Optimal. Leaf size=102 \[ \frac {\sqrt [3]{1+x^4} \left (9-3 x^4-4 x^8\right )}{36 x^{12}}+\frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2}{27} \log \left (-1+\sqrt [3]{1+x^4}\right )+\frac {1}{27} \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {457, 79, 43, 44,
59, 632, 210, 31} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {2 \sqrt [3]{x^4+1}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\sqrt [3]{x^4+1}}{9 x^4}-\frac {1}{9} \log \left (1-\sqrt [3]{x^4+1}\right )+\frac {\left (x^4+1\right )^{4/3}}{4 x^{12}}-\frac {\sqrt [3]{x^4+1}}{3 x^8}+\frac {4 \log (x)}{27} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 44
Rule 59
Rule 79
Rule 210
Rule 457
Rule 632
Rubi steps
\begin {align*} \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^{13}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(-3+x) \sqrt [3]{1+x}}{x^4} \, dx,x,x^4\right )\\ &=\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}+\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x^3} \, dx,x,x^4\right )\\ &=-\frac {\sqrt [3]{1+x^4}}{3 x^8}+\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{x^2 (1+x)^{2/3}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt [3]{1+x^4}}{3 x^8}-\frac {\sqrt [3]{1+x^4}}{9 x^4}+\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}-\frac {2}{27} \text {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt [3]{1+x^4}}{3 x^8}-\frac {\sqrt [3]{1+x^4}}{9 x^4}+\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}+\frac {4 \log (x)}{27}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^4}\right )+\frac {1}{9} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^4}\right )\\ &=-\frac {\sqrt [3]{1+x^4}}{3 x^8}-\frac {\sqrt [3]{1+x^4}}{9 x^4}+\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}+\frac {4 \log (x)}{27}-\frac {1}{9} \log \left (1-\sqrt [3]{1+x^4}\right )-\frac {2}{9} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^4}\right )\\ &=-\frac {\sqrt [3]{1+x^4}}{3 x^8}-\frac {\sqrt [3]{1+x^4}}{9 x^4}+\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}+\frac {2 \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {4 \log (x)}{27}-\frac {1}{9} \log \left (1-\sqrt [3]{1+x^4}\right )\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 95, normalized size = 0.93 \begin {gather*} \frac {1}{108} \left (-\frac {3 \sqrt [3]{1+x^4} \left (-9+3 x^4+4 x^8\right )}{x^{12}}+8 \sqrt {3} \text {ArcTan}\left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )-8 \log \left (-1+\sqrt [3]{1+x^4}\right )+4 \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 3.70, size = 76, normalized size = 0.75
method | result | size |
risch | \(-\frac {4 x^{12}+7 x^{8}-6 x^{4}-9}{36 x^{12} \left (x^{4}+1\right )^{\frac {2}{3}}}-\frac {2 \left (-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{4}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )\right )}{27 \Gamma \left (\frac {2}{3}\right )}\) | \(76\) |
meijerg | \(\frac {\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], -x^{4}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{27}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{12}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{8}}-\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}-\frac {-\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], -x^{4}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{8}}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{4}}}{12 \Gamma \left (\frac {2}{3}\right )}\) | \(128\) |
trager | \(-\frac {\left (4 x^{8}+3 x^{4}-9\right ) \left (x^{4}+1\right )^{\frac {1}{3}}}{36 x^{12}}+\frac {2 \ln \left (\frac {180 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}-74 x^{4}+351 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-180 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+351 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-165 \left (x^{4}+1\right )^{\frac {2}{3}}+411 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-165 \left (x^{4}+1\right )^{\frac {1}{3}}-185}{x^{4}}\right )}{27}-\frac {2 \ln \left (\frac {180 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}-74 x^{4}+351 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-180 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+351 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-165 \left (x^{4}+1\right )^{\frac {2}{3}}+411 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-165 \left (x^{4}+1\right )^{\frac {1}{3}}-185}{x^{4}}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{9}+\frac {2 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {180 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-129 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}-51 x^{4}-351 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-180 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-351 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-48 \left (x^{4}+1\right )^{\frac {2}{3}}-291 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-48 \left (x^{4}+1\right )^{\frac {1}{3}}-68}{x^{4}}\right )}{9}\) | \(450\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 146, normalized size = 1.43 \begin {gather*} \frac {2}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} - 13 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{72 \, {\left (3 \, x^{4} + {\left (x^{4} + 1\right )}^{3} - 3 \, {\left (x^{4} + 1\right )}^{2} + 2\right )}} + \frac {{\left (x^{4} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{24 \, {\left (2 \, x^{4} - {\left (x^{4} + 1\right )}^{2} + 1\right )}} + \frac {1}{27} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{27} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 91, normalized size = 0.89 \begin {gather*} \frac {8 \, \sqrt {3} x^{12} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 4 \, x^{12} \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - 8 \, x^{12} \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (4 \, x^{8} + 3 \, x^{4} - 9\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{108 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 87, normalized size = 0.85 \begin {gather*} \frac {2}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {4 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} - 5 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} - 8 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{36 \, x^{12}} + \frac {1}{27} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{27} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 231, normalized size = 2.26 \begin {gather*} \frac {\frac {5\,{\left (x^4+1\right )}^{1/3}}{36}+\frac {13\,{\left (x^4+1\right )}^{4/3}}{72}-\frac {5\,{\left (x^4+1\right )}^{7/3}}{72}}{{\left (x^4+1\right )}^3-3\,{\left (x^4+1\right )}^2+3\,x^4+2}-\frac {5\,\ln \left (\frac {25\,{\left (x^4+1\right )}^{1/3}}{1296}-\frac {25}{1296}\right )}{108}-\frac {\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{144}-\frac {1}{144}\right )}{36}+\frac {\frac {{\left (x^4+1\right )}^{1/3}}{12}+\frac {{\left (x^4+1\right )}^{4/3}}{24}}{2\,x^4-{\left (x^4+1\right )}^2+1}-\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{4}+\frac {1}{8}-\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )\,\left (-\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{72}\right )+\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{4}+\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )\,\left (\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{72}\right )-\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{12}+\frac {5}{24}-\frac {\sqrt {3}\,5{}\mathrm {i}}{24}\right )\,\left (-\frac {5}{216}+\frac {\sqrt {3}\,5{}\mathrm {i}}{216}\right )+\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{12}+\frac {5}{24}+\frac {\sqrt {3}\,5{}\mathrm {i}}{24}\right )\,\left (\frac {5}{216}+\frac {\sqrt {3}\,5{}\mathrm {i}}{216}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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