Integrand size = 18, antiderivative size = 100 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{13}} \, dx=\frac {\sqrt [3]{1+x^4} \left (-9-6 x^4+x^8\right )}{36 x^{12}}-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{18 \sqrt {3}}+\frac {1}{54} \log \left (-1+\sqrt [3]{1+x^4}\right )-\frac {1}{108} \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {457, 79, 43, 44, 59, 632, 210, 31} \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{13}} \, dx=-\frac {\arctan \left (\frac {2 \sqrt [3]{x^4+1}+1}{\sqrt {3}}\right )}{18 \sqrt {3}}+\frac {\sqrt [3]{x^4+1}}{36 x^4}+\frac {1}{36} \log \left (1-\sqrt [3]{x^4+1}\right )-\frac {\left (x^4+1\right )^{4/3}}{4 x^{12}}+\frac {\sqrt [3]{x^4+1}}{12 x^8}-\frac {\log (x)}{27} \]
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Rule 31
Rule 43
Rule 44
Rule 59
Rule 79
Rule 210
Rule 457
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {\sqrt [3]{1+x} (3+x)}{x^4} \, dx,x,x^4\right ) \\ & = -\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}-\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x^3} \, dx,x,x^4\right ) \\ & = \frac {\sqrt [3]{1+x^4}}{12 x^8}-\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}-\frac {1}{36} \text {Subst}\left (\int \frac {1}{x^2 (1+x)^{2/3}} \, dx,x,x^4\right ) \\ & = \frac {\sqrt [3]{1+x^4}}{12 x^8}+\frac {\sqrt [3]{1+x^4}}{36 x^4}-\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}+\frac {1}{54} \text {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^4\right ) \\ & = \frac {\sqrt [3]{1+x^4}}{12 x^8}+\frac {\sqrt [3]{1+x^4}}{36 x^4}-\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}-\frac {\log (x)}{27}-\frac {1}{36} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^4}\right )-\frac {1}{36} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^4}\right ) \\ & = \frac {\sqrt [3]{1+x^4}}{12 x^8}+\frac {\sqrt [3]{1+x^4}}{36 x^4}-\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}-\frac {\log (x)}{27}+\frac {1}{36} \log \left (1-\sqrt [3]{1+x^4}\right )+\frac {1}{18} \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}}{12 x^8}+\frac {\sqrt [3]{1+x^4}}{36 x^4}-\frac {\left (1+x^4\right )^{4/3}}{4 x^{12}}-\frac {\arctan \left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{18 \sqrt {3}}-\frac {\log (x)}{27}+\frac {1}{36} \log \left (1-\sqrt [3]{1+x^4}\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{13}} \, dx=\frac {1}{108} \left (\frac {3 \sqrt [3]{1+x^4} \left (-9-6 x^4+x^8\right )}{x^{12}}-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+x^4}\right )-\log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.45 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {x^{12}-5 x^{8}-15 x^{4}-9}{36 x^{12} \left (x^{4}+1\right )^{\frac {2}{3}}}+\frac {-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{4}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{54 \Gamma \left (\frac {2}{3}\right )}\) | \(74\) |
pseudoelliptic | \(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}} \left (x^{8}-6 x^{4}-9\right )-x^{12} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{4}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )+\ln \left (1+\left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}\right )-2 \ln \left (-1+\left (x^{4}+1\right )^{\frac {1}{3}}\right )\right )}{108 {\left (1+\left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}\right )}^{3} {\left (-1+\left (x^{4}+1\right )^{\frac {1}{3}}\right )}^{3}}\) | \(109\) |
meijerg | \(-\frac {-\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], -x^{4}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{8}}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{4}}}{12 \Gamma \left (\frac {2}{3}\right )}-\frac {\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], -x^{4}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{27}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{12}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{8}}-\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}\) | \(128\) |
trager | \(\frac {\left (x^{4}+1\right )^{\frac {1}{3}} \left (x^{8}-6 x^{4}-9\right )}{36 x^{12}}-\frac {\ln \left (-\frac {-180 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+74 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}+165 \left (x^{4}+1\right )^{\frac {2}{3}}+351 \left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+180 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+165 \left (x^{4}+1\right )^{\frac {1}{3}}+411 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+185}{x^{4}}\right )}{54}-\frac {\ln \left (-\frac {-180 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+74 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}+165 \left (x^{4}+1\right )^{\frac {2}{3}}+351 \left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+180 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+165 \left (x^{4}+1\right )^{\frac {1}{3}}+411 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+185}{x^{4}}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{18}+\frac {\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {180 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+129 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-51 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-48 \left (x^{4}+1\right )^{\frac {2}{3}}+351 \left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-180 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-48 \left (x^{4}+1\right )^{\frac {1}{3}}+291 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-68}{x^{4}}\right )}{18}\) | \(450\) |
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Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{13}} \, dx=-\frac {2 \, \sqrt {3} x^{12} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{12} \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{12} \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{8} - 6 \, x^{4} - 9\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{108 \, x^{12}} \]
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Timed out. \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{13}} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{13}} \, dx=-\frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {5 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} - 13 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{72 \, {\left (3 \, x^{4} + {\left (x^{4} + 1\right )}^{3} - 3 \, {\left (x^{4} + 1\right )}^{2} + 2\right )}} + \frac {{\left (x^{4} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{24 \, {\left (2 \, x^{4} - {\left (x^{4} + 1\right )}^{2} + 1\right )}} - \frac {1}{108} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{54} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{13}} \, dx=-\frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {{\left (x^{4} + 1\right )}^{\frac {7}{3}} - 8 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{36 \, x^{12}} - \frac {1}{108} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{54} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 7.11 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{13}} \, dx=\frac {5\,\ln \left (\frac {25\,{\left (x^4+1\right )}^{1/3}}{1296}-\frac {25}{1296}\right )}{108}-\frac {\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{144}-\frac {1}{144}\right )}{36}-\frac {\frac {5\,{\left (x^4+1\right )}^{1/3}}{36}+\frac {13\,{\left (x^4+1\right )}^{4/3}}{72}-\frac {5\,{\left (x^4+1\right )}^{7/3}}{72}}{{\left (x^4+1\right )}^3-3\,{\left (x^4+1\right )}^2+3\,x^4+2}+\frac {\frac {{\left (x^4+1\right )}^{1/3}}{12}+\frac {{\left (x^4+1\right )}^{4/3}}{24}}{2\,x^4-{\left (x^4+1\right )}^2+1}-\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{4}+\frac {1}{8}-\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )\,\left (-\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{72}\right )+\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{4}+\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )\,\left (\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{72}\right )+\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{12}+\frac {5}{24}-\frac {\sqrt {3}\,5{}\mathrm {i}}{24}\right )\,\left (-\frac {5}{216}+\frac {\sqrt {3}\,5{}\mathrm {i}}{216}\right )-\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{12}+\frac {5}{24}+\frac {\sqrt {3}\,5{}\mathrm {i}}{24}\right )\,\left (\frac {5}{216}+\frac {\sqrt {3}\,5{}\mathrm {i}}{216}\right ) \]
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