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 = 0.0223402, antiderivative size = 33, normalized size of antiderivative = 1., 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 275
Rule 203
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 = 0.0082419, size = 33, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x^2}{\sqrt{2} \sqrt{a+b}}\right )}{2 \sqrt{2} \sqrt{a+b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 26, normalized size = 0.8 \begin{align*}{\frac{1}{2}\arctan \left ({{x}^{2}{\frac{1}{\sqrt{2\,a+2\,b}}}} \right ){\frac{1}{\sqrt{2\,a+2\,b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.49796, size = 230, normalized size = 6.97 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.205193, size = 110, normalized size = 3.33 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11739, size = 32, normalized size = 0.97 \begin{align*} \frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} x^{2}}{2 \, \sqrt{a + b}}\right )}{4 \, \sqrt{a + b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]