Integrand size = 13, antiderivative size = 38 \[ \int \frac {\sqrt {1+x^3}}{x^7} \, dx=\frac {\left (-2-x^3\right ) \sqrt {1+x^3}}{12 x^6}+\frac {1}{12} \text {arctanh}\left (\sqrt {1+x^3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 43, 44, 65, 213} \[ \int \frac {\sqrt {1+x^3}}{x^7} \, dx=\frac {1}{12} \text {arctanh}\left (\sqrt {x^3+1}\right )-\frac {\sqrt {x^3+1}}{12 x^3}-\frac {\sqrt {x^3+1}}{6 x^6} \]
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Rule 43
Rule 44
Rule 65
Rule 213
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {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} \text {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} \text {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} \text {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} \text {arctanh}\left (\sqrt {1+x^3}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {1+x^3}}{x^7} \, dx=\frac {1}{12} \left (-\frac {\sqrt {1+x^3} \left (2+x^3\right )}{x^6}+\text {arctanh}\left (\sqrt {1+x^3}\right )\right ) \]
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Time = 1.53 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {x^{6}+3 x^{3}+2}{12 x^{6} \sqrt {x^{3}+1}}+\frac {\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{12}\) | \(34\) |
default | \(-\frac {\sqrt {x^{3}+1}}{6 x^{6}}-\frac {\sqrt {x^{3}+1}}{12 x^{3}}+\frac {\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{12}\) | \(36\) |
elliptic | \(-\frac {\sqrt {x^{3}+1}}{6 x^{6}}-\frac {\sqrt {x^{3}+1}}{12 x^{3}}+\frac {\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{12}\) | \(36\) |
trager | \(-\frac {\left (x^{3}+2\right ) \sqrt {x^{3}+1}}{12 x^{6}}-\frac {\ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+1}-2}{x^{3}}\right )}{24}\) | \(43\) |
pseudoelliptic | \(\frac {-\ln \left (\sqrt {x^{3}+1}-1\right ) x^{6}+\ln \left (\sqrt {x^{3}+1}+1\right ) x^{6}-2 \sqrt {x^{3}+1}\, x^{3}-4 \sqrt {x^{3}+1}}{24 \left (\sqrt {x^{3}+1}-1\right )^{2} \left (\sqrt {x^{3}+1}+1\right )^{2}}\) | \(76\) |
meijerg | \(-\frac {-\frac {\sqrt {\pi }\, \left (x^{6}+8 x^{3}+8\right )}{8 x^{6}}+\frac {\sqrt {\pi }\, \left (4 x^{3}+8\right ) \sqrt {x^{3}+1}}{8 x^{6}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )}{2}+\frac {\left (\frac {1}{2}-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{x^{6}}+\frac {\sqrt {\pi }}{x^{3}}}{6 \sqrt {\pi }}\) | \(93\) |
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Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {1+x^3}}{x^7} \, dx=\frac {x^{6} \log \left (\sqrt {x^{3} + 1} + 1\right ) - x^{6} \log \left (\sqrt {x^{3} + 1} - 1\right ) - 2 \, {\left (x^{3} + 2\right )} \sqrt {x^{3} + 1}}{24 \, x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (32) = 64\).
Time = 1.70 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {1+x^3}}{x^7} \, dx=\frac {\operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{12} - \frac {1}{12 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {1}{4 x^{\frac {9}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {1}{6 x^{\frac {15}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (28) = 56\).
Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {1+x^3}}{x^7} \, dx=\frac {{\left (x^{3} + 1\right )}^{\frac {3}{2}} + \sqrt {x^{3} + 1}}{12 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {1}{24} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {1}{24} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {1+x^3}}{x^7} \, dx=-\frac {\sqrt {x^{3} + 1} + \frac {1}{\sqrt {x^{3} + 1}}}{12 \, {\left ({\left (\sqrt {x^{3} + 1} + \frac {1}{\sqrt {x^{3} + 1}}\right )}^{2} - 4\right )}} + \frac {1}{48} \, \log \left (\sqrt {x^{3} + 1} + \frac {1}{\sqrt {x^{3} + 1}} + 2\right ) - \frac {1}{48} \, \log \left ({\left | \sqrt {x^{3} + 1} + \frac {1}{\sqrt {x^{3} + 1}} - 2 \right |}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 189, normalized size of antiderivative = 4.97 \[ \int \frac {\sqrt {1+x^3}}{x^7} \, dx=-\frac {\sqrt {x^3+1}}{12\,x^3}-\frac {\sqrt {x^3+1}}{6\,x^6}+\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{4\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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