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 = 0.14, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1790, 1403, 211, 1165, 628, 1162, 617, 204, 1584, 1478, 275, 321, 207} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 207
Rule 211
Rule 275
Rule 321
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1403
Rule 1478
Rule 1584
Rule 1790
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*}
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Mathematica [A] time = 0.07, size = 159, normalized size = 1.07 \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.29, size = 178, normalized size = 1.20 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 122, normalized size = 0.82 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 110, normalized size = 0.74 \begin {gather*} \frac {x^{2}}{2}+\frac {7^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, 7^{\frac {3}{4}} x}{7}-1\right )}{28}+\frac {7^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, 7^{\frac {3}{4}} x}{7}+1\right )}{28}+\frac {7^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+7^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {7}}{x^{2}-7^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {7}}\right )}{56}+\frac {\ln \left (x -1\right )}{4}+\frac {\ln \left (x +1\right )}{4}-\frac {\ln \left (x^{2}+1\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.01, size = 132, normalized size = 0.89 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.19, size = 124, normalized size = 0.84 \begin {gather*} \frac {\mathrm {atan}\left (x^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {x^2}{2}+\sqrt {2}\,7^{1/4}\,\mathrm {atan}\left (\frac {\sqrt {2}\,7^{1/4}\,x\,\left (\frac {89653248}{2401}+\frac {89653248}{2401}{}\mathrm {i}\right )}{-\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}+\frac {\sqrt {2}\,7^{3/4}\,x\,\left (-\frac {524288}{343}+\frac {524288}{343}{}\mathrm {i}\right )}{-\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}\right )\,\left (\frac {1}{28}+\frac {1}{28}{}\mathrm {i}\right )+\sqrt {2}\,7^{1/4}\,\mathrm {atan}\left (\frac {\sqrt {2}\,7^{1/4}\,x\,\left (\frac {89653248}{2401}-\frac {89653248}{2401}{}\mathrm {i}\right )}{\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}+\frac {\sqrt {2}\,7^{3/4}\,x\,\left (-\frac {524288}{343}-\frac {524288}{343}{}\mathrm {i}\right )}{\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}\right )\,\left (-\frac {1}{28}+\frac {1}{28}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 146, normalized size = 0.99 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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