Optimal. Leaf size=22 \[ \frac {1}{4 \left (x^4+1\right )}+\frac {1}{4} \log \left (x^4+1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {28, 266, 43} \[ \frac {1}{4 \left (x^4+1\right )}+\frac {1}{4} \log \left (x^4+1\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^7}{1+2 x^4+x^8} \, dx &=\int \frac {x^7}{\left (1+x^4\right )^2} \, dx\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{(1+x)^2} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (-\frac {1}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx,x,x^4\right )\\ &=\frac {1}{4 \left (1+x^4\right )}+\frac {1}{4} \log \left (1+x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.82 \[ \frac {1}{4} \left (\frac {1}{x^4+1}+\log \left (x^4+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 23, normalized size = 1.05 \[ \frac {{\left (x^{4} + 1\right )} \log \left (x^{4} + 1\right ) + 1}{4 \, {\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 18, normalized size = 0.82 \[ \frac {1}{4 \, {\left (x^{4} + 1\right )}} + \frac {1}{4} \, \log \left (x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 19, normalized size = 0.86 \[ \frac {\ln \left (x^{4}+1\right )}{4}+\frac {1}{4 x^{4}+4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 18, normalized size = 0.82 \[ \frac {1}{4 \, {\left (x^{4} + 1\right )}} + \frac {1}{4} \, \log \left (x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 18, normalized size = 0.82 \[ \frac {\ln \left (x^4+1\right )}{4}+\frac {1}{4\,\left (x^4+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 15, normalized size = 0.68 \[ \frac {\log {\left (x^{4} + 1 \right )}}{4} + \frac {1}{4 x^{4} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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