Optimal. Leaf size=41 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}-c x^2}}\right )}{a \sqrt{c}} \]
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Rubi [A] time = 0.14157, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2128, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}-c x^2}}\right )}{a \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 2128
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^4\right ) \sqrt{-c x^2+d \sqrt{a+b x^4}}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{1+c x^2} \, dx,x,\frac{x}{\sqrt{-c x^2+d \sqrt{a+b x^4}}}\right )}{a}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{-c x^2+d \sqrt{a+b x^4}}}\right )}{a \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.447791, size = 47, normalized size = 1.15 \[ \frac{\sqrt{\frac{1}{c}} \cot ^{-1}\left (\frac{\sqrt{\frac{1}{c}} \sqrt{d \sqrt{a+b x^4}-c x^2}}{x}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{4}+a}{\frac{1}{\sqrt{-c{x}^{2}+d\sqrt{b{x}^{4}+a}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{-c x^{2} + \sqrt{b x^{4} + a} d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{4}\right ) \sqrt{- c x^{2} + d \sqrt{a + b x^{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{-c x^{2} + \sqrt{b x^{4} + a} d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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