Optimal. Leaf size=60 \[ \frac {\sqrt {x^4+1} \left (1-2 x^2\right )}{4 x^4}+\frac {1}{2} \log \left (\sqrt {x^4+1}+x^2\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}+x^2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 0.77, 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, 811, 844, 215, 266, 63, 207} \begin {gather*} \frac {1}{4} \tanh ^{-1}\left (\sqrt {x^4+1}\right )+\frac {1}{2} \sinh ^{-1}\left (x^2\right )+\frac {\sqrt {x^4+1} \left (1-2 x^2\right )}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 215
Rule 266
Rule 811
Rule 844
Rule 1252
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-1+x) \sqrt {1+x^2}}{x^3} \, dx,x,x^2\right )\\ &=\frac {\left (1-2 x^2\right ) \sqrt {1+x^4}}{4 x^4}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {2-4 x}{x \sqrt {1+x^2}} \, dx,x,x^2\right )\\ &=\frac {\left (1-2 x^2\right ) \sqrt {1+x^4}}{4 x^4}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{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 {\left (1-2 x^2\right ) \sqrt {1+x^4}}{4 x^4}+\frac {1}{2} \sinh ^{-1}\left (x^2\right )-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {\left (1-2 x^2\right ) \sqrt {1+x^4}}{4 x^4}+\frac {1}{2} \sinh ^{-1}\left (x^2\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^4}\right )\\ &=\frac {\left (1-2 x^2\right ) \sqrt {1+x^4}}{4 x^4}+\frac {1}{2} \sinh ^{-1}\left (x^2\right )+\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^4}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 41, normalized size = 0.68 \begin {gather*} \frac {1}{4} \left (\tanh ^{-1}\left (\sqrt {x^4+1}\right )+2 \sinh ^{-1}\left (x^2\right )+\frac {\sqrt {x^4+1} \left (1-2 x^2\right )}{x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 64, normalized size = 1.07 \begin {gather*} \frac {\sqrt {x^4+1} \left (1-2 x^2\right )}{4 x^4}-\frac {1}{2} \log \left (\sqrt {x^4+1}-x^2\right )-\frac {1}{2} \tanh ^{-1}\left (x^2-\sqrt {x^4+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 85, normalized size = 1.42 \begin {gather*} \frac {x^{4} \log \left (-x^{2} + \sqrt {x^{4} + 1} + 1\right ) - 2 \, x^{4} \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) - x^{4} \log \left (-x^{2} + \sqrt {x^{4} + 1} - 1\right ) - 2 \, x^{4} - \sqrt {x^{4} + 1} {\left (2 \, x^{2} - 1\right )}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.71, size = 118, normalized size = 1.97 \begin {gather*} -\frac {{\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{3} - 2 \, {\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} + x^{2} - \sqrt {x^{4} + 1} + 2}{2 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1\right )}^{2}} - \frac {1}{4} \, \log \left (x^{2} - \sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \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 = 0.03, size = 63, normalized size = 1.05 \begin {gather*} -\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}+\frac {\left (x^{4}+1\right )^{\frac {3}{2}}}{4 x^{4}}-\frac {\sqrt {x^{4}+1}}{4}+\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 81, normalized size = 1.35 \begin {gather*} -\frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} + \frac {\sqrt {x^{4} + 1}}{4 \, x^{4}} + \frac {1}{8} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) - \frac {1}{8} \, \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 = 0.97, size = 45, normalized size = 0.75 \begin {gather*} \frac {\mathrm {asinh}\left (x^2\right )}{2}-\frac {\sqrt {x^4+1}}{2\,x^2}+\frac {\sqrt {x^4+1}}{4\,x^4}-\frac {\mathrm {atan}\left (\sqrt {x^4+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.70, size = 58, normalized size = 0.97 \begin {gather*} - \frac {x^{2}}{2 \sqrt {x^{4} + 1}} + \frac {\operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{4} + \frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} + \frac {\sqrt {1 + \frac {1}{x^{4}}}}{4 x^{2}} - \frac {1}{2 x^{2} \sqrt {x^{4} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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