Optimal. Leaf size=23 \[ 2 x-3 e \left (1-\frac {4 x}{4+x^2}\right )-\log (x) \]
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Rubi [A]
time = 0.05, antiderivative size = 19, normalized size of antiderivative = 0.83, number of steps
used = 6, number of rules used = 5, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {1608, 28, 1819,
1600, 45} \begin {gather*} \frac {12 e x}{x^2+4}+2 x-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 45
Rule 1600
Rule 1608
Rule 1819
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-16+32 x-8 x^2+16 x^3-x^4+2 x^5+e \left (48 x-12 x^3\right )}{x \left (16+8 x^2+x^4\right )} \, dx\\ &=\int \frac {-16+32 x-8 x^2+16 x^3-x^4+2 x^5+e \left (48 x-12 x^3\right )}{x \left (4+x^2\right )^2} \, dx\\ &=\frac {12 e x}{4+x^2}-\frac {1}{8} \int \frac {32-64 x+8 x^2-16 x^3}{x \left (4+x^2\right )} \, dx\\ &=\frac {12 e x}{4+x^2}-\frac {1}{8} \int \frac {8-16 x}{x} \, dx\\ &=\frac {12 e x}{4+x^2}-\frac {1}{8} \int \left (-16+\frac {8}{x}\right ) \, dx\\ &=2 x+\frac {12 e x}{4+x^2}-\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 19, normalized size = 0.83 \begin {gather*} 2 x+\frac {12 e x}{4+x^2}-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 21, normalized size = 0.91
method | result | size |
default | \(2 x +\frac {12 \,{\mathrm e} x}{x^{2}+4}-\ln \left (x \right )\) | \(21\) |
risch | \(2 x +\frac {12 \,{\mathrm e} x}{x^{2}+4}-\ln \left (x \right )\) | \(21\) |
norman | \(\frac {\left (8+12 \,{\mathrm e}\right ) x +2 x^{3}}{x^{2}+4}-\ln \left (x \right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 20, normalized size = 0.87 \begin {gather*} 2 \, x + \frac {12 \, x e}{x^{2} + 4} - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 31, normalized size = 1.35 \begin {gather*} \frac {2 \, x^{3} + 12 \, x e - {\left (x^{2} + 4\right )} \log \left (x\right ) + 8 \, x}{x^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 17, normalized size = 0.74 \begin {gather*} 2 x + \frac {12 e x}{x^{2} + 4} - \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 21, normalized size = 0.91 \begin {gather*} 2 \, x + \frac {12 \, x e}{x^{2} + 4} - \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 20, normalized size = 0.87 \begin {gather*} 2\,x-\ln \left (x\right )+\frac {12\,x\,\mathrm {e}}{x^2+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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