Optimal. Leaf size=109 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 \sqrt [4]{a} \sqrt{b} \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 \sqrt [4]{a} \sqrt{b} \sqrt{\sqrt{a}+\sqrt{b}}} \]
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Rubi [A] time = 0.0467918, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1093, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 \sqrt [4]{a} \sqrt{b} \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 \sqrt [4]{a} \sqrt{b} \sqrt{\sqrt{a}+\sqrt{b}}} \]
Antiderivative was successfully verified.
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Rule 1093
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{a-b+2 a x^2+a x^4} \, dx &=\frac{\sqrt{a} \int \frac{1}{a-\sqrt{a} \sqrt{b}+a x^2} \, dx}{2 \sqrt{b}}-\frac{\sqrt{a} \int \frac{1}{a+\sqrt{a} \sqrt{b}+a x^2} \, dx}{2 \sqrt{b}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 \sqrt [4]{a} \sqrt{\sqrt{a}-\sqrt{b}} \sqrt{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 \sqrt [4]{a} \sqrt{\sqrt{a}+\sqrt{b}} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0628166, size = 105, normalized size = 0.96 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a-\sqrt{a} \sqrt{b}}}\right )}{2 \sqrt{b} \sqrt{a-\sqrt{a} \sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{2 \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.139, size = 74, normalized size = 0.7 \begin{align*} -{\frac{a}{2}{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}}-{\frac{a}{2}\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \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}{a x^{4} + 2 \, a x^{2} + a - b}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50551, size = 1115, normalized size = 10.23 \begin{align*} -\frac{1}{4} \, \sqrt{-\frac{\frac{a b - b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} \log \left ({\left (b - \frac{a^{2} b - a b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt{-\frac{\frac{a b - b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} + x\right ) + \frac{1}{4} \, \sqrt{-\frac{\frac{a b - b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} \log \left (-{\left (b - \frac{a^{2} b - a b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt{-\frac{\frac{a b - b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} + x\right ) - \frac{1}{4} \, \sqrt{\frac{\frac{a b - b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} \log \left ({\left (b + \frac{a^{2} b - a b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt{\frac{\frac{a b - b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} + x\right ) + \frac{1}{4} \, \sqrt{\frac{\frac{a b - b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} \log \left (-{\left (b + \frac{a^{2} b - a b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt{\frac{\frac{a b - b^{2}}{\sqrt{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} + x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.639959, size = 63, normalized size = 0.58 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (256 a^{2} b^{2} - 256 a b^{3}\right ) + 32 t^{2} a b + 1, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} b + 64 t^{3} a b^{2} - 4 t a - 4 t b + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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