Optimal. Leaf size=46 \[ -\frac{a^2}{4 c^3 \left (a+c x^4\right )}-\frac{a \log \left (a+c x^4\right )}{2 c^3}+\frac{x^4}{4 c^2} \]
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Rubi [A] time = 0.0325363, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{a^2}{4 c^3 \left (a+c x^4\right )}-\frac{a \log \left (a+c x^4\right )}{2 c^3}+\frac{x^4}{4 c^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a+c x^4\right )^2} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(a+c x)^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{a^2}{c^2 (a+c x)^2}-\frac{2 a}{c^2 (a+c x)}\right ) \, dx,x,x^4\right )\\ &=\frac{x^4}{4 c^2}-\frac{a^2}{4 c^3 \left (a+c x^4\right )}-\frac{a \log \left (a+c x^4\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.0180651, size = 38, normalized size = 0.83 \[ \frac{-\frac{a^2}{a+c x^4}-2 a \log \left (a+c x^4\right )+c x^4}{4 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 41, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4\,{c}^{2}}}-{\frac{{a}^{2}}{4\,{c}^{3} \left ( c{x}^{4}+a \right ) }}-{\frac{a\ln \left ( c{x}^{4}+a \right ) }{2\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03688, size = 58, normalized size = 1.26 \begin{align*} \frac{x^{4}}{4 \, c^{2}} - \frac{a^{2}}{4 \,{\left (c^{4} x^{4} + a c^{3}\right )}} - \frac{a \log \left (c x^{4} + a\right )}{2 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68867, size = 113, normalized size = 2.46 \begin{align*} \frac{c^{2} x^{8} + a c x^{4} - a^{2} - 2 \,{\left (a c x^{4} + a^{2}\right )} \log \left (c x^{4} + a\right )}{4 \,{\left (c^{4} x^{4} + a c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.697961, size = 41, normalized size = 0.89 \begin{align*} - \frac{a^{2}}{4 a c^{3} + 4 c^{4} x^{4}} - \frac{a \log{\left (a + c x^{4} \right )}}{2 c^{3}} + \frac{x^{4}}{4 c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13941, size = 66, normalized size = 1.43 \begin{align*} \frac{x^{4}}{4 \, c^{2}} - \frac{a \log \left ({\left | c x^{4} + a \right |}\right )}{2 \, c^{3}} + \frac{2 \, a c x^{4} + a^{2}}{4 \,{\left (c x^{4} + a\right )} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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