Optimal. Leaf size=60 \[ \frac {\sqrt {x^4+1} \left (x^2+1\right )}{2 x^2}-\frac {1}{2} \log \left (\sqrt {x^4+1}+x^2+1\right )+\tanh ^{-1}\left (-2 \sqrt {x^4+1}-2 x^2+1\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 44, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1252, 813, 844, 215, 266, 63, 207} \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right )-\frac {1}{2} \sinh ^{-1}\left (x^2\right )+\frac {\sqrt {x^4+1} \left (x^2+1\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 215
Rule 266
Rule 813
Rule 844
Rule 1252
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-1+x) \sqrt {1+x^2}}{x^2} \, dx,x,x^2\right )\\ &=\frac {\left (1+x^2\right ) \sqrt {1+x^4}}{2 x^2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {-2+2 x}{x \sqrt {1+x^2}} \, dx,x,x^2\right )\\ &=\frac {\left (1+x^2\right ) \sqrt {1+x^4}}{2 x^2}-\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 {\left (1+x^2\right ) \sqrt {1+x^4}}{2 x^2}-\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 {\left (1+x^2\right ) \sqrt {1+x^4}}{2 x^2}-\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 {\left (1+x^2\right ) \sqrt {1+x^4}}{2 x^2}-\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 = 0.03, size = 41, normalized size = 0.68 \begin {gather*} \frac {1}{2} \left (-\tanh ^{-1}\left (\sqrt {x^4+1}\right )-\sinh ^{-1}\left (x^2\right )+\frac {\sqrt {x^4+1} \left (x^2+1\right )}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 58, normalized size = 0.97 \begin {gather*} \frac {\left (1+x^2\right ) \sqrt {1+x^4}}{2 x^2}+\tanh ^{-1}\left (x^2-\sqrt {1+x^4}\right )+\frac {1}{2} \log \left (-x^2+\sqrt {1+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 73, normalized size = 1.22 \begin {gather*} \frac {x^{2} \log \left (2 \, x^{4} + x^{2} - \sqrt {x^{4} + 1} {\left (2 \, x^{2} + 1\right )} + 1\right ) - x^{2} \log \left (-x^{2} + \sqrt {x^{4} + 1} + 1\right ) + x^{2} + \sqrt {x^{4} + 1} {\left (x^{2} + 1\right )}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 81, normalized size = 1.35 \begin {gather*} \frac {1}{2} \, \sqrt {x^{4} + 1} - \frac {1}{{\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1} + \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 38, normalized size = 0.63
method | result | size |
trager | \(\frac {\left (x^{2}+1\right ) \sqrt {x^{4}+1}}{2 x^{2}}-\ln \left (\frac {1+x^{2}+\sqrt {x^{4}+1}}{x}\right )\) | \(38\) |
risch | \(\frac {\sqrt {x^{4}+1}}{2 x^{2}}+\frac {\sqrt {x^{4}+1}}{2}-\frac {\arcsinh \left (x^{2}\right )}{2}-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(39\) |
elliptic | \(\frac {\sqrt {x^{4}+1}}{2 x^{2}}+\frac {\sqrt {x^{4}+1}}{2}-\frac {\arcsinh \left (x^{2}\right )}{2}-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(39\) |
default | \(\frac {\sqrt {x^{4}+1}}{2}-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}+\frac {\left (x^{4}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {x^{2} \sqrt {x^{4}+1}}{2}-\frac {\arcsinh \left (x^{2}\right )}{2}\) | \(51\) |
meijerg | \(\frac {\frac {4 \sqrt {\pi }\, \sqrt {x^{4}+1}}{x^{2}}-4 \sqrt {\pi }\, \arcsinh \left (x^{2}\right )}{8 \sqrt {\pi }}-\frac {4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {x^{4}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{4}+1}}{2}\right )-2 \left (2-2 \ln \relax (2)+4 \ln \relax (x )\right ) \sqrt {\pi }}{8 \sqrt {\pi }}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 78, normalized size = 1.30 \begin {gather*} \frac {1}{2} \, \sqrt {x^{4} + 1} + \frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 42, normalized size = 0.70 \begin {gather*} \frac {\sqrt {x^4+1}}{2}-\frac {\mathrm {asinh}\left (x^2\right )}{2}+\frac {\sqrt {x^4+1}}{2\,x^2}+\frac {\mathrm {atan}\left (\sqrt {x^4+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.90, size = 75, normalized size = 1.25 \begin {gather*} \frac {x^{2}}{2 \sqrt {x^{4} + 1}} + \frac {x^{2}}{2 \sqrt {1 + \frac {1}{x^{4}}}} - \frac {\operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{2} - \frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} + \frac {1}{2 x^{2} \sqrt {x^{4} + 1}} + \frac {1}{2 x^{2} \sqrt {1 + \frac {1}{x^{4}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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