Optimal. Leaf size=16 \[ \tanh ^{-1}\left (\frac {x^2-1}{\sqrt {x^4+1}}\right ) \]
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Rubi [A] time = 0.02, 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 63
Rule 207
Rule 215
Rule 266
Rule 844
Rule 1252
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*}
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Mathematica [A] time = 0.02, 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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IntegrateAlgebraic [B] time = 0.19, size = 35, normalized size = 2.19 \[ \frac {1}{2} \log \left (\sqrt {x^4+1}+x^2\right )-\tanh ^{-1}\left (\sqrt {x^4+1}+x^2\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 49, normalized size = 3.06 \[ -\frac {1}{2} \, \log \left (2 \, x^{4} - x^{2} - \sqrt {x^{4} + 1} {\left (2 \, x^{2} - 1\right )} + 1\right ) + \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 51, normalized size = 3.19 \[ \frac {1}{2} \, \log \left (x^{2} - \sqrt {x^{4} + 1} + 1\right ) - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1} + 1\right ) - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 18, normalized size = 1.12
method | result | size |
default | \(-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}+\frac {\arcsinh \left (x^{2}\right )}{2}\) | \(18\) |
trager | \(\ln \left (\frac {x^{2}+\sqrt {x^{4}+1}-1}{x}\right )\) | \(18\) |
elliptic | \(-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}+\frac {\arcsinh \left (x^{2}\right )}{2}\) | \(18\) |
meijerg | \(\frac {\arcsinh \left (x^{2}\right )}{2}+\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{4}+1}}{2}\right )+\left (-2 \ln \relax (2)+4 \ln \relax (x )\right ) \sqrt {\pi }}{4 \sqrt {\pi }}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.17, size = 57, normalized size = 3.56 \[ -\frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) + \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} + 1\right ) - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 17, normalized size = 1.06 \[ \frac {\mathrm {asinh}\left (x^2\right )}{2}-\frac {\mathrm {atanh}\left (\sqrt {x^4+1}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.51, size = 14, normalized size = 0.88 \[ - \frac {\operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{2} + \frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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