Optimal. Leaf size=32 \[ \frac {x^7}{7}-\frac {2 x^6}{3}+x^5-\frac {4 x^3}{3}+4 x-4 \tan ^{-1}(x) \]
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Rubi [A] time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1629, 203} \begin {gather*} \frac {x^7}{7}-\frac {2 x^6}{3}+x^5-\frac {4 x^3}{3}+4 x-4 \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1629
Rubi steps
\begin {align*} \int \frac {(-1+x)^4 x^4}{1+x^2} \, dx &=\int \left (4-4 x^2+5 x^4-4 x^5+x^6-\frac {4}{1+x^2}\right ) \, dx\\ &=4 x-\frac {4 x^3}{3}+x^5-\frac {2 x^6}{3}+\frac {x^7}{7}-4 \int \frac {1}{1+x^2} \, dx\\ &=4 x-\frac {4 x^3}{3}+x^5-\frac {2 x^6}{3}+\frac {x^7}{7}-4 \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 1.00 \begin {gather*} \frac {x^7}{7}-\frac {2 x^6}{3}+x^5-\frac {4 x^3}{3}+4 x-4 \tan ^{-1}(x) \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 x^4}{1+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.61, size = 26, normalized size = 0.81 \begin {gather*} \frac {1}{7} \, x^{7} - \frac {2}{3} \, x^{6} + x^{5} - \frac {4}{3} \, x^{3} + 4 \, x - 4 \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 26, normalized size = 0.81 \begin {gather*} \frac {1}{7} \, x^{7} - \frac {2}{3} \, x^{6} + x^{5} - \frac {4}{3} \, x^{3} + 4 \, x - 4 \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.84 \begin {gather*} \frac {x^{7}}{7}-\frac {2 x^{6}}{3}+x^{5}-\frac {4 x^{3}}{3}+4 x -4 \arctan \relax (x ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 26, normalized size = 0.81 \begin {gather*} \frac {1}{7} \, x^{7} - \frac {2}{3} \, x^{6} + x^{5} - \frac {4}{3} \, x^{3} + 4 \, x - 4 \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 26, normalized size = 0.81 \begin {gather*} 4\,x-4\,\mathrm {atan}\relax (x)-\frac {4\,x^3}{3}+x^5-\frac {2\,x^6}{3}+\frac {x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 29, normalized size = 0.91 \begin {gather*} \frac {x^{7}}{7} - \frac {2 x^{6}}{3} + x^{5} - \frac {4 x^{3}}{3} + 4 x - 4 \operatorname {atan}{\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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