Integrand size = 32, antiderivative size = 153 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\frac {2 \text {arctanh}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \text {arctanh}\left (\frac {d+4 e x}{\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}} \]
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Time = 0.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1120, 1107, 214} \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\frac {2 \text {arctanh}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \text {arctanh}\left (\frac {d+4 e x}{\sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}} \]
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Rule 214
Rule 1107
Rule 1120
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\frac {1}{32} \left (\frac {5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4} \, dx,x,\frac {d}{4 e}+x\right ) \\ & = \frac {\left (4 e^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {3 d^2 e}{2}-e \sqrt {d^4-64 a e^3}+8 e^3 x^2} \, dx,x,\frac {d}{4 e}+x\right )}{\sqrt {d^4-64 a e^3}}-\frac {\left (4 e^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {3 d^2 e}{2}+e \sqrt {d^4-64 a e^3}+8 e^3 x^2} \, dx,x,\frac {d}{4 e}+x\right )}{\sqrt {d^4-64 a e^3}} \\ & = \frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.46 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=-\text {RootSum}\left [8 a e^2-d^3 \text {$\#$1}+8 d e^2 \text {$\#$1}^3+8 e^3 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{d^3-24 d e^2 \text {$\#$1}^2-32 e^3 \text {$\#$1}^3}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.44
method | result | size |
default | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 e^{3} \textit {\_Z}^{4}+8 d \,e^{2} \textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3} e^{3}+24 \textit {\_R}^{2} d \,e^{2}-d^{3}}\) | \(67\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 e^{3} \textit {\_Z}^{4}+8 d \,e^{2} \textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3} e^{3}+24 \textit {\_R}^{2} d \,e^{2}-d^{3}}\) | \(67\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1115 vs. \(2 (133) = 266\).
Time = 0.31 (sec) , antiderivative size = 1115, normalized size of antiderivative = 7.29 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\text {Too large to display} \]
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Time = 0.99 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.80 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (1048576 a^{3} e^{9} - 12288 a^{2} d^{4} e^{6} - 384 a d^{8} e^{3} + 5 d^{12}\right ) + t^{2} \cdot \left (384 a d^{2} e^{3} - 6 d^{6}\right ) + 1, \left ( t \mapsto t \log {\left (x + \frac {- 49152 t^{3} a^{2} d^{2} e^{6} - 192 t^{3} a d^{6} e^{3} + 15 t^{3} d^{10} + 256 t a e^{3} - 13 t d^{4} + 2 d}{8 e} \right )} \right )\right )} \]
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\[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int { \frac {1}{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (133) = 266\).
Time = 0.28 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.77 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=-\frac {2 \, \log \left (x + \frac {1}{4} \, \sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{4 \, e}\right )}{e^{3} {\left (\sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{e}\right )}^{3} - 3 \, d e^{2} {\left (\sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{e}\right )}^{2} + 2 \, d^{3}} + \frac {2 \, \log \left (x - \frac {1}{4} \, \sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{4 \, e}\right )}{e^{3} {\left (\sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} - \frac {d}{e}\right )}^{3} + 3 \, d e^{2} {\left (\sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} - \frac {d}{e}\right )}^{2} - 2 \, d^{3}} - \frac {2 \, \log \left (x + \frac {1}{4} \, \sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{4 \, e}\right )}{e^{3} {\left (\sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{e}\right )}^{3} - 3 \, d e^{2} {\left (\sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{e}\right )}^{2} + 2 \, d^{3}} + \frac {2 \, \log \left (x - \frac {1}{4} \, \sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{4 \, e}\right )}{e^{3} {\left (\sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} - \frac {d}{e}\right )}^{3} + 3 \, d e^{2} {\left (\sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} - \frac {d}{e}\right )}^{2} - 2 \, d^{3}} \]
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Time = 11.30 (sec) , antiderivative size = 1264, normalized size of antiderivative = 8.26 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=-\mathrm {atan}\left (\frac {d^3\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,3{}\mathrm {i}+d^9\,2{}\mathrm {i}-a\,d^5\,e^3\,256{}\mathrm {i}+a^2\,d\,e^6\,8192{}\mathrm {i}+a^2\,e^7\,x\,32768{}\mathrm {i}+d^8\,e\,x\,8{}\mathrm {i}+d^2\,e\,x\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,12{}\mathrm {i}-a\,d^4\,e^4\,x\,1024{}\mathrm {i}}{5\,d^{12}\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}+1048576\,a^3\,e^9\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-384\,a\,d^8\,e^3\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-12288\,a^2\,d^4\,e^6\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}}\right )\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {d^3\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,3{}\mathrm {i}-d^9\,2{}\mathrm {i}+a\,d^5\,e^3\,256{}\mathrm {i}-a^2\,d\,e^6\,8192{}\mathrm {i}-a^2\,e^7\,x\,32768{}\mathrm {i}-d^8\,e\,x\,8{}\mathrm {i}+d^2\,e\,x\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,12{}\mathrm {i}+a\,d^4\,e^4\,x\,1024{}\mathrm {i}}{5\,d^{12}\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}+1048576\,a^3\,e^9\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-384\,a\,d^8\,e^3\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-12288\,a^2\,d^4\,e^6\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}}\right )\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}\,2{}\mathrm {i} \]
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