Optimal. Leaf size=140 \[ \frac {\sqrt {x^8+x^6+x^4+x^2+1} \left (8 x^8+26 x^6+65 x^4+26 x^2+8\right )}{48 x^6}-\frac {65}{32} \log \left (-2 x^4-x^2+2 \sqrt {x^8+x^6+x^4+x^2+1}-2\right )+2 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x^2}{x^4-2 x^2-\sqrt {x^8+x^6+x^4+x^2+1}+1}\right )+\frac {65 \log (x)}{16} \]
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Rubi [F] time = 1.14, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+x^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx &=\int \left (-\frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^7}-\frac {2 \sqrt {1+x^2+x^4+x^6+x^8}}{x^5}-\frac {2 \sqrt {1+x^2+x^4+x^6+x^8}}{x^3}-\frac {2 \sqrt {1+x^2+x^4+x^6+x^8}}{x}+x \sqrt {1+x^2+x^4+x^6+x^8}+\frac {4 x \sqrt {1+x^2+x^4+x^6+x^8}}{-1+x^2}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^5} \, dx\right )-2 \int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^3} \, dx-2 \int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x} \, dx+4 \int \frac {x \sqrt {1+x^2+x^4+x^6+x^8}}{-1+x^2} \, dx-\int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^7} \, dx+\int x \sqrt {1+x^2+x^4+x^6+x^8} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \sqrt {1+x+x^2+x^3+x^4} \, dx,x,x^2\right )-2 \int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^5} \, dx-2 \int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^3} \, dx+2 \operatorname {Subst}\left (\int \frac {\sqrt {1+x+x^2+x^3+x^4}}{-1+x} \, dx,x,x^2\right )-\int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^7} \, dx-\operatorname {Subst}\left (\int \frac {\sqrt {1+x+x^2+x^3+x^4}}{x} \, dx,x,x^2\right )\\ \end {align*}
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Mathematica [C] time = 3.16, size = 687, normalized size = 4.91 \begin {gather*} \frac {\frac {15 (-1)^{3/5} \sqrt {-\frac {\sqrt [5]{-1} \left (\sqrt [5]{-1}-1\right ) \left ((-1)^{4/5}-x^2\right ) \left (x^2+\sqrt [5]{-1}\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right )^2 \left ((-1)^{2/5}-x^2\right )^2}} \sqrt {\frac {x^2+(-1)^{4/5}+(-1)^{2/5}-\sqrt [5]{-1}+1}{\left (\sqrt [5]{-1}-1\right ) \left ((-1)^{2/5}-x^2\right )}} \left ((-1)^{2/5}-x^2\right )^2 \left (\left (45-32 \sqrt [5]{-1}+38 (-1)^{2/5}-32 (-1)^{3/5}+45 (-1)^{4/5}\right ) F\left (\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (x^2+\sqrt [5]{-1}\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left ((-1)^{2/5}-x^2\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )+13 \left (1-\sqrt [5]{-1}+2 (-1)^{3/5}\right ) \Pi \left (\frac {-1+\sqrt [5]{-1}-(-1)^{2/5}}{-1+\sqrt [5]{-1}};\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (x^2+\sqrt [5]{-1}\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left ((-1)^{2/5}-x^2\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )+\sqrt [5]{-1} \left (13 \left (2-(-1)^{2/5}+(-1)^{3/5}\right ) \Pi \left (\frac {-1+\sqrt [5]{-1}-(-1)^{4/5}}{-1+\sqrt [5]{-1}};\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (x^2+\sqrt [5]{-1}\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left ((-1)^{2/5}-x^2\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )-64 \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \Pi \left (1-\sqrt [5]{-1}+(-1)^{4/5};\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (x^2+\sqrt [5]{-1}\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left ((-1)^{2/5}-x^2\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )\right )\right )}{\left ((-1)^{2/5}-1\right )^2}+\frac {\left (x^8+x^6+x^4+x^2+1\right ) \left (8 x^8+26 x^6+65 x^4+26 x^2+8\right )}{x^6}}{48 \sqrt {x^8+x^6+x^4+x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.50, size = 140, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^8+x^6+x^4+x^2+1} \left (8 x^8+26 x^6+65 x^4+26 x^2+8\right )}{48 x^6}-\frac {65}{32} \log \left (-2 x^4-x^2+2 \sqrt {x^8+x^6+x^4+x^2+1}-2\right )+2 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x^2}{x^4-2 x^2-\sqrt {x^8+x^6+x^4+x^2+1}+1}\right )+\frac {65 \log (x)}{16} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 165, normalized size = 1.18 \begin {gather*} \frac {48 \, \sqrt {5} x^{6} \log \left (-\frac {9 \, x^{8} + 4 \, x^{6} + 14 \, x^{4} - 4 \, \sqrt {5} \sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} {\left (x^{4} + 1\right )} + 4 \, x^{2} + 9}{x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1}\right ) + 195 \, x^{6} \log \left (\frac {2 \, x^{4} + x^{2} + 2 \, \sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} + 2}{x^{2}}\right ) + 2 \, {\left (8 \, x^{8} + 26 \, x^{6} + 65 \, x^{4} + 26 \, x^{2} + 8\right )} \sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1}}{96 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} {\left (x^{8} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{2} - 1\right )} x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.98, size = 172, normalized size = 1.23 \begin {gather*} \frac {65 x^{12}+91 x^{10}+99 x^{8}+99 x^{6}+99 x^{4}+34 x^{2}+8}{48 x^{6} \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}+\frac {\left (8 x^{2}+26\right ) \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{48}-\frac {65 \ln \left (\frac {-2-x^{2}-2 x^{4}+2 \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{x^{2}}\right )}{32}-\RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-5\right ) x^{4}+2 \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}+\RootOf \left (\textit {\_Z}^{2}-5\right )}{\left (1+x \right )^{2} \left (-1+x \right )^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} {\left (x^{8} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{2} - 1\right )} x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2+1\right )\,\left (x^8+1\right )\,\sqrt {x^8+x^6+x^4+x^2+1}}{x^7\,\left (x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x^{4} - x^{3} + x^{2} - x + 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (x^{2} + 1\right ) \left (x^{8} + 1\right )}{x^{7} \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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