Integrand size = 15, antiderivative size = 35 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \text {arctanh}\left (\sqrt {1-x^4}\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 44, 65, 212} \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {1}{4} \text {arctanh}\left (\sqrt {1-x^4}\right )-\frac {\sqrt {1-x^4}}{4 x^4} \]
[In]
[Out]
Rule 44
Rule 65
Rule 212
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^2} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt {1-x^4}}{4 x^4}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^4}\right ) \\ & = -\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \text {arctanh}\left (\sqrt {1-x^4}\right ) \]
[In]
[Out]
Time = 4.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {\sqrt {-x^{4}+1}}{4 x^{4}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{4}\) | \(28\) |
elliptic | \(-\frac {\sqrt {-x^{4}+1}}{4 x^{4}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{4}\) | \(28\) |
risch | \(\frac {x^{4}-1}{4 x^{4} \sqrt {-x^{4}+1}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{4}\) | \(33\) |
pseudoelliptic | \(\frac {-\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{4}+1}}\right ) x^{4}-\sqrt {-x^{4}+1}}{4 x^{4}}\) | \(33\) |
trager | \(-\frac {\sqrt {-x^{4}+1}}{4 x^{4}}+\frac {\ln \left (\frac {-1+\sqrt {-x^{4}+1}}{x^{2}}\right )}{4}\) | \(34\) |
meijerg | \(-\frac {\frac {\sqrt {\pi }}{x^{4}}-\frac {\left (1-2 \ln \left (2\right )+4 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 x^{4}+8\right )}{8 x^{4}}+\frac {\sqrt {\pi }\, \sqrt {-x^{4}+1}}{x^{4}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{4}+1}}{2}\right )}{4 \sqrt {\pi }}\) | \(82\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {x^{4} \log \left (\sqrt {-x^{4} + 1} + 1\right ) - x^{4} \log \left (\sqrt {-x^{4} + 1} - 1\right ) + 2 \, \sqrt {-x^{4} + 1}}{8 \, x^{4}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.09 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=\begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x^{2}} \right )}}{4} + \frac {1}{4 x^{2} \sqrt {-1 + \frac {1}{x^{4}}}} - \frac {1}{4 x^{6} \sqrt {-1 + \frac {1}{x^{4}}}} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {1}{x^{2}} \right )}}{4} - \frac {i \sqrt {1 - \frac {1}{x^{4}}}}{4 x^{2}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {-x^{4} + 1}}{4 \, x^{4}} - \frac {1}{8} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {-x^{4} + 1} - 1\right ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {-x^{4} + 1}}{4 \, x^{4}} - \frac {1}{8} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{8} \, \log \left (-\sqrt {-x^{4} + 1} + 1\right ) \]
[In]
[Out]
Time = 5.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {\mathrm {atanh}\left (\sqrt {1-x^4}\right )}{4}-\frac {\sqrt {1-x^4}}{4\,x^4} \]
[In]
[Out]