Optimal. Leaf size=43 \[ \frac {\sqrt {1+x^3} \left (-8+10 x^3-15 x^6\right )}{72 x^9}+\frac {5}{24} \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.47, number of steps
used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 44, 65,
213} \begin {gather*} -\frac {5 \sqrt {x^3+1}}{24 x^3}+\frac {5}{24} \tanh ^{-1}\left (\sqrt {x^3+1}\right )-\frac {\sqrt {x^3+1}}{9 x^9}+\frac {5 \sqrt {x^3+1}}{36 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^{10} \sqrt {1+x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^4 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{9 x^9}-\frac {5}{18} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{9 x^9}+\frac {5 \sqrt {1+x^3}}{36 x^6}+\frac {5}{24} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{9 x^9}+\frac {5 \sqrt {1+x^3}}{36 x^6}-\frac {5 \sqrt {1+x^3}}{24 x^3}-\frac {5}{48} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{9 x^9}+\frac {5 \sqrt {1+x^3}}{36 x^6}-\frac {5 \sqrt {1+x^3}}{24 x^3}-\frac {5}{24} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=-\frac {\sqrt {1+x^3}}{9 x^9}+\frac {5 \sqrt {1+x^3}}{36 x^6}-\frac {5 \sqrt {1+x^3}}{24 x^3}+\frac {5}{24} \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^3} \left (-8+10 x^3-15 x^6\right )}{72 x^9}+\frac {5}{24} \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.12, size = 48, normalized size = 1.12
method | result | size |
risch | \(-\frac {15 x^{9}+5 x^{6}-2 x^{3}+8}{72 x^{9} \sqrt {x^{3}+1}}+\frac {5 \arctanh \left (\sqrt {x^{3}+1}\right )}{24}\) | \(41\) |
default | \(-\frac {\sqrt {x^{3}+1}}{9 x^{9}}+\frac {5 \sqrt {x^{3}+1}}{36 x^{6}}-\frac {5 \sqrt {x^{3}+1}}{24 x^{3}}+\frac {5 \arctanh \left (\sqrt {x^{3}+1}\right )}{24}\) | \(48\) |
elliptic | \(-\frac {\sqrt {x^{3}+1}}{9 x^{9}}+\frac {5 \sqrt {x^{3}+1}}{36 x^{6}}-\frac {5 \sqrt {x^{3}+1}}{24 x^{3}}+\frac {5 \arctanh \left (\sqrt {x^{3}+1}\right )}{24}\) | \(48\) |
trager | \(-\frac {\left (15 x^{6}-10 x^{3}+8\right ) \sqrt {x^{3}+1}}{72 x^{9}}-\frac {5 \ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+1}-2}{x^{3}}\right )}{48}\) | \(50\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \left (148 x^{9}+144 x^{6}-96 x^{3}+128\right )}{384 x^{9}}-\frac {\sqrt {\pi }\, \left (240 x^{6}-160 x^{3}+128\right ) \sqrt {x^{3}+1}}{384 x^{9}}+\frac {5 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )}{8}-\frac {5 \left (\frac {37}{30}-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }}{16}-\frac {\sqrt {\pi }}{3 x^{9}}+\frac {\sqrt {\pi }}{4 x^{6}}-\frac {3 \sqrt {\pi }}{8 x^{3}}}{3 \sqrt {\pi }}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs.
\(2 (35) = 70\).
time = 0.26, size = 80, normalized size = 1.86 \begin {gather*} -\frac {15 \, {\left (x^{3} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{3} + 1}}{72 \, {\left ({\left (x^{3} + 1\right )}^{3} + 3 \, x^{3} - 3 \, {\left (x^{3} + 1\right )}^{2} + 2\right )}} + \frac {5}{48} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {5}{48} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 57, normalized size = 1.33 \begin {gather*} \frac {15 \, x^{9} \log \left (\sqrt {x^{3} + 1} + 1\right ) - 15 \, x^{9} \log \left (\sqrt {x^{3} + 1} - 1\right ) - 2 \, {\left (15 \, x^{6} - 10 \, x^{3} + 8\right )} \sqrt {x^{3} + 1}}{144 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs.
\(2 (37) = 74\).
time = 7.50, size = 85, normalized size = 1.98 \begin {gather*} \frac {5 \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{24} - \frac {5}{24 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {5}{72 x^{\frac {9}{2}} \sqrt {1 + \frac {1}{x^{3}}}} + \frac {1}{36 x^{\frac {15}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {1}{9 x^{\frac {21}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 59, normalized size = 1.37 \begin {gather*} -\frac {15 \, {\left (x^{3} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{3} + 1}}{72 \, x^{9}} + \frac {5}{48} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {5}{48} \, \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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Mupad [B]
time = 0.05, size = 201, normalized size = 4.67 \begin {gather*} \frac {5\,\sqrt {x^3+1}}{36\,x^6}-\frac {5\,\sqrt {x^3+1}}{24\,x^3}-\frac {\sqrt {x^3+1}}{9\,x^9}+\frac {5\,\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 )}{8\,\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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