Integrand size = 15, antiderivative size = 27 \[ \int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx=\frac {x^2}{2 \sqrt {1-x^4}}-\frac {\arcsin \left (x^2\right )}{2} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {281, 294, 222} \[ \int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx=\frac {x^2}{2 \sqrt {1-x^4}}-\frac {\arcsin \left (x^2\right )}{2} \]
[In]
[Out]
Rule 222
Rule 281
Rule 294
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^{3/2}} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2 \sqrt {1-x^4}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2 \sqrt {1-x^4}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2} \left (\frac {x^2}{\sqrt {1-x^4}}-\arcsin \left (x^2\right )\right ) \]
[In]
[Out]
Time = 4.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {\arcsin \left (x^{2}\right )}{2}+\frac {x^{2}}{2 \sqrt {-x^{4}+1}}\) | \(22\) |
pseudoelliptic | \(\frac {-\arcsin \left (x^{2}\right ) \sqrt {-x^{4}+1}+x^{2}}{2 \sqrt {-x^{4}+1}}\) | \(31\) |
meijerg | \(\frac {i \left (-\frac {i \sqrt {\pi }\, x^{2}}{\sqrt {-x^{4}+1}}+i \sqrt {\pi }\, \arcsin \left (x^{2}\right )\right )}{2 \sqrt {\pi }}\) | \(36\) |
trager | \(-\frac {x^{2} \sqrt {-x^{4}+1}}{2 \left (x^{4}-1\right )}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}+1}+x^{2}\right )}{2}\) | \(53\) |
default | \(-\frac {\arcsin \left (x^{2}\right )}{2}-\frac {\sqrt {-\left (x^{2}-1\right )^{2}+2-2 x^{2}}}{4 \left (x^{2}-1\right )}-\frac {\sqrt {-\left (x^{2}+1\right )^{2}+2+2 x^{2}}}{4 \left (x^{2}+1\right )}\) | \(62\) |
elliptic | \(-\frac {\arcsin \left (x^{2}\right )}{2}-\frac {\sqrt {-\left (x^{2}-1\right )^{2}+2-2 x^{2}}}{4 \left (x^{2}-1\right )}-\frac {\sqrt {-\left (x^{2}+1\right )^{2}+2+2 x^{2}}}{4 \left (x^{2}+1\right )}\) | \(62\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx=-\frac {\sqrt {-x^{4} + 1} x^{2} - 2 \, {\left (x^{4} - 1\right )} \arctan \left (\frac {\sqrt {-x^{4} + 1} - 1}{x^{2}}\right )}{2 \, {\left (x^{4} - 1\right )}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx=\begin {cases} - \frac {i x^{2}}{2 \sqrt {x^{4} - 1}} + \frac {i \operatorname {acosh}{\left (x^{2} \right )}}{2} & \text {for}\: \left |{x^{4}}\right | > 1 \\\frac {x^{2}}{2 \sqrt {1 - x^{4}}} - \frac {\operatorname {asin}{\left (x^{2} \right )}}{2} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx=\frac {x^{2}}{2 \, \sqrt {-x^{4} + 1}} + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{4} + 1}}{x^{2}}\right ) \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx=-\frac {\sqrt {-x^{4} + 1} x^{2}}{2 \, {\left (x^{4} - 1\right )}} - \frac {1}{2} \, \arcsin \left (x^{2}\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx=\int \frac {x^5}{{\left (1-x^4\right )}^{3/2}} \,d x \]
[In]
[Out]