Integrand size = 13, antiderivative size = 45 \[ \int \frac {\sqrt [4]{1+x^6}}{x^7} \, dx=-\frac {\sqrt [4]{1+x^6}}{6 x^6}-\frac {1}{12} \arctan \left (\sqrt [4]{1+x^6}\right )-\frac {1}{12} \text {arctanh}\left (\sqrt [4]{1+x^6}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 43, 65, 218, 212, 209} \[ \int \frac {\sqrt [4]{1+x^6}}{x^7} \, dx=-\frac {1}{12} \arctan \left (\sqrt [4]{x^6+1}\right )-\frac {1}{12} \text {arctanh}\left (\sqrt [4]{x^6+1}\right )-\frac {\sqrt [4]{x^6+1}}{6 x^6} \]
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Rule 43
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {\sqrt [4]{1+x}}{x^2} \, dx,x,x^6\right ) \\ & = -\frac {\sqrt [4]{1+x^6}}{6 x^6}+\frac {1}{24} \text {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^6\right ) \\ & = -\frac {\sqrt [4]{1+x^6}}{6 x^6}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^6}\right ) \\ & = -\frac {\sqrt [4]{1+x^6}}{6 x^6}-\frac {1}{12} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^6}\right )-\frac {1}{12} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^6}\right ) \\ & = -\frac {\sqrt [4]{1+x^6}}{6 x^6}-\frac {1}{12} \arctan \left (\sqrt [4]{1+x^6}\right )-\frac {1}{12} \text {arctanh}\left (\sqrt [4]{1+x^6}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{1+x^6}}{x^7} \, dx=-\frac {\sqrt [4]{1+x^6}}{6 x^6}-\frac {1}{12} \arctan \left (\sqrt [4]{1+x^6}\right )-\frac {1}{12} \text {arctanh}\left (\sqrt [4]{1+x^6}\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 4.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16
method | result | size |
meijerg | \(-\frac {\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{4}\right ], \left [2, 3\right ], -x^{6}\right )}{8}-\left (-3 \ln \left (2\right )+\frac {\pi }{2}-1+6 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )+\frac {4 \Gamma \left (\frac {3}{4}\right )}{x^{6}}}{24 \Gamma \left (\frac {3}{4}\right )}\) | \(52\) |
risch | \(-\frac {\left (x^{6}+1\right )^{\frac {1}{4}}}{6 x^{6}}+\frac {-\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], -x^{6}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+6 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )}{24 \Gamma \left (\frac {3}{4}\right )}\) | \(56\) |
pseudoelliptic | \(\frac {\ln \left (\left (x^{6}+1\right )^{\frac {1}{4}}-1\right ) x^{6}-2 \arctan \left (\left (x^{6}+1\right )^{\frac {1}{4}}\right ) x^{6}-\ln \left (\left (x^{6}+1\right )^{\frac {1}{4}}+1\right ) x^{6}-4 \left (x^{6}+1\right )^{\frac {1}{4}}}{24 x^{6}}\) | \(58\) |
trager | \(-\frac {\left (x^{6}+1\right )^{\frac {1}{4}}}{6 x^{6}}-\frac {\ln \left (\frac {x^{6}+2 \left (x^{6}+1\right )^{\frac {3}{4}}+2 \sqrt {x^{6}+1}+2 \left (x^{6}+1\right )^{\frac {1}{4}}+2}{x^{6}}\right )}{24}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \left (x^{6}+1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+2 \left (x^{6}+1\right )^{\frac {1}{4}}}{x^{6}}\right )}{24}\) | \(120\) |
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Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [4]{1+x^6}}{x^7} \, dx=-\frac {2 \, x^{6} \arctan \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}}\right ) + x^{6} \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} + 1\right ) - x^{6} \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} - 1\right ) + 4 \, {\left (x^{6} + 1\right )}^{\frac {1}{4}}}{24 \, x^{6}} \]
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Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt [4]{1+x^6}}{x^7} \, dx=- \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{6}}} \right )}}{6 x^{\frac {9}{2}} \Gamma \left (\frac {7}{4}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [4]{1+x^6}}{x^7} \, dx=-\frac {{\left (x^{6} + 1\right )}^{\frac {1}{4}}}{6 \, x^{6}} - \frac {1}{12} \, \arctan \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{24} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{24} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [4]{1+x^6}}{x^7} \, dx=-\frac {{\left (x^{6} + 1\right )}^{\frac {1}{4}}}{6 \, x^{6}} - \frac {1}{12} \, \arctan \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{24} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{24} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 5.54 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt [4]{1+x^6}}{x^7} \, dx=-\frac {\mathrm {atan}\left ({\left (x^6+1\right )}^{1/4}\right )}{12}-\frac {\mathrm {atanh}\left ({\left (x^6+1\right )}^{1/4}\right )}{12}-\frac {{\left (x^6+1\right )}^{1/4}}{6\,x^6} \]
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