \sqrt [4]{1+x^4}} \]
[In]
[Out]
Time = 0.85 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
gosper | \(-\frac {4 x^{4}+1}{3 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(20\) |
trager | \(-\frac {4 x^{4}+1}{3 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(20\) |
meijerg | \(-\frac {4 x^{4}+1}{3 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(20\) |
risch | \(-\frac {4 x^{4}+1}{3 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(20\) |
pseudoelliptic | \(\frac {-4 x^{4}-1}{3 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(20\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^4 \left (1+x^4\right )^{5/4}} \, dx=-\frac {{\left (4 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}}}{3 \, {\left (x^{7} + x^{3}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (22) = 44\).
Time = 0.46 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.26 \[ \int \frac {1}{x^4 \left (1+x^4\right )^{5/4}} \, dx=\frac {4 x^{4} \left (x^{4} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{16 x^{7} \Gamma \left (\frac {5}{4}\right ) + 16 x^{3} \Gamma \left (\frac {5}{4}\right )} + \frac {\left (x^{4} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{16 x^{7} \Gamma \left (\frac {5}{4}\right ) + 16 x^{3} \Gamma \left (\frac {5}{4}\right )} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \left (1+x^4\right )^{5/4}} \, dx=-\frac {x}{{\left (x^{4} + 1\right )}^{\frac {1}{4}}} - \frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
[In]
[Out]
\[ \int \frac {1}{x^4 \left (1+x^4\right )^{5/4}} \, dx=\int { \frac {1}{{\left (x^{4} + 1\right )}^{\frac {5}{4}} x^{4}} \,d x } \]
[In]
[Out]
Time = 5.57 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^4 \left (1+x^4\right )^{5/4}} \, dx=-\frac {4\,x^4+1}{3\,x^3\,{\left (x^4+1\right )}^{1/4}} \]
[In]
[Out]