Optimal. Leaf size=16 \[ \tanh ^{-1}\left (\frac{x^2-1}{\sqrt{x^4+1}}\right ) \]
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Rubi [A] time = 0.0241054, antiderivative size = 23, normalized size of antiderivative = 1.44, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1252, 844, 215, 266, 63, 207} \[ \frac{1}{2} \sinh ^{-1}\left (x^2\right )-\frac{1}{2} \tanh ^{-1}\left (\sqrt{x^4+1}\right ) \]
Antiderivative was successfully verified.
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Rule 1252
Rule 844
Rule 215
Rule 266
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{1+x^2}{x \sqrt{1+x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+x}{x \sqrt{1+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^2\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \sinh ^{-1}\left (x^2\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,x^4\right )\\ &=\frac{1}{2} \sinh ^{-1}\left (x^2\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x^4}\right )\\ &=\frac{1}{2} \sinh ^{-1}\left (x^2\right )-\frac{1}{2} \tanh ^{-1}\left (\sqrt{1+x^4}\right )\\ \end{align*}
Mathematica [A] time = 0.0169876, size = 21, normalized size = 1.31 \[ \frac{1}{2} \left (\sinh ^{-1}\left (x^2\right )-\tanh ^{-1}\left (\sqrt{x^4+1}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 18, normalized size = 1.1 \begin{align*} -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{4}+1}}} \right ) }+{\frac{{\it Arcsinh} \left ({x}^{2} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.51227, size = 77, normalized size = 4.81 \begin{align*} -\frac{1}{4} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) + \frac{1}{4} \, \log \left (\sqrt{x^{4} + 1} - 1\right ) + \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) - \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.55958, size = 123, normalized size = 7.69 \begin{align*} -\frac{1}{2} \, \log \left (2 \, x^{4} - x^{2} - \sqrt{x^{4} + 1}{\left (2 \, x^{2} - 1\right )} + 1\right ) + \frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.95985, size = 14, normalized size = 0.88 \begin{align*} - \frac{\operatorname{asinh}{\left (\frac{1}{x^{2}} \right )}}{2} + \frac{\operatorname{asinh}{\left (x^{2} \right )}}{2} \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{x^{2} + 1}{\sqrt{x^{4} + 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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