Optimal. Leaf size=25 \[ -e^{2 x+2 x^2 \left (x+\log \left (4+x^2\right )\right )}+4 x \]
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Rubi [B] time = 0.75, antiderivative size = 51, normalized size of antiderivative = 2.04, number of steps used = 3, number of rules used = 2, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6725, 2288} \begin {gather*} 4 x-\frac {e^{2 x^3+2 x} \left (x^2+4\right )^{2 x^2-1} \left (3 x^4+13 x^2+4\right )}{3 x^2+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 2288
Rule 6725
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4-2 e^{2 x+2 x^3} \left (4+x^2\right )^{-1+2 x^2} \left (4+13 x^2+2 x^3+3 x^4+8 x \log \left (4+x^2\right )+2 x^3 \log \left (4+x^2\right )\right )\right ) \, dx\\ &=4 x-2 \int e^{2 x+2 x^3} \left (4+x^2\right )^{-1+2 x^2} \left (4+13 x^2+2 x^3+3 x^4+8 x \log \left (4+x^2\right )+2 x^3 \log \left (4+x^2\right )\right ) \, dx\\ &=4 x-\frac {e^{2 x+2 x^3} \left (4+x^2\right )^{-1+2 x^2} \left (4+13 x^2+3 x^4\right )}{1+3 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.41, size = 26, normalized size = 1.04 \begin {gather*} 4 x-e^{2 \left (x+x^3\right )} \left (4+x^2\right )^{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 27, normalized size = 1.08 \begin {gather*} 4 \, x - e^{\left (2 \, x^{3} + 2 \, x^{2} \log \left (x^{2} + 4\right ) + 2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 27, normalized size = 1.08 \begin {gather*} 4 \, x - e^{\left (2 \, x^{3} + 2 \, x^{2} \log \left (x^{2} + 4\right ) + 2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 27, normalized size = 1.08
method | result | size |
risch | \(-\left (x^{2}+4\right )^{2 x^{2}} {\mathrm e}^{2 x \left (x^{2}+1\right )}+4 x\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 27, normalized size = 1.08 \begin {gather*} 4 \, x - e^{\left (2 \, x^{3} + 2 \, x^{2} \log \left (x^{2} + 4\right ) + 2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.41, size = 27, normalized size = 1.08 \begin {gather*} 4\,x-{\mathrm {e}}^{2\,x^3+2\,x}\,{\left (x^2+4\right )}^{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 24, normalized size = 0.96 \begin {gather*} 4 x - e^{2 x^{3} + 2 x^{2} \log {\left (x^{2} + 4 \right )} + 2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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