Optimal. Leaf size=70 \[ -\frac {\left (x^7+1\right )^{2/3}}{7 x^7}+\frac {1}{7} \log \left (1-\sqrt [3]{x^7+1}\right )+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^7+1}+1}{\sqrt {3}}\right )}{7 \sqrt {3}}-\frac {\log (x)}{3} \]
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Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 47, 55, 618, 204, 31} \[ -\frac {\left (x^7+1\right )^{2/3}}{7 x^7}+\frac {1}{7} \log \left (1-\sqrt [3]{x^7+1}\right )+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^7+1}+1}{\sqrt {3}}\right )}{7 \sqrt {3}}-\frac {\log (x)}{3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 55
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\left (1+x^7\right )^{2/3}}{x^8} \, dx &=\frac {1}{7} \operatorname {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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^7}\right )+\frac {1}{7} \operatorname {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} \operatorname {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 \tan ^{-1}\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*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.37 \[ \frac {3}{35} \left (x^7+1\right )^{5/3} \, _2F_1\left (\frac {5}{3},2;\frac {8}{3};x^7+1\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 90, normalized size = 1.29 \[ -\frac {\left (x^7+1\right )^{2/3}}{7 x^7}+\frac {2}{21} \log \left (\sqrt [3]{x^7+1}-1\right )-\frac {1}{21} \log \left (\left (x^7+1\right )^{2/3}+\sqrt [3]{x^7+1}+1\right )+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^7+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{7 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 79, normalized size = 1.13 \[ \frac {2 \, \sqrt {3} x^{7} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{7} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - x^{7} \log \left ({\left (x^{7} + 1\right )}^{\frac {2}{3}} + {\left (x^{7} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, x^{7} \log \left ({\left (x^{7} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{7} + 1\right )}^{\frac {2}{3}}}{21 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 67, normalized size = 0.96 \[ \frac {2}{21} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{7} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{7} + 1\right )}^{\frac {2}{3}}}{7 \, x^{7}} - \frac {1}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac {2}{3}} + {\left (x^{7} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{21} \, \log \left ({\left | {\left (x^{7} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 9.89, size = 76, normalized size = 1.09
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {\pi \sqrt {3}\, x^{7} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], -x^{7}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+7 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{7}}\right )}{21 \pi }\) | \(76\) |
risch | \(-\frac {\left (x^{7}+1\right )^{\frac {2}{3}}}{7 x^{7}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{7} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{7}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+7 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{21 \pi }\) | \(76\) |
trager | \(-\frac {\left (x^{7}+1\right )^{\frac {2}{3}}}{7 x^{7}}+\frac {2 \ln \left (-\frac {3914010 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{7}-2502441 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{7}-71266 x^{7}+6095754 \left (x^{7}+1\right )^{\frac {2}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3914010 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-770175 \left (x^{7}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1775193 \left (x^{7}+1\right )^{\frac {2}{3}}-6630249 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2031918 \left (x^{7}+1\right )^{\frac {1}{3}}-178165}{x^{7}}\right )}{21}+\frac {2 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {320697 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{7}+1625367 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{7}+1304670 x^{7}+6095754 \left (x^{7}+1\right )^{\frac {2}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-320697 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-5325579 \left (x^{7}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+256725 \left (x^{7}+1\right )^{\frac {2}{3}}-877074 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2031918 \left (x^{7}+1\right )^{\frac {1}{3}}+1739560}{x^{7}}\right )}{7}\) | \(294\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 66, normalized size = 0.94 \[ \frac {2}{21} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{7} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{7} + 1\right )}^{\frac {2}{3}}}{7 \, x^{7}} - \frac {1}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac {2}{3}} + {\left (x^{7} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 92, normalized size = 1.31 \[ \frac {2\,\ln \left (\frac {4\,{\left (x^7+1\right )}^{1/3}}{49}-\frac {4}{49}\right )}{21}+\ln \left (\frac {4\,{\left (x^7+1\right )}^{1/3}}{49}-9\,{\left (-\frac {1}{21}+\frac {\sqrt {3}\,1{}\mathrm {i}}{21}\right )}^2\right )\,\left (-\frac {1}{21}+\frac {\sqrt {3}\,1{}\mathrm {i}}{21}\right )-\ln \left (\frac {4\,{\left (x^7+1\right )}^{1/3}}{49}-9\,{\left (\frac {1}{21}+\frac {\sqrt {3}\,1{}\mathrm {i}}{21}\right )}^2\right )\,\left (\frac {1}{21}+\frac {\sqrt {3}\,1{}\mathrm {i}}{21}\right )-\frac {{\left (x^7+1\right )}^{2/3}}{7\,x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.42, size = 34, normalized size = 0.49 \[ - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{7}}} \right )}}{7 x^{\frac {7}{3}} \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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