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 = 0.0378973, antiderivative size = 70, normalized size of antiderivative = 1., 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 266
Rule 47
Rule 55
Rule 618
Rule 204
Rule 31
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*}
Mathematica [C] time = 0.0057785, 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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Maple [C] time = 0.039, size = 76, normalized size = 1.1 \begin{align*} -{\frac{1}{7\,{x}^{7}} \left ({x}^{7}+1 \right ) ^{{\frac{2}{3}}}}+{\frac{\sqrt{3}\Gamma \left ({\frac{2}{3}} \right ) }{21\,\pi } \left ({\frac{2\,\pi \,\sqrt{3}}{3\,\Gamma \left ( 2/3 \right ) } \left ( -{\frac{\pi \,\sqrt{3}}{6}}-{\frac{3\,\ln \left ( 3 \right ) }{2}}+7\,\ln \left ( x \right ) \right ) }-{\frac{2\,\pi \,\sqrt{3}{x}^{7}}{9\,\Gamma \left ( 2/3 \right ) }{\mbox{$_3$F$_2$}(1,1,{\frac{4}{3}};\,2,2;\,-{x}^{7})}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45476, size = 89, normalized size = 1.27 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81303, size = 240, normalized size = 3.43 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.89561, size = 34, normalized size = 0.49 \begin{align*} - \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09018, size = 90, normalized size = 1.29 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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