Optimal. Leaf size=33 \[ \frac {\tan ^{-1}\left (\frac {x^2}{\sqrt {2} \sqrt {a+b}}\right )}{2 \sqrt {2} \sqrt {a+b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {275, 203} \[ \frac {\tan ^{-1}\left (\frac {x^2}{\sqrt {2} \sqrt {a+b}}\right )}{2 \sqrt {2} \sqrt {a+b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 275
Rubi steps
\begin {align*} \int \frac {x}{2 a+2 b+x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{2 a+2 b+x^2} \, dx,x,x^2\right )\\ &=\frac {\tan ^{-1}\left (\frac {x^2}{\sqrt {2} \sqrt {a+b}}\right )}{2 \sqrt {2} \sqrt {a+b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {x^2}{\sqrt {2} \sqrt {a+b}}\right )}{2 \sqrt {2} \sqrt {a+b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 91, normalized size = 2.76 \[ \left [-\frac {\sqrt {-2 \, a - 2 \, b} \log \left (\frac {x^{4} - 2 \, \sqrt {-2 \, a - 2 \, b} x^{2} - 2 \, a - 2 \, b}{x^{4} + 2 \, a + 2 \, b}\right )}{8 \, {\left (a + b\right )}}, \frac {\sqrt {2 \, a + 2 \, b} \arctan \left (\frac {\sqrt {2 \, a + 2 \, b} x^{2}}{2 \, {\left (a + b\right )}}\right )}{4 \, {\left (a + b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 24, normalized size = 0.73 \[ \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} x^{2}}{2 \, \sqrt {a + b}}\right )}{4 \, \sqrt {a + b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 26, normalized size = 0.79 \[ \frac {\arctan \left (\frac {x^{2}}{\sqrt {2 a +2 b}}\right )}{2 \sqrt {2 a +2 b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.03, size = 25, normalized size = 0.76 \[ \frac {\arctan \left (\frac {x^{2}}{\sqrt {2 \, a + 2 \, b}}\right )}{2 \, \sqrt {2 \, a + 2 \, b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.04, size = 32, normalized size = 0.97 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^2\,\sqrt {a+b}}{2\,a+2\,b}\right )}{4\,\sqrt {a+b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.49, size = 110, normalized size = 3.33 \[ - \frac {\sqrt {2} \sqrt {- \frac {1}{a + b}} \log {\left (- \sqrt {2} a \sqrt {- \frac {1}{a + b}} - \sqrt {2} b \sqrt {- \frac {1}{a + b}} + x^{2} \right )}}{8} + \frac {\sqrt {2} \sqrt {- \frac {1}{a + b}} \log {\left (\sqrt {2} a \sqrt {- \frac {1}{a + b}} + \sqrt {2} b \sqrt {- \frac {1}{a + b}} + x^{2} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________