Integrand size = 13, antiderivative size = 70 \[ \int \frac {\left (1+x^7\right )^{2/3}}{x^8} \, dx=-\frac {\left (1+x^7\right )^{2/3}}{7 x^7}+\frac {2 \arctan \left (\frac {1+2 \sqrt [3]{1+x^7}}{\sqrt {3}}\right )}{7 \sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{7} \log \left (1-\sqrt [3]{1+x^7}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 70, 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, 57, 632, 210, 31} \[ \int \frac {\left (1+x^7\right )^{2/3}}{x^8} \, dx=\frac {2 \arctan \left (\frac {2 \sqrt [3]{x^7+1}+1}{\sqrt {3}}\right )}{7 \sqrt {3}}-\frac {\left (x^7+1\right )^{2/3}}{7 x^7}+\frac {1}{7} \log \left (1-\sqrt [3]{x^7+1}\right )-\frac {\log (x)}{3} \]
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Rule 31
Rule 43
Rule 57
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} \text {Subst}\left (\int \frac {(1+x)^{2/3}}{x^2} \, dx,x,x^7\right ) \\ & = -\frac {\left (1+x^7\right )^{2/3}}{7 x^7}+\frac {2}{21} \text {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^7\right ) \\ & = -\frac {\left (1+x^7\right )^{2/3}}{7 x^7}-\frac {\log (x)}{3}-\frac {1}{7} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^7}\right )+\frac {1}{7} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^7}\right ) \\ & = -\frac {\left (1+x^7\right )^{2/3}}{7 x^7}-\frac {\log (x)}{3}+\frac {1}{7} \log \left (1-\sqrt [3]{1+x^7}\right )-\frac {2}{7} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^7}\right ) \\ & = -\frac {\left (1+x^7\right )^{2/3}}{7 x^7}+\frac {2 \arctan \left (\frac {1+2 \sqrt [3]{1+x^7}}{\sqrt {3}}\right )}{7 \sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{7} \log \left (1-\sqrt [3]{1+x^7}\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.19 \[ \int \frac {\left (1+x^7\right )^{2/3}}{x^8} \, dx=\frac {1}{21} \left (-\frac {3 \left (1+x^7\right )^{2/3}}{x^7}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^7}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+x^7}\right )-\log \left (1+\sqrt [3]{1+x^7}+\left (1+x^7\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 6.64 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.09
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{7}}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+7 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}\, x^{7} {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (1,1,\frac {4}{3};2,3;-x^{7}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\right )}{21 \pi }\) | \(76\) |
risch | \(-\frac {\left (x^{7}+1\right )^{\frac {2}{3}}}{7 x^{7}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+7 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \pi \sqrt {3}\, x^{7} {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (1,1,\frac {4}{3};2,2;-x^{7}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\right )}{21 \pi }\) | \(76\) |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\left (1+2 \left (x^{7}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, x^{7}-\ln \left (\left (x^{7}+1\right )^{\frac {2}{3}}+\left (x^{7}+1\right )^{\frac {1}{3}}+1\right ) x^{7}+2 \ln \left (\left (x^{7}+1\right )^{\frac {1}{3}}-1\right ) x^{7}-3 \left (x^{7}+1\right )^{\frac {2}{3}}}{21 \left (\left (x^{7}+1\right )^{\frac {2}{3}}+\left (x^{7}+1\right )^{\frac {1}{3}}+1\right ) \left (\left (x^{7}+1\right )^{\frac {1}{3}}-1\right )}\) | \(104\) |
trager | \(-\frac {\left (x^{7}+1\right )^{\frac {2}{3}}}{7 x^{7}}-\frac {2 \ln \left (\frac {3593313 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{7}+3486414 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{7}-106899 x^{7}+6095754 \left (x^{7}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3593313 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6095754 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+1\right )^{\frac {1}{3}}+256725 \left (x^{7}+1\right )^{\frac {2}{3}}+4897983 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+256725 \left (x^{7}+1\right )^{\frac {1}{3}}-142532}{x^{7}}\right )}{21}-\frac {2 \ln \left (\frac {3593313 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{7}+3486414 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{7}-106899 x^{7}+6095754 \left (x^{7}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3593313 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6095754 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+1\right )^{\frac {1}{3}}+256725 \left (x^{7}+1\right )^{\frac {2}{3}}+4897983 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+256725 \left (x^{7}+1\right )^{\frac {1}{3}}-142532}{x^{7}}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{7}+\frac {2 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {3593313 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{7}-1090872 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{7}-869780 x^{7}-6095754 \left (x^{7}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3593313 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-6095754 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+1\right )^{\frac {1}{3}}-1775193 \left (x^{7}+1\right )^{\frac {2}{3}}-7293525 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1775193 \left (x^{7}+1\right )^{\frac {1}{3}}-2174450}{x^{7}}\right )}{7}\) | \(438\) |
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Time = 0.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.13 \[ \int \frac {\left (1+x^7\right )^{2/3}}{x^8} \, dx=\frac {2 \, \sqrt {3} x^{7} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{7} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - x^{7} \log \left ({\left (x^{7} + 1\right )}^{\frac {2}{3}} + {\left (x^{7} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, x^{7} \log \left ({\left (x^{7} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{7} + 1\right )}^{\frac {2}{3}}}{21 \, x^{7}} \]
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Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^7\right )^{2/3}}{x^8} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{7}}} \right )}}{7 x^{\frac {7}{3}} \Gamma \left (\frac {4}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {\left (1+x^7\right )^{2/3}}{x^8} \, dx=\frac {2}{21} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{7} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{7} + 1\right )}^{\frac {2}{3}}}{7 \, x^{7}} - \frac {1}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac {2}{3}} + {\left (x^{7} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1+x^7\right )^{2/3}}{x^8} \, dx=\frac {2}{21} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{7} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{7} + 1\right )}^{\frac {2}{3}}}{7 \, x^{7}} - \frac {1}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac {2}{3}} + {\left (x^{7} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{21} \, \log \left ({\left | {\left (x^{7} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 0.39 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.31 \[ \int \frac {\left (1+x^7\right )^{2/3}}{x^8} \, dx=\frac {2\,\ln \left (\frac {4\,{\left (x^7+1\right )}^{1/3}}{49}-\frac {4}{49}\right )}{21}+\ln \left (\frac {4\,{\left (x^7+1\right )}^{1/3}}{49}-9\,{\left (-\frac {1}{21}+\frac {\sqrt {3}\,1{}\mathrm {i}}{21}\right )}^2\right )\,\left (-\frac {1}{21}+\frac {\sqrt {3}\,1{}\mathrm {i}}{21}\right )-\ln \left (\frac {4\,{\left (x^7+1\right )}^{1/3}}{49}-9\,{\left (\frac {1}{21}+\frac {\sqrt {3}\,1{}\mathrm {i}}{21}\right )}^2\right )\,\left (\frac {1}{21}+\frac {\sqrt {3}\,1{}\mathrm {i}}{21}\right )-\frac {{\left (x^7+1\right )}^{2/3}}{7\,x^7} \]
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