Optimal. Leaf size=77 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{2} a \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
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Rubi [A] time = 0.0358898, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {1152, 377, 208} \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{2} a \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1152
Rule 377
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}} \, dx &=\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \int \frac{1}{\left (a-b x^2\right ) \sqrt{a+b x^2}} \, dx}{\sqrt{a^2-b^2 x^4}}\\ &=\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{a-2 a b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{\sqrt{a^2-b^2 x^4}}\\ &=\frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{2} a \sqrt{b} \sqrt{a^2-b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0509518, size = 77, normalized size = 1. \[ \frac{\sqrt{a^2-b^2 x^4} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{2} a \sqrt{b} \sqrt{a-b x^2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 267, normalized size = 3.5 \begin{align*}{\frac{1}{2\,b{x}^{2}-2\,a}\sqrt{-b{x}^{2}+a}\sqrt{-{b}^{2}{x}^{4}+{a}^{2}}\sqrt{b} \left ( \sqrt{a}\sqrt{2}\ln \left ( 2\,{\frac{b \left ( \sqrt{2}\sqrt{a}\sqrt{b{x}^{2}+a}+\sqrt{ab}x+a \right ) }{bx-\sqrt{ab}}} \right ) \sqrt{b}-\sqrt{a}\sqrt{2}\ln \left ( 2\,{\frac{b \left ( \sqrt{2}\sqrt{a}\sqrt{b{x}^{2}+a}-\sqrt{ab}x+a \right ) }{bx+\sqrt{ab}}} \right ) \sqrt{b}+2\,\ln \left ({\frac{1}{\sqrt{b}} \left ( \sqrt{b}\sqrt{-{\frac{ \left ( bx+\sqrt{-ab} \right ) \left ( -bx+\sqrt{-ab} \right ) }{b}}}+bx \right ) } \right ) \sqrt{ab}-2\,\sqrt{ab}\ln \left ({\frac{\sqrt{b}\sqrt{b{x}^{2}+a}+bx}{\sqrt{b}}} \right ) \right ){\frac{1}{\sqrt{b{x}^{2}+a}}} \left ( \sqrt{-ab}+\sqrt{ab} \right ) ^{-1} \left ( \sqrt{-ab}-\sqrt{ab} \right ) ^{-1}{\frac{1}{\sqrt{ab}}}} \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}{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99997, size = 359, normalized size = 4.66 \begin{align*} \left [\frac{\sqrt{2} \log \left (-\frac{3 \, b^{2} x^{4} - 2 \, a b x^{2} - 2 \, \sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{b} x - a^{2}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}\right )}{4 \, a \sqrt{b}}, \frac{\sqrt{2} \sqrt{-b} \arctan \left (\frac{\sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{-b}}{2 \,{\left (b^{2} x^{3} - a b x\right )}}\right )}{2 \, a b}\right ] \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}{\sqrt{- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \sqrt{a - b x^{2}}}\, 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}{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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