Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]
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Rubi [A] time = 0.0210504, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {275, 199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 275
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{x}{\left (a+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{4 a \left (a+c x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{4 a}\\ &=\frac{x^2}{4 a \left (a+c x^4\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0255249, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 40, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{1}{4\,a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.80212, size = 274, normalized size = 5.59 \begin{align*} \left [\frac{2 \, a c x^{2} -{\left (c x^{4} + a\right )} \sqrt{-a c} \log \left (\frac{c x^{4} - 2 \, \sqrt{-a c} x^{2} - a}{c x^{4} + a}\right )}{8 \,{\left (a^{2} c^{2} x^{4} + a^{3} c\right )}}, \frac{a c x^{2} -{\left (c x^{4} + a\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c}}{c x^{2}}\right )}{4 \,{\left (a^{2} c^{2} x^{4} + a^{3} c\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.613966, size = 83, normalized size = 1.69 \begin{align*} \frac{x^{2}}{4 a^{2} + 4 a c x^{4}} - \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} c}} + x^{2} \right )}}{8} + \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} c}} + x^{2} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16712, size = 53, normalized size = 1.08 \begin{align*} \frac{x^{2}}{4 \,{\left (c x^{4} + a\right )} a} + \frac{\arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \, \sqrt{a c} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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