Optimal. Leaf size=41 \[ \frac {2 \sqrt {x^3+1} \left (x^6+7 x^3-3\right )}{9 x^3}-\frac {4}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 57, normalized size of antiderivative = 1.39, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1821, 1612, 51, 63, 207} \begin {gather*} \frac {2}{9} \left (x^3+1\right )^{3/2}-\frac {2 \sqrt {x^3+1}}{3 x^3}+\frac {4 \sqrt {x^3+1}}{3}-\frac {4}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 207
Rule 1612
Rule 1821
Rubi steps
\begin {align*} \int \frac {\left (2+x^3\right ) \left (1+x^3+x^6\right )}{x^4 \sqrt {1+x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(2+x) \left (1+x+x^2\right )}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {2}{\sqrt {1+x}}+\frac {2}{x^2 \sqrt {1+x}}+\frac {3}{x \sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,x^3\right )\\ &=\frac {4 \sqrt {1+x^3}}{3}+\frac {2}{9} \left (1+x^3\right )^{3/2}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )+\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {4 \sqrt {1+x^3}}{3}-\frac {2 \sqrt {1+x^3}}{3 x^3}+\frac {2}{9} \left (1+x^3\right )^{3/2}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )+2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=\frac {4 \sqrt {1+x^3}}{3}-\frac {2 \sqrt {1+x^3}}{3 x^3}+\frac {2}{9} \left (1+x^3\right )^{3/2}-2 \tanh ^{-1}\left (\sqrt {1+x^3}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=\frac {4 \sqrt {1+x^3}}{3}-\frac {2 \sqrt {1+x^3}}{3 x^3}+\frac {2}{9} \left (1+x^3\right )^{3/2}-\frac {4}{3} \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 40, normalized size = 0.98 \begin {gather*} \frac {2}{9} \left (\frac {\sqrt {x^3+1} \left (x^6+7 x^3-3\right )}{x^3}-6 \tanh ^{-1}\left (\sqrt {x^3+1}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 41, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+x^3} \left (-3+7 x^3+x^6\right )}{9 x^3}-\frac {4}{3} \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 55, normalized size = 1.34 \begin {gather*} -\frac {2 \, {\left (3 \, x^{3} \log \left (\sqrt {x^{3} + 1} + 1\right ) - 3 \, x^{3} \log \left (\sqrt {x^{3} + 1} - 1\right ) - {\left (x^{6} + 7 \, x^{3} - 3\right )} \sqrt {x^{3} + 1}\right )}}{9 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 56, normalized size = 1.37 \begin {gather*} \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} + \frac {4}{3} \, \sqrt {x^{3} + 1} - \frac {2 \, \sqrt {x^{3} + 1}}{3 \, x^{3}} - \frac {2}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {2}{3} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 45, normalized size = 1.10
method | result | size |
default | \(\frac {2 \sqrt {x^{3}+1}\, x^{3}}{9}+\frac {14 \sqrt {x^{3}+1}}{9}-\frac {4 \arctanh \left (\sqrt {x^{3}+1}\right )}{3}-\frac {2 \sqrt {x^{3}+1}}{3 x^{3}}\) | \(45\) |
risch | \(\frac {2 \sqrt {x^{3}+1}\, x^{3}}{9}+\frac {14 \sqrt {x^{3}+1}}{9}-\frac {4 \arctanh \left (\sqrt {x^{3}+1}\right )}{3}-\frac {2 \sqrt {x^{3}+1}}{3 x^{3}}\) | \(45\) |
elliptic | \(\frac {2 \sqrt {x^{3}+1}\, x^{3}}{9}+\frac {14 \sqrt {x^{3}+1}}{9}-\frac {4 \arctanh \left (\sqrt {x^{3}+1}\right )}{3}-\frac {2 \sqrt {x^{3}+1}}{3 x^{3}}\) | \(45\) |
trager | \(\frac {2 \sqrt {x^{3}+1}\, \left (x^{6}+7 x^{3}-3\right )}{9 x^{3}}-\frac {2 \ln \left (-\frac {x^{3}+2 \sqrt {x^{3}+1}+2}{x^{3}}\right )}{3}\) | \(46\) |
meijerg | \(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right ) \sqrt {x^{3}+1}}{6}}{3 \sqrt {\pi }}+\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{3}+1}}{\sqrt {\pi }}+\frac {\left (-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }-2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{\sqrt {\pi }}+\frac {-\frac {2 \sqrt {\pi }}{3 x^{3}}-\frac {\left (1-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }}{3}+\frac {\sqrt {\pi }\, \left (4 x^{3}+8\right )}{12 x^{3}}-\frac {2 \sqrt {\pi }\, \sqrt {x^{3}+1}}{3 x^{3}}+\frac {2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{3}}{\sqrt {\pi }}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 55, normalized size = 1.34 \begin {gather*} \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} + \frac {4}{3} \, \sqrt {x^{3} + 1} - \frac {2 \, \sqrt {x^{3} + 1}}{3 \, x^{3}} - \frac {2}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {2}{3} \, \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 = 0.05, size = 198, normalized size = 4.83 \begin {gather*} \frac {14\,\sqrt {x^3+1}}{9}-\frac {2\,\sqrt {x^3+1}}{3\,x^3}+\frac {2\,x^3\,\sqrt {x^3+1}}{9}-\frac {4\,\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 )}{\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 = 74.24, size = 83, normalized size = 2.02 \begin {gather*} \frac {2 \left (x^{3} + 1\right )^{\frac {3}{2}}}{9} + \frac {4 \sqrt {x^{3} + 1}}{3} + \frac {2 \log {\left (-1 + \frac {1}{\sqrt {x^{3} + 1}} \right )}}{3} - \frac {2 \log {\left (1 + \frac {1}{\sqrt {x^{3} + 1}} \right )}}{3} + \frac {1}{3 \left (1 + \frac {1}{\sqrt {x^{3} + 1}}\right )} + \frac {1}{3 \left (-1 + \frac {1}{\sqrt {x^{3} + 1}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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