Optimal. Leaf size=22 \[ \frac {\left (x-\frac {4}{x^4 \log ^8(5)}\right )^2 \log (x)}{\log \left (x^2\right )} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).
time = 2.24, antiderivative size = 47, normalized size of antiderivative = 2.14, number
of steps used = 57, number of rules used = 13, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.153, Rules
used = {12, 6873, 6874, 2395, 2343, 2347, 2209, 2344, 2335, 2413, 6617, 15, 6631}
\begin {gather*} \frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 15
Rule 2209
Rule 2335
Rule 2343
Rule 2344
Rule 2347
Rule 2395
Rule 2413
Rule 6617
Rule 6631
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^2\left (x^2\right )} \, dx}{\log ^{16}(5)}\\ &=\frac {\int \frac {\left (4-x^5 \log ^8(5)\right ) \left (-8 \log (x)+2 x^5 \log ^8(5) \log (x)+4 \log \left (x^2\right )-x^5 \log ^8(5) \log \left (x^2\right )-32 \log (x) \log \left (x^2\right )-2 x^5 \log ^8(5) \log (x) \log \left (x^2\right )\right )}{x^9 \log ^2\left (x^2\right )} \, dx}{\log ^{16}(5)}\\ &=\frac {\int \left (-\frac {2 \left (-4+x^5 \log ^8(5)\right )^2 \log (x)}{x^9 \log ^2\left (x^2\right )}+\frac {\left (-4+x^5 \log ^8(5)\right ) \left (-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)\right )}{x^9 \log \left (x^2\right )}\right ) \, dx}{\log ^{16}(5)}\\ &=\frac {\int \frac {\left (-4+x^5 \log ^8(5)\right ) \left (-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)\right )}{x^9 \log \left (x^2\right )} \, dx}{\log ^{16}(5)}-\frac {2 \int \frac {\left (-4+x^5 \log ^8(5)\right )^2 \log (x)}{x^9 \log ^2\left (x^2\right )} \, dx}{\log ^{16}(5)}\\ &=\frac {\int \left (-\frac {4 \left (-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)\right )}{x^9 \log \left (x^2\right )}+\frac {\log ^8(5) \left (-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)\right )}{x^4 \log \left (x^2\right )}\right ) \, dx}{\log ^{16}(5)}-\frac {2 \int \left (\frac {16 \log (x)}{x^9 \log ^2\left (x^2\right )}-\frac {8 \log ^8(5) \log (x)}{x^4 \log ^2\left (x^2\right )}+\frac {x \log ^{16}(5) \log (x)}{\log ^2\left (x^2\right )}\right ) \, dx}{\log ^{16}(5)}\\ &=-\left (2 \int \frac {x \log (x)}{\log ^2\left (x^2\right )} \, dx\right )-\frac {4 \int \frac {-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)}{x^9 \log \left (x^2\right )} \, dx}{\log ^{16}(5)}-\frac {32 \int \frac {\log (x)}{x^9 \log ^2\left (x^2\right )} \, dx}{\log ^{16}(5)}+\frac {\int \frac {-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)}{x^4 \log \left (x^2\right )} \, dx}{\log ^8(5)}+\frac {16 \int \frac {\log (x)}{x^4 \log ^2\left (x^2\right )} \, dx}{\log ^8(5)}\\ &=\frac {64 \text {Ei}\left (-4 \log \left (x^2\right )\right ) \log (x)}{\log ^{16}(5)}-\frac {12 \left (x^2\right )^{3/2} \text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right ) \log (x)}{x^3 \log ^8(5)}+\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}-\log (x) \text {li}\left (x^2\right )+2 \int \left (-\frac {x}{2 \log \left (x^2\right )}+\frac {\text {li}\left (x^2\right )}{2 x}\right ) \, dx-\frac {4 \int \left (-\frac {4}{x^9 \log \left (x^2\right )}+\frac {\log ^8(5)}{x^4 \log \left (x^2\right )}+\frac {32 \log (x)}{x^9 \log \left (x^2\right )}+\frac {2 \log ^8(5) \log (x)}{x^4 \log \left (x^2\right )}\right ) \, dx}{\log ^{16}(5)}+\frac {32 \int \left (-\frac {2 \text {Ei}\left (-4 \log \left (x^2\right )\right )}{x}-\frac {1}{2 x^9 \log \left (x^2\right )}\right ) \, dx}{\log ^{16}(5)}+\frac {\int \left (-\frac {4}{x^4 \log \left (x^2\right )}+\frac {x \log ^8(5)}{\log \left (x^2\right )}+\frac {32 \log (x)}{x^4 \log \left (x^2\right )}+\frac {2 x \log ^8(5) \log (x)}{\log \left (x^2\right )}\right ) \, dx}{\log ^8(5)}-\frac {16 \int \left (-\frac {3 \text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{4 \sqrt {x^2}}-\frac {1}{2 x^4 \log \left (x^2\right )}\right ) \, dx}{\log ^8(5)}\\ &=\frac {64 \text {Ei}\left (-4 \log \left (x^2\right )\right ) \log (x)}{\log ^{16}(5)}-\frac {12 \left (x^2\right )^{3/2} \text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right ) \log (x)}{x^3 \log ^8(5)}+\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}-\log (x) \text {li}\left (x^2\right )+2 \int \frac {x \log (x)}{\log \left (x^2\right )} \, dx-\frac {64 \int \frac {\text {Ei}\left (-4 \log \left (x^2\right )\right )}{x} \, dx}{\log ^{16}(5)}-\frac {128 \int \frac {\log (x)}{x^9 \log \left (x^2\right )} \, dx}{\log ^{16}(5)}-2 \frac {4 \int \frac {1}{x^4 \log \left (x^2\right )} \, dx}{\log ^8(5)}+\frac {8 \int \frac {1}{x^4 \log \left (x^2\right )} \, dx}{\log ^8(5)}-\frac {8 \int \frac {\log (x)}{x^4 \log \left (x^2\right )} \, dx}{\log ^8(5)}+\frac {12 \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{\sqrt {x^2}} \, dx}{\log ^8(5)}+\frac {32 \int \frac {\log (x)}{x^4 \log \left (x^2\right )} \, dx}{\log ^8(5)}+\int \frac {\text {li}\left (x^2\right )}{x} \, dx\\ &=\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}+\frac {1}{2} \text {Subst}\left (\int \frac {\text {li}(x)}{x} \, dx,x,x^2\right )-2 \int \frac {\text {li}\left (x^2\right )}{2 x} \, dx-\frac {32 \text {Subst}\left (\int \text {Ei}(-4 x) \, dx,x,\log \left (x^2\right )\right )}{\log ^{16}(5)}+\frac {128 \int \frac {\text {Ei}\left (-4 \log \left (x^2\right )\right )}{2 x} \, dx}{\log ^{16}(5)}+\frac {8 \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{2 \sqrt {x^2}} \, dx}{\log ^8(5)}-\frac {32 \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{2 \sqrt {x^2}} \, dx}{\log ^8(5)}+\frac {(12 x) \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{x} \, dx}{\sqrt {x^2} \log ^8(5)}-2 \frac {\left (2 \left (x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {e^{-3 x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{x^3 \log ^8(5)}+\frac {\left (4 \left (x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {e^{-3 x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{x^3 \log ^8(5)}\\ &=-\frac {x^2}{2}-\frac {8}{x^8 \log ^{16}(5)}+\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}-\frac {32 \text {Ei}\left (-4 \log \left (x^2\right )\right ) \log \left (x^2\right )}{\log ^{16}(5)}+\frac {1}{2} \log \left (x^2\right ) \text {li}\left (x^2\right )+\frac {64 \int \frac {\text {Ei}\left (-4 \log \left (x^2\right )\right )}{x} \, dx}{\log ^{16}(5)}+\frac {4 \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{\sqrt {x^2}} \, dx}{\log ^8(5)}-\frac {16 \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{\sqrt {x^2}} \, dx}{\log ^8(5)}+\frac {(6 x) \text {Subst}\left (\int \text {Ei}\left (-\frac {3 x}{2}\right ) \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2} \log ^8(5)}-\int \frac {\text {li}\left (x^2\right )}{x} \, dx\\ &=-\frac {x^2}{2}-\frac {8}{x^8 \log ^{16}(5)}+\frac {4}{x^3 \log ^8(5)}+\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}-\frac {32 \text {Ei}\left (-4 \log \left (x^2\right )\right ) \log \left (x^2\right )}{\log ^{16}(5)}+\frac {6 x \text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right ) \log \left (x^2\right )}{\sqrt {x^2} \log ^8(5)}+\frac {1}{2} \log \left (x^2\right ) \text {li}\left (x^2\right )-\frac {1}{2} \text {Subst}\left (\int \frac {\text {li}(x)}{x} \, dx,x,x^2\right )+\frac {32 \text {Subst}\left (\int \text {Ei}(-4 x) \, dx,x,\log \left (x^2\right )\right )}{\log ^{16}(5)}+\frac {(4 x) \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{x} \, dx}{\sqrt {x^2} \log ^8(5)}-\frac {(16 x) \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{x} \, dx}{\sqrt {x^2} \log ^8(5)}\\ &=\frac {4}{x^3 \log ^8(5)}+\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}+\frac {6 x \text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right ) \log \left (x^2\right )}{\sqrt {x^2} \log ^8(5)}+\frac {(2 x) \text {Subst}\left (\int \text {Ei}\left (-\frac {3 x}{2}\right ) \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2} \log ^8(5)}-\frac {(8 x) \text {Subst}\left (\int \text {Ei}\left (-\frac {3 x}{2}\right ) \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2} \log ^8(5)}\\ &=\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 28, normalized size = 1.27 \begin {gather*} \frac {\left (-4+x^5 \log ^8(5)\right )^2 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )} \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 = 6.86, size = 242, normalized size = 11.00
method | result | size |
risch | \(\frac {x^{10} \ln \left (5\right )^{16}-8 x^{5} \ln \left (5\right )^{8}+16}{2 \ln \left (5\right )^{16} x^{8}}-\frac {\pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (\ln \left (5\right )^{16} x^{10} \mathrm {csgn}\left (i x \right )^{2}-2 \ln \left (5\right )^{16} x^{10} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )+\ln \left (5\right )^{16} x^{10} \mathrm {csgn}\left (i x^{2}\right )^{2}-8 \ln \left (5\right )^{8} x^{5} \mathrm {csgn}\left (i x \right )^{2}+16 \ln \left (5\right )^{8} x^{5} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )-8 \ln \left (5\right )^{8} x^{5} \mathrm {csgn}\left (i x^{2}\right )^{2}+16 \mathrm {csgn}\left (i x \right )^{2}-32 \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )+16 \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{2 \ln \left (5\right )^{16} x^{8} \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (x \right )\right )}\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 28, normalized size = 1.27 \begin {gather*} \frac {x^{10} \log \left (5\right )^{16} - 8 \, x^{5} \log \left (5\right )^{8} + 16}{2 \, x^{8} \log \left (5\right )^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 28, normalized size = 1.27 \begin {gather*} \frac {x^{10} \log \left (5\right )^{16} - 8 \, x^{5} \log \left (5\right )^{8} + 16}{2 \, x^{8} \log \left (5\right )^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 29, normalized size = 1.32 \begin {gather*} \frac {\frac {x^{2} \log {\left (5 \right )}^{16}}{2} + \frac {- 4 x^{5} \log {\left (5 \right )}^{8} + 8}{x^{8}}}{\log {\left (5 \right )}^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 30, normalized size = 1.36 \begin {gather*} \frac {x^{2} \log \left (5\right )^{16} - \frac {8 \, {\left (x^{5} \log \left (5\right )^{8} - 2\right )}}{x^{8}}}{2 \, \log \left (5\right )^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.00, size = 28, normalized size = 1.27 \begin {gather*} \frac {\ln \left (x\right )\,{\left (x^5\,{\ln \left (5\right )}^8-4\right )}^2}{x^8\,\ln \left (x^2\right )\,{\ln \left (5\right )}^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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