Optimal. Leaf size=148 \[ \frac{x^2}{2}-\frac{\log \left (x^2-\sqrt{2} \sqrt [4]{7} x+\sqrt{7}\right )}{4 \sqrt{2} 7^{3/4}}+\frac{\log \left (x^2+\sqrt{2} \sqrt [4]{7} x+\sqrt{7}\right )}{4 \sqrt{2} 7^{3/4}}-\frac{1}{2} \tanh ^{-1}\left (x^2\right )-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{7}}\right )}{2 \sqrt{2} 7^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{7}}+1\right )}{2 \sqrt{2} 7^{3/4}} \]
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Rubi [A] time = 0.135981, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1790, 1403, 211, 1165, 628, 1162, 617, 204, 1584, 1478, 275, 321, 207} \[ \frac{x^2}{2}-\frac{\log \left (x^2-\sqrt{2} \sqrt [4]{7} x+\sqrt{7}\right )}{4 \sqrt{2} 7^{3/4}}+\frac{\log \left (x^2+\sqrt{2} \sqrt [4]{7} x+\sqrt{7}\right )}{4 \sqrt{2} 7^{3/4}}-\frac{1}{2} \tanh ^{-1}\left (x^2\right )-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{7}}\right )}{2 \sqrt{2} 7^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{7}}+1\right )}{2 \sqrt{2} 7^{3/4}} \]
Antiderivative was successfully verified.
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Rule 1790
Rule 1403
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 1584
Rule 1478
Rule 275
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \frac{-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx &=\int \left (\frac{-1+x^4}{-7+6 x^4+x^8}+\frac{x \left (7 x^4+x^8\right )}{-7+6 x^4+x^8}\right ) \, dx\\ &=\int \frac{-1+x^4}{-7+6 x^4+x^8} \, dx+\int \frac{x \left (7 x^4+x^8\right )}{-7+6 x^4+x^8} \, dx\\ &=\int \frac{1}{7+x^4} \, dx+\int \frac{x^5 \left (7+x^4\right )}{-7+6 x^4+x^8} \, dx\\ &=\frac{\int \frac{\sqrt{7}-x^2}{7+x^4} \, dx}{2 \sqrt{7}}+\frac{\int \frac{\sqrt{7}+x^2}{7+x^4} \, dx}{2 \sqrt{7}}+\int \frac{x^5}{-1+x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,x^2\right )-\frac{\int \frac{\sqrt{2} \sqrt [4]{7}+2 x}{-\sqrt{7}-\sqrt{2} \sqrt [4]{7} x-x^2} \, dx}{4 \sqrt{2} 7^{3/4}}-\frac{\int \frac{\sqrt{2} \sqrt [4]{7}-2 x}{-\sqrt{7}+\sqrt{2} \sqrt [4]{7} x-x^2} \, dx}{4 \sqrt{2} 7^{3/4}}+\frac{\int \frac{1}{\sqrt{7}-\sqrt{2} \sqrt [4]{7} x+x^2} \, dx}{4 \sqrt{7}}+\frac{\int \frac{1}{\sqrt{7}+\sqrt{2} \sqrt [4]{7} x+x^2} \, dx}{4 \sqrt{7}}\\ &=\frac{x^2}{2}-\frac{\log \left (\sqrt{7}-\sqrt{2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt{2} 7^{3/4}}+\frac{\log \left (\sqrt{7}+\sqrt{2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt{2} 7^{3/4}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,x^2\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} x}{\sqrt [4]{7}}\right )}{2 \sqrt{2} 7^{3/4}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} x}{\sqrt [4]{7}}\right )}{2 \sqrt{2} 7^{3/4}}\\ &=\frac{x^2}{2}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{7}}\right )}{2 \sqrt{2} 7^{3/4}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} x}{\sqrt [4]{7}}\right )}{2 \sqrt{2} 7^{3/4}}-\frac{1}{2} \tanh ^{-1}\left (x^2\right )-\frac{\log \left (\sqrt{7}-\sqrt{2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt{2} 7^{3/4}}+\frac{\log \left (\sqrt{7}+\sqrt{2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt{2} 7^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0685093, size = 159, normalized size = 1.07 \[ \frac{1}{56} \left (28 x^2-14 \log \left (x^2+1\right )-\sqrt{2} \sqrt [4]{7} \log \left (\sqrt{7} x^2-\sqrt{2} 7^{3/4} x+7\right )+\sqrt{2} \sqrt [4]{7} \log \left (\sqrt{7} x^2+\sqrt{2} 7^{3/4} x+7\right )+14 \log (1-x)+14 \log (x+1)-2 \sqrt{2} \sqrt [4]{7} \tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{7}}\right )+2 \sqrt{2} \sqrt [4]{7} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{7}}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 110, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}}{2}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{4}}+{\frac{\ln \left ( x-1 \right ) }{4}}+{\frac{\sqrt [4]{7}\sqrt{2}}{28}\arctan \left ( -1+{\frac{x\sqrt{2}{7}^{{\frac{3}{4}}}}{7}} \right ) }+{\frac{\sqrt [4]{7}\sqrt{2}}{56}\ln \left ({\frac{{x}^{2}+\sqrt [4]{7}x\sqrt{2}+\sqrt{7}}{{x}^{2}-\sqrt [4]{7}x\sqrt{2}+\sqrt{7}}} \right ) }+{\frac{\sqrt [4]{7}\sqrt{2}}{28}\arctan \left ( 1+{\frac{x\sqrt{2}{7}^{{\frac{3}{4}}}}{7}} \right ) }+{\frac{\ln \left ( 1+x \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68216, size = 178, normalized size = 1.2 \begin{align*} \frac{1}{2} \, x^{2} + \frac{1}{28} \cdot 7^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{1}{14} \cdot 7^{\frac{3}{4}} \sqrt{2}{\left (2 \, x + 7^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{28} \cdot 7^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{1}{14} \cdot 7^{\frac{3}{4}} \sqrt{2}{\left (2 \, x - 7^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{56} \cdot 7^{\frac{1}{4}} \sqrt{2} \log \left (x^{2} + 7^{\frac{1}{4}} \sqrt{2} x + \sqrt{7}\right ) - \frac{1}{56} \cdot 7^{\frac{1}{4}} \sqrt{2} \log \left (x^{2} - 7^{\frac{1}{4}} \sqrt{2} x + \sqrt{7}\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58688, size = 635, normalized size = 4.29 \begin{align*} -\frac{1}{686} \cdot 343^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{7} \cdot 343^{\frac{1}{4}} \sqrt{2} x + \frac{1}{49} \cdot 343^{\frac{1}{4}} \sqrt{2} \sqrt{343^{\frac{3}{4}} \sqrt{2} x + 49 \, x^{2} + 49 \, \sqrt{7}} - 1\right ) - \frac{1}{686} \cdot 343^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{7} \cdot 343^{\frac{1}{4}} \sqrt{2} x + \frac{1}{49} \cdot 343^{\frac{1}{4}} \sqrt{2} \sqrt{-343^{\frac{3}{4}} \sqrt{2} x + 49 \, x^{2} + 49 \, \sqrt{7}} + 1\right ) + \frac{1}{2744} \cdot 343^{\frac{3}{4}} \sqrt{2} \log \left (343^{\frac{3}{4}} \sqrt{2} x + 49 \, x^{2} + 49 \, \sqrt{7}\right ) - \frac{1}{2744} \cdot 343^{\frac{3}{4}} \sqrt{2} \log \left (-343^{\frac{3}{4}} \sqrt{2} x + 49 \, x^{2} + 49 \, \sqrt{7}\right ) + \frac{1}{2} \, x^{2} - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{4} \, \log \left (x^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.420059, size = 146, normalized size = 0.99 \begin{align*} \frac{x^{2}}{2} + \frac{\log{\left (x^{2} - 1 \right )}}{4} - \frac{\log{\left (x^{2} + 1 \right )}}{4} - \frac{\sqrt{2} \sqrt [4]{7} \log{\left (x^{2} - \sqrt{2} \sqrt [4]{7} x + \sqrt{7} \right )}}{56} + \frac{\sqrt{2} \sqrt [4]{7} \log{\left (x^{2} + \sqrt{2} \sqrt [4]{7} x + \sqrt{7} \right )}}{56} + \frac{\sqrt{2} \sqrt [4]{7} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 7^{\frac{3}{4}} x}{7} - 1 \right )}}{28} + \frac{\sqrt{2} \sqrt [4]{7} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 7^{\frac{3}{4}} x}{7} + 1 \right )}}{28} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33989, size = 165, normalized size = 1.11 \begin{align*} \frac{1}{2} \, x^{2} + \frac{1}{28} \cdot 28^{\frac{1}{4}} \arctan \left (\frac{1}{14} \cdot 7^{\frac{3}{4}} \sqrt{2}{\left (2 \, x + 7^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{28} \cdot 28^{\frac{1}{4}} \arctan \left (\frac{1}{14} \cdot 7^{\frac{3}{4}} \sqrt{2}{\left (2 \, x - 7^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{56} \cdot 28^{\frac{1}{4}} \log \left (x^{2} + 7^{\frac{1}{4}} \sqrt{2} x + \sqrt{7}\right ) - \frac{1}{56} \cdot 28^{\frac{1}{4}} \log \left (x^{2} - 7^{\frac{1}{4}} \sqrt{2} x + \sqrt{7}\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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