Optimal. Leaf size=26 \[ \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(1-x)^4}\right )^{\frac {1}{x}}}{1+x} \]
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Rubi [F] time = 4.70, 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 {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} \left (x \left (4+3 x+x^2\right )+\left (-1+x^2\right ) \log \left (\frac {4}{(-1+x)^4}\right )\right )}{(1-x) x^2 (1+x)^2} \, dx\\ &=\int \left (\frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} \left (-4-3 x-x^2\right )}{(-1+x) x (1+x)^2}-\frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} \log \left (\frac {4}{(-1+x)^4}\right )}{x^2 (1+x)}\right ) \, dx\\ &=\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} \left (-4-3 x-x^2\right )}{(-1+x) x (1+x)^2} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} \log \left (\frac {4}{(-1+x)^4}\right )}{x^2 (1+x)} \, dx\\ &=-\left (\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx\right )+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx+\int \left (\frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x}-\frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2}-\frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} x}{-1+x^2}\right ) \, dx+\int -\frac {4 \left (\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx+\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx\right )}{-1+x} \, dx\\ &=-\left (4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx+\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx}{-1+x} \, dx\right )-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx+\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2} \, dx-\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} x}{-1+x^2} \, dx\\ &=-\left (4 \int \left (\frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx}{-1+x}+\frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx}{-1+x}\right ) \, dx\right )-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx+\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2} \, dx-\int \left (\frac {2^{2/x} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{-1+x}+\frac {2^{2/x} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x}\right ) \, dx\\ &=-\left (4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx}{-1+x} \, dx\right )-4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx}{-1+x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx-\int \frac {2^{2/x} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{-1+x} \, dx+\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2} \, dx-\int \frac {2^{2/x} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx\\ &=-\left (4 \int \left (\frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx}{-1+x}-\frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx}{-1+x}\right ) \, dx\right )-4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx}{-1+x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx-\int \frac {4^{\frac {1}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{-1+x} \, dx+\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2} \, dx-\int \frac {4^{\frac {1}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx\\ &=-\left (4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx}{-1+x} \, dx\right )+4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx}{-1+x} \, dx-4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx}{-1+x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx-\int \frac {4^{\frac {1}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{-1+x} \, dx+\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2} \, dx-\int \frac {4^{\frac {1}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.69, size = 24, normalized size = 0.92 \begin {gather*} \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 33, normalized size = 1.27 \begin {gather*} \frac {\left (\frac {4}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )^{\left (\frac {1}{x}\right )}}{2 \, {\left (x + 1\right )}} \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 {{\left (x^{3} + 3 \, x^{2} + {\left (x^{2} - 1\right )} \log \left (\frac {4}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) + 4 \, x\right )} \left (\frac {4}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )^{\left (\frac {1}{x}\right )}}{2 \, {\left (x^{5} + x^{4} - x^{3} - x^{2}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 34, normalized size = 1.31
method | result | size |
risch | \(\frac {\left (\frac {4}{x^{4}-4 x^{3}+6 x^{2}-4 x +1}\right )^{\frac {1}{x}}}{2 x +2}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 25, normalized size = 0.96 \begin {gather*} \frac {e^{\left (\frac {2 \, \log \relax (2)}{x} - \frac {4 \, \log \left (x - 1\right )}{x}\right )}}{2 \, {\left (x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.86, size = 34, normalized size = 1.31 \begin {gather*} \frac {{\left (\frac {4}{x^4-4\,x^3+6\,x^2-4\,x+1}\right )}^{1/x}}{2\,\left (x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 29, normalized size = 1.12 \begin {gather*} \frac {e^{\frac {\log {\left (\frac {4}{x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1} \right )}}{x}}}{2 x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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