Integrand size = 13, antiderivative size = 14 \[ \int \frac {1}{x \sqrt {1+x^8}} \, dx=-\frac {1}{4} \text {arctanh}\left (\sqrt {1+x^8}\right ) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 65, 213} \[ \int \frac {1}{x \sqrt {1+x^8}} \, dx=-\frac {1}{4} \text {arctanh}\left (\sqrt {x^8+1}\right ) \]
[In]
[Out]
Rule 65
Rule 213
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^8\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^8}\right ) \\ & = -\frac {1}{4} \text {arctanh}\left (\sqrt {1+x^8}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {1+x^8}} \, dx=-\frac {1}{4} \text {arctanh}\left (\sqrt {1+x^8}\right ) \]
[In]
[Out]
Time = 0.40 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{8}+1}}\right )}{4}\) | \(11\) |
trager | \(-\frac {\ln \left (\frac {\sqrt {x^{8}+1}+1}{x^{4}}\right )}{4}\) | \(17\) |
meijerg | \(\frac {\left (-2 \ln \left (2\right )+8 \ln \left (x \right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{8}+1}}{2}\right )}{8 \sqrt {\pi }}\) | \(37\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (10) = 20\).
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x \sqrt {1+x^8}} \, dx=-\frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \]
[In]
[Out]
Time = 0.55 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x \sqrt {1+x^8}} \, dx=- \frac {\operatorname {asinh}{\left (\frac {1}{x^{4}} \right )}}{4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (10) = 20\).
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x \sqrt {1+x^8}} \, dx=-\frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (10) = 20\).
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x \sqrt {1+x^8}} \, dx=-\frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \sqrt {1+x^8}} \, dx=-\frac {\mathrm {atanh}\left (\sqrt {x^8+1}\right )}{4} \]
[In]
[Out]