Integrand size = 11, antiderivative size = 25 \[ \int \frac {x}{a+b x^4} \, dx=\frac {\arctan \left (\sqrt {\frac {b}{a}} x^2\right )}{2 \sqrt {a b}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {281, 211} \[ \int \frac {x}{a+b x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \]
[In]
[Out]
Rule 211
Rule 281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right ) \\ & = \frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {x}{a+b x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {\arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\) | \(19\) |
risch | \(-\frac {\ln \left (x^{2} \sqrt {-a b}-a \right )}{4 \sqrt {-a b}}+\frac {\ln \left (x^{2} \sqrt {-a b}+a \right )}{4 \sqrt {-a b}}\) | \(46\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.88 \[ \int \frac {x}{a+b x^4} \, dx=\left [-\frac {\sqrt {-a b} \log \left (\frac {b x^{4} - 2 \, \sqrt {-a b} x^{2} - a}{b x^{4} + a}\right )}{4 \, a b}, -\frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b x^{2}}\right )}{2 \, a b}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {x}{a+b x^4} \, dx=- \frac {\sqrt {- \frac {1}{a b}} \log {\left (- a \sqrt {- \frac {1}{a b}} + x^{2} \right )}}{4} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (a \sqrt {- \frac {1}{a b}} + x^{2} \right )}}{4} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {x}{a+b x^4} \, dx=\frac {\arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {x}{a+b x^4} \, dx=\frac {\arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} \]
[In]
[Out]
Time = 13.95 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {x}{a+b x^4} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x^2}{\sqrt {a}}\right )}{2\,\sqrt {a}\,\sqrt {b}} \]
[In]
[Out]