Optimal. Leaf size=38 \[ \frac {1}{12} \tanh ^{-1}\left (\sqrt {x^3+1}\right )+\frac {\sqrt {x^3+1} \left (-x^3-2\right )}{12 x^6} \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {266, 47, 51, 63, 207} \begin {gather*} -\frac {\sqrt {x^3+1}}{12 x^3}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {x^3+1}\right )-\frac {\sqrt {x^3+1}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^3}}{x^7} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{6 x^6}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{6 x^6}-\frac {\sqrt {1+x^3}}{12 x^3}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{6 x^6}-\frac {\sqrt {1+x^3}}{12 x^3}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=-\frac {\sqrt {1+x^3}}{6 x^6}-\frac {\sqrt {1+x^3}}{12 x^3}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.68 \begin {gather*} -\frac {2}{9} \left (x^3+1\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};x^3+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 38, normalized size = 1.00 \begin {gather*} \frac {\left (-2-x^3\right ) \sqrt {1+x^3}}{12 x^6}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 49, normalized size = 1.29 \begin {gather*} \frac {x^{6} \log \left (\sqrt {x^{3} + 1} + 1\right ) - x^{6} \log \left (\sqrt {x^{3} + 1} - 1\right ) - 2 \, {\left (x^{3} + 2\right )} \sqrt {x^{3} + 1}}{24 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 78, normalized size = 2.05 \begin {gather*} -\frac {\sqrt {x^{3} + 1} + \frac {1}{\sqrt {x^{3} + 1}}}{12 \, {\left ({\left (\sqrt {x^{3} + 1} + \frac {1}{\sqrt {x^{3} + 1}}\right )}^{2} - 4\right )}} + \frac {1}{48} \, \log \left (\sqrt {x^{3} + 1} + \frac {1}{\sqrt {x^{3} + 1}} + 2\right ) - \frac {1}{48} \, \log \left ({\left | \sqrt {x^{3} + 1} + \frac {1}{\sqrt {x^{3} + 1}} - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 34, normalized size = 0.89
method | result | size |
risch | \(-\frac {x^{6}+3 x^{3}+2}{12 x^{6} \sqrt {x^{3}+1}}+\frac {\arctanh \left (\sqrt {x^{3}+1}\right )}{12}\) | \(34\) |
default | \(-\frac {\sqrt {x^{3}+1}}{6 x^{6}}-\frac {\sqrt {x^{3}+1}}{12 x^{3}}+\frac {\arctanh \left (\sqrt {x^{3}+1}\right )}{12}\) | \(36\) |
elliptic | \(-\frac {\sqrt {x^{3}+1}}{6 x^{6}}-\frac {\sqrt {x^{3}+1}}{12 x^{3}}+\frac {\arctanh \left (\sqrt {x^{3}+1}\right )}{12}\) | \(36\) |
trager | \(-\frac {\left (x^{3}+2\right ) \sqrt {x^{3}+1}}{12 x^{6}}+\frac {\ln \left (-\frac {x^{3}+2 \sqrt {x^{3}+1}+2}{x^{3}}\right )}{24}\) | \(41\) |
meijerg | \(-\frac {\frac {\sqrt {\pi }}{x^{6}}+\frac {\sqrt {\pi }}{x^{3}}+\frac {\left (\frac {1}{2}-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }\, \left (x^{6}+8 x^{3}+8\right )}{8 x^{6}}+\frac {\sqrt {\pi }\, \left (4 x^{3}+8\right ) \sqrt {x^{3}+1}}{8 x^{6}}-\frac {\ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{2}}{6 \sqrt {\pi }}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 60, normalized size = 1.58 \begin {gather*} \frac {{\left (x^{3} + 1\right )}^{\frac {3}{2}} + \sqrt {x^{3} + 1}}{12 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {1}{24} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {1}{24} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 189, normalized size = 4.97 \begin {gather*} -\frac {\sqrt {x^3+1}}{12\,x^3}-\frac {\sqrt {x^3+1}}{6\,x^6}+\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{4\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.26, size = 65, normalized size = 1.71 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{12} - \frac {1}{12 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {1}{4 x^{\frac {9}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {1}{6 x^{\frac {15}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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