Optimal. Leaf size=31 \[ \frac {\left (\left (-3+16 \left (5-(5-x)^2\right )\right ) x\right )^{2-5^{\frac {1}{x}}}}{x} \]
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Rubi [F]
time = 13.42, 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 {\left (-323 x+160 x^2-16 x^3\right )^{2-5^{\frac {1}{x}}} \left (323 x-480 x^2+80 x^3+5^{\frac {1}{x}} \left (-323 x+320 x^2-48 x^3\right )+5^{\frac {1}{x}} \left (323-160 x+16 x^2\right ) \log (5) \log \left (-323 x+160 x^2-16 x^3\right )\right )}{323 x^3-160 x^4+16 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 {\left (-323 x+160 x^2-16 x^3\right )^{2-5^{\frac {1}{x}}} \left (323 x-480 x^2+80 x^3+5^{\frac {1}{x}} \left (-323 x+320 x^2-48 x^3\right )+5^{\frac {1}{x}} \left (323-160 x+16 x^2\right ) \log (5) \log \left (-323 x+160 x^2-16 x^3\right )\right )}{x^3 \left (323-160 x+16 x^2\right )} \, dx\\ &=\int \left (\frac {\left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}} \left (323-480 x+80 x^2\right )}{x^2 \left (323-160 x+16 x^2\right )}-\frac {5^{\frac {1}{x}} \left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}} \left (323 x-320 x^2+48 x^3-323 \log (5) \log \left (x \left (-323+160 x-16 x^2\right )\right )+160 x \log (5) \log \left (x \left (-323+160 x-16 x^2\right )\right )-16 x^2 \log (5) \log \left (x \left (-323+160 x-16 x^2\right )\right )\right )}{x^3 \left (323-160 x+16 x^2\right )}\right ) \, dx\\ &=\int \frac {\left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}} \left (323-480 x+80 x^2\right )}{x^2 \left (323-160 x+16 x^2\right )} \, dx-\int \frac {5^{\frac {1}{x}} \left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}} \left (323 x-320 x^2+48 x^3-323 \log (5) \log \left (x \left (-323+160 x-16 x^2\right )\right )+160 x \log (5) \log \left (x \left (-323+160 x-16 x^2\right )\right )-16 x^2 \log (5) \log \left (x \left (-323+160 x-16 x^2\right )\right )\right )}{x^3 \left (323-160 x+16 x^2\right )} \, dx\\ &=\int \left (\frac {\left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}}}{x^2}-\frac {320 \left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}}}{323 x}+\frac {64 (-477+80 x) \left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}}}{323 \left (323-160 x+16 x^2\right )}\right ) \, dx-\int \frac {5^{\frac {1}{x}} \left (x \left (-323+160 x-16 x^2\right )\right )^{-5^{\frac {1}{x}}} \left (323-160 x+16 x^2\right ) \left (x \left (323-320 x+48 x^2\right )+\left (-323+160 x-16 x^2\right ) \log (5) \log \left (x \left (-323+160 x-16 x^2\right )\right )\right )}{x} \, dx\\ &=\frac {64}{323} \int \frac {(-477+80 x) \left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}}}{323-160 x+16 x^2} \, dx-\frac {320}{323} \int \frac {\left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}}}{x} \, dx+\int \frac {\left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}}}{x^2} \, dx-\int \left (5^{\frac {1}{x}} \left (x \left (-323+160 x-16 x^2\right )\right )^{-5^{\frac {1}{x}}} \left (323-160 x+16 x^2\right ) \left (323-320 x+48 x^2\right )-\frac {5^{\frac {1}{x}} \left (x \left (-323+160 x-16 x^2\right )\right )^{-5^{\frac {1}{x}}} \left (323-160 x+16 x^2\right )^2 \log (5) \log \left (x \left (-323+160 x-16 x^2\right )\right )}{x}\right ) \, dx\\ &=\frac {64}{323} \int \left (\frac {\left (80-4 \sqrt {77}\right ) \left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}}}{-160-8 \sqrt {77}+32 x}+\frac {\left (80+4 \sqrt {77}\right ) \left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}}}{-160+8 \sqrt {77}+32 x}\right ) \, dx-\frac {320}{323} \int \frac {\left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}}}{x} \, dx+\log (5) \int \frac {5^{\frac {1}{x}} \left (x \left (-323+160 x-16 x^2\right )\right )^{-5^{\frac {1}{x}}} \left (323-160 x+16 x^2\right )^2 \log \left (x \left (-323+160 x-16 x^2\right )\right )}{x} \, dx+\int \frac {\left (x \left (-323+160 x-16 x^2\right )\right )^{2-5^{\frac {1}{x}}}}{x^2} \, dx-\int 5^{\frac {1}{x}} \left (x \left (-323+160 x-16 x^2\right )\right )^{-5^{\frac {1}{x}}} \left (323-160 x+16 x^2\right ) \left (323-320 x+48 x^2\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.14, size = 34, normalized size = 1.10 \begin {gather*} x \left (x \left (-323+160 x-16 x^2\right )\right )^{-5^{\frac {1}{x}}} \left (323-160 x+16 x^2\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.47, size = 162, normalized size = 5.23
method | result | size |
risch | \(\frac {{\mathrm e}^{-\frac {\left (5^{\frac {1}{x}}-2\right ) \left (i \pi \mathrm {csgn}\left (i x \left (x^{2}-10 x +\frac {323}{16}\right )\right )^{3}+i \pi \mathrm {csgn}\left (i x \left (x^{2}-10 x +\frac {323}{16}\right )\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \mathrm {csgn}\left (i x \left (x^{2}-10 x +\frac {323}{16}\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{2}-10 x +\frac {323}{16}\right )\right )-i \pi \,\mathrm {csgn}\left (i x \left (x^{2}-10 x +\frac {323}{16}\right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x^{2}-10 x +\frac {323}{16}\right )\right )-2 i \pi \mathrm {csgn}\left (i x \left (x^{2}-10 x +\frac {323}{16}\right )\right )^{2}+2 i \pi +2 \ln \left (x \right )+2 \ln \left (x^{2}-10 x +\frac {323}{16}\right )\right )}{2}}}{x}\) | \(162\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (26) = 52\).
time = 0.57, size = 54, normalized size = 1.74 \begin {gather*} {\left (256 \, x^{5} - 5120 \, x^{4} + 35936 \, x^{3} - 103360 \, x^{2} + 104329 \, x\right )} e^{\left (-5^{\left (\frac {1}{x}\right )} \log \left (-16 \, x^{2} + 160 \, x - 323\right ) - 5^{\left (\frac {1}{x}\right )} \log \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 28, normalized size = 0.90 \begin {gather*} \frac {{\left (-16 \, x^{3} + 160 \, x^{2} - 323 \, x\right )}^{-5^{\left (\frac {1}{x}\right )} + 2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.68, size = 26, normalized size = 0.84 \begin {gather*} \frac {e^{\left (2 - e^{\frac {\log {\left (5 \right )}}{x}}\right ) \log {\left (- 16 x^{3} + 160 x^{2} - 323 x \right )}}}{x} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.47, size = 134, normalized size = 4.32 \begin {gather*} \frac {104329\,x}{{\left (-16\,x^3+160\,x^2-323\,x\right )}^{5^{1/x}}}-\frac {103360\,x^2}{{\left (-16\,x^3+160\,x^2-323\,x\right )}^{5^{1/x}}}+\frac {35936\,x^3}{{\left (-16\,x^3+160\,x^2-323\,x\right )}^{5^{1/x}}}-\frac {5120\,x^4}{{\left (-16\,x^3+160\,x^2-323\,x\right )}^{5^{1/x}}}+\frac {256\,x^5}{{\left (-16\,x^3+160\,x^2-323\,x\right )}^{5^{1/x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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