Optimal. Leaf size=35 \[ \frac {1}{3} \log \left (\sqrt {x^6+1}+x^3\right )-\frac {\sqrt {x^6+1}}{3 x^3} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 277, 215} \begin {gather*} \frac {1}{3} \sinh ^{-1}\left (x^3\right )-\frac {\sqrt {x^6+1}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 275
Rule 277
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^6}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^6}}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^6}}{3 x^3}+\frac {1}{3} \sinh ^{-1}\left (x^3\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.71 \begin {gather*} \frac {1}{3} \sinh ^{-1}\left (x^3\right )-\frac {\sqrt {x^6+1}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 35, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+x^6}}{3 x^3}+\frac {1}{3} \log \left (x^3+\sqrt {1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 34, normalized size = 0.97 \begin {gather*} -\frac {x^{3} \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) + x^{3} + \sqrt {x^{6} + 1}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 38, normalized size = 1.09 \begin {gather*} -\frac {2 \, \sqrt {\frac {1}{x^{6}} + 1} - \log \left (\sqrt {\frac {1}{x^{6}} + 1} + 1\right ) + \log \left (\sqrt {\frac {1}{x^{6}} + 1} - 1\right )}{6 \, \mathrm {sgn}\relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 20, normalized size = 0.57
method | result | size |
risch | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}+\frac {\arcsinh \left (x^{3}\right )}{3}\) | \(20\) |
trager | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}+1}\right )}{3}\) | \(30\) |
meijerg | \(-\frac {\frac {4 \sqrt {\pi }\, \sqrt {x^{6}+1}}{x^{3}}-4 \sqrt {\pi }\, \arcsinh \left (x^{3}\right )}{12 \sqrt {\pi }}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 45, normalized size = 1.29 \begin {gather*} -\frac {\sqrt {x^{6} + 1}}{3 \, x^{3}} + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {x^6+1}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.98, size = 34, normalized size = 0.97 \begin {gather*} - \frac {x^{3}}{3 \sqrt {x^{6} + 1}} + \frac {\operatorname {asinh}{\left (x^{3} \right )}}{3} - \frac {1}{3 x^{3} \sqrt {x^{6} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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