Optimal. Leaf size=227 \[ \frac {1}{16} \left (4+(1-i) 2^{3/4}\right ) \operatorname {PolyLog}\left (2,-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{16} \left (4+(1-i) 2^{3/4}\right ) \log (x) \log \left (1+\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \log (x) \log \left (\frac {(-1)^{3/4} x}{\sqrt [4]{2}}+1\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2357, 2317, 2391} \[ \frac {1}{16} \left (4+(1-i) 2^{3/4}\right ) \text {PolyLog}\left (2,-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \text {PolyLog}\left (2,\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \text {PolyLog}\left (2,-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \text {PolyLog}\left (2,\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{16} \left (4+(1-i) 2^{3/4}\right ) \log (x) \log \left (1+\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \log (x) \log \left (\frac {(-1)^{3/4} x}{\sqrt [4]{2}}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2317
Rule 2357
Rule 2391
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right ) \log (x)}{2+x^4} \, dx &=\int \left (\frac {\left (-2+\sqrt [4]{-2}\right ) \log (x)}{8 \left (\sqrt [4]{-2}-x\right )}+\frac {\left (-2 i+\sqrt [4]{-2}\right ) \log (x)}{8 \left (\sqrt [4]{-2}-i x\right )}+\frac {\left (2 i+\sqrt [4]{-2}\right ) \log (x)}{8 \left (\sqrt [4]{-2}+i x\right )}+\frac {\left (2+\sqrt [4]{-2}\right ) \log (x)}{8 \left (\sqrt [4]{-2}+x\right )}\right ) \, dx\\ &=\frac {1}{8} \left (-2+\sqrt [4]{-2}\right ) \int \frac {\log (x)}{\sqrt [4]{-2}-x} \, dx+\frac {1}{8} \left (-2 i+\sqrt [4]{-2}\right ) \int \frac {\log (x)}{\sqrt [4]{-2}-i x} \, dx+\frac {1}{8} \left (2 i+\sqrt [4]{-2}\right ) \int \frac {\log (x)}{\sqrt [4]{-2}+i x} \, dx+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \int \frac {\log (x)}{\sqrt [4]{-2}+x} \, dx\\ &=\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2-i \sqrt [4]{-2}\right ) \log (x) \log \left (1+\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \log (x) \log \left (1+\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (-2-\sqrt [4]{-2}\right ) \int \frac {\log \left (1-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )}{x} \, dx+\frac {1}{8} \left (i \left (2 i-\sqrt [4]{-2}\right )\right ) \int \frac {\log \left (1-\sqrt [4]{-\frac {1}{2}} x\right )}{x} \, dx+\frac {1}{8} \left (-2+i \sqrt [4]{-2}\right ) \int \frac {\log \left (1+\sqrt [4]{-\frac {1}{2}} x\right )}{x} \, dx+\frac {1}{8} \left (-2+\sqrt [4]{-2}\right ) \int \frac {\log \left (1+\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )}{x} \, dx\\ &=\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2-i \sqrt [4]{-2}\right ) \log (x) \log \left (1+\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \log (x) \log \left (1+\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{16} \left (4+(1-i) 2^{3/4}\right ) \text {Li}_2\left (-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \text {Li}_2\left (\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \text {Li}_2\left (-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \text {Li}_2\left (\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.28, size = 194, normalized size = 0.85 \[ \frac {1}{8} \left (\left (2+\frac {1-i}{\sqrt [4]{2}}\right ) \operatorname {PolyLog}\left (2,-\frac {(1+i) x}{2^{3/4}}\right )+\left (2+\sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,-\frac {(1-i) x}{2^{3/4}}\right )-\left (\sqrt [4]{-2}-2\right ) \operatorname {PolyLog}\left (2,\frac {(1-i) x}{2^{3/4}}\right )+\left (2+i \sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,\frac {(1+i) x}{2^{3/4}}\right )+\left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\sqrt [4]{-\frac {1}{2}} x\right )+\left (2+\frac {1-i}{\sqrt [4]{2}}\right ) \log (x) \log \left (\sqrt [4]{-\frac {1}{2}} x+1\right )-\left (\sqrt [4]{-2}-2\right ) \log (x) \log \left (1-\frac {(1-i) x}{2^{3/4}}\right )+\left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1+\frac {(1-i) x}{2^{3/4}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{3} + 1\right )} \log \relax (x)}{x^{4} + 2}, x\right ) \]
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 \[ \int \frac {{\left (x^{3} + 1\right )} \log \relax (x)}{x^{4} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1210, normalized size = 5.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{3} + 1\right )} \log \relax (x)}{x^{4} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \relax (x)\,\left (x^3+1\right )}{x^4+2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x + 1\right ) \left (x^{2} - x + 1\right ) \log {\relax (x )}}{x^{4} + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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