Optimal. Leaf size=59 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac{\sqrt{b x^2+c x^4}}{2 b x^3} \]
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Rubi [A] time = 0.05555, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3, 2025, 2008, 206} \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac{\sqrt{b x^2+c x^4}}{2 b x^3} \]
Antiderivative was successfully verified.
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Rule 3
Rule 2025
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx &=\int \frac{1}{x^2 \sqrt{b x^2+c x^4}} \, dx\\ &=-\frac{\sqrt{b x^2+c x^4}}{2 b x^3}-\frac{c \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx}{2 b}\\ &=-\frac{\sqrt{b x^2+c x^4}}{2 b x^3}+\frac{c \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )}{2 b}\\ &=-\frac{\sqrt{b x^2+c x^4}}{2 b x^3}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0629977, size = 68, normalized size = 1.15 \[ \frac{c \sqrt{x^2 \left (b+c x^2\right )} \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{c x^2}{b}+1}\right )}{2 \sqrt{\frac{c x^2}{b}+1}}-\frac{b}{2 c x^2}\right )}{b^2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 73, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,x}\sqrt{c{x}^{2}+b} \left ( -c\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{2}b+\sqrt{c{x}^{2}+b}{b}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{b}^{-{\frac{5}{2}}}} \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{c x^{4} + b x^{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52262, size = 306, normalized size = 5.19 \begin{align*} \left [\frac{\sqrt{b} c x^{3} \log \left (-\frac{c x^{3} + 2 \, b x + 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}} b}{4 \, b^{2} x^{3}}, -\frac{\sqrt{-b} c x^{3} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}{c x^{3} + b x}\right ) + \sqrt{c x^{4} + b x^{2}} b}{2 \, b^{2} x^{3}}\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}{x^{2} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \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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