Integrand size = 18, antiderivative size = 107 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {\sqrt [3]{1+x^4} \left (-81-45 x^4+6 x^8-10 x^{12}\right )}{432 x^{16}}+\frac {5 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{108 \sqrt {3}}-\frac {5}{324} \log \left (-1+\sqrt [3]{1+x^4}\right )+\frac {5}{648} \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.10, number of steps used = 9, 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} \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {5 \arctan \left (\frac {2 \sqrt [3]{x^4+1}+1}{\sqrt {3}}\right )}{108 \sqrt {3}}-\frac {5 \sqrt [3]{x^4+1}}{216 x^4}-\frac {5}{216} \log \left (1-\sqrt [3]{x^4+1}\right )-\frac {3 \left (x^4+1\right )^{4/3}}{16 x^{16}}+\frac {\sqrt [3]{x^4+1}}{12 x^{12}}+\frac {\sqrt [3]{x^4+1}}{72 x^8}+\frac {5 \log (x)}{162} \]
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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^5} \, dx,x,x^4\right ) \\ & = -\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}-\frac {1}{4} \text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x^4} \, dx,x,x^4\right ) \\ & = \frac {\sqrt [3]{1+x^4}}{12 x^{12}}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}-\frac {1}{36} \text {Subst}\left (\int \frac {1}{x^3 (1+x)^{2/3}} \, dx,x,x^4\right ) \\ & = \frac {\sqrt [3]{1+x^4}}{12 x^{12}}+\frac {\sqrt [3]{1+x^4}}{72 x^8}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}+\frac {5}{216} \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^{12}}+\frac {\sqrt [3]{1+x^4}}{72 x^8}-\frac {5 \sqrt [3]{1+x^4}}{216 x^4}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}-\frac {5}{324} \text {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^4\right ) \\ & = \frac {\sqrt [3]{1+x^4}}{12 x^{12}}+\frac {\sqrt [3]{1+x^4}}{72 x^8}-\frac {5 \sqrt [3]{1+x^4}}{216 x^4}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}+\frac {5 \log (x)}{162}+\frac {5}{216} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^4}\right )+\frac {5}{216} \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^{12}}+\frac {\sqrt [3]{1+x^4}}{72 x^8}-\frac {5 \sqrt [3]{1+x^4}}{216 x^4}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}+\frac {5 \log (x)}{162}-\frac {5}{216} \log \left (1-\sqrt [3]{1+x^4}\right )-\frac {5}{108} \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^{12}}+\frac {\sqrt [3]{1+x^4}}{72 x^8}-\frac {5 \sqrt [3]{1+x^4}}{216 x^4}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}+\frac {5 \arctan \left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{108 \sqrt {3}}+\frac {5 \log (x)}{162}-\frac {5}{216} \log \left (1-\sqrt [3]{1+x^4}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {-\frac {3 \sqrt [3]{1+x^4} \left (81+45 x^4-6 x^8+10 x^{12}\right )}{x^{16}}+20 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )-20 \log \left (-1+\sqrt [3]{1+x^4}\right )+10 \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right )}{1296} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {10 x^{16}+4 x^{12}+39 x^{8}+126 x^{4}+81}{432 x^{16} \left (x^{4}+1\right )^{\frac {2}{3}}}-\frac {5 \left (-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{4}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )\right )}{324 \Gamma \left (\frac {2}{3}\right )}\) | \(81\) |
pseudoelliptic | \(\frac {\left (-30 x^{12}+18 x^{8}-135 x^{4}-243\right ) \left (x^{4}+1\right )^{\frac {1}{3}}+10 x^{16} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{4}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )+\ln \left (1+\left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}\right )-2 \ln \left (-1+\left (x^{4}+1\right )^{\frac {1}{3}}\right )\right )}{1296 {\left (1+\left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}\right )}^{4} {\left (-1+\left (x^{4}+1\right )^{\frac {1}{3}}\right )}^{4}}\) | \(115\) |
meijerg | \(-\frac {\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], -x^{4}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{27}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{12}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{8}}-\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{4}}}{12 \Gamma \left (\frac {2}{3}\right )}-\frac {-\frac {22 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {14}{3}\right ], \left [2, 6\right ], -x^{4}\right )}{243}+\frac {10 \left (\frac {47}{120}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{81}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{4 x^{16}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{12}}-\frac {\Gamma \left (\frac {2}{3}\right )}{6 x^{8}}+\frac {5 \Gamma \left (\frac {2}{3}\right )}{27 x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}\) | \(144\) |
trager | \(-\frac {\left (10 x^{12}-6 x^{8}+45 x^{4}+81\right ) \left (x^{4}+1\right )^{\frac {1}{3}}}{432 x^{16}}-\frac {5 \ln \left (-\frac {333 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-393 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+60 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}+48 \left (x^{4}+1\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-333 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-165 \left (x^{4}+1\right )^{\frac {1}{3}}-384 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+80}{x^{4}}\right )}{324}+\frac {5 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {153 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+162 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+40 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-165 \left (x^{4}+1\right )^{\frac {2}{3}}-495 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-153 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+48 \left (x^{4}+1\right )^{\frac {1}{3}}+195 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+100}{x^{4}}\right )}{108}\) | \(311\) |
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Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {20 \, \sqrt {3} x^{16} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 10 \, x^{16} \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - 20 \, x^{16} \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (10 \, x^{12} - 6 \, x^{8} + 45 \, x^{4} + 81\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{1296 \, x^{16}} \]
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Timed out. \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (85) = 170\).
Time = 0.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {5}{324} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {20 \, {\left (x^{4} + 1\right )}^{\frac {10}{3}} - 72 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} + 93 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} + 40 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{432 \, {\left ({\left (x^{4} + 1\right )}^{4} - 4 \, x^{4} - 4 \, {\left (x^{4} + 1\right )}^{3} + 6 \, {\left (x^{4} + 1\right )}^{2} - 3\right )}} + \frac {5 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} - 13 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{216 \, {\left (3 \, x^{4} + {\left (x^{4} + 1\right )}^{3} - 3 \, {\left (x^{4} + 1\right )}^{2} + 2\right )}} + \frac {5}{648} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {5}{324} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {5}{324} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {10 \, {\left (x^{4} + 1\right )}^{\frac {10}{3}} - 36 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} + 87 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} + 20 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{432 \, x^{16}} + \frac {5}{648} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {5}{324} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 6.67 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.50 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {5\,\ln \left (\frac {25\,{\left (x^4+1\right )}^{1/3}}{11664}-\frac {25}{11664}\right )}{324}-\frac {5\,\ln \left (\frac {25\,{\left (x^4+1\right )}^{1/3}}{2916}-\frac {25}{2916}\right )}{162}-\frac {\frac {5\,{\left (x^4+1\right )}^{1/3}}{108}+\frac {13\,{\left (x^4+1\right )}^{4/3}}{216}-\frac {5\,{\left (x^4+1\right )}^{7/3}}{216}}{{\left (x^4+1\right )}^3-3\,{\left (x^4+1\right )}^2+3\,x^4+2}+\frac {\frac {5\,{\left (x^4+1\right )}^{1/3}}{54}+\frac {31\,{\left (x^4+1\right )}^{4/3}}{144}-\frac {{\left (x^4+1\right )}^{7/3}}{6}+\frac {5\,{\left (x^4+1\right )}^{10/3}}{108}}{4\,{\left (x^4+1\right )}^3-6\,{\left (x^4+1\right )}^2-{\left (x^4+1\right )}^4+4\,x^4+3}-\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{18}+\frac {5}{36}-\frac {\sqrt {3}\,5{}\mathrm {i}}{36}\right )\,\left (-\frac {5}{324}+\frac {\sqrt {3}\,5{}\mathrm {i}}{324}\right )+\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{18}+\frac {5}{36}+\frac {\sqrt {3}\,5{}\mathrm {i}}{36}\right )\,\left (\frac {5}{324}+\frac {\sqrt {3}\,5{}\mathrm {i}}{324}\right )+\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{36}+\frac {5}{72}-\frac {\sqrt {3}\,5{}\mathrm {i}}{72}\right )\,\left (-\frac {5}{648}+\frac {\sqrt {3}\,5{}\mathrm {i}}{648}\right )-\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{36}+\frac {5}{72}+\frac {\sqrt {3}\,5{}\mathrm {i}}{72}\right )\,\left (\frac {5}{648}+\frac {\sqrt {3}\,5{}\mathrm {i}}{648}\right ) \]
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