Optimal. Leaf size=147 \[ -\text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )-2 \log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]+\sqrt [4]{x^4-x^3}-\frac {7}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )+\frac {7}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right ) \]
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Rubi [C] time = 0.73, antiderivative size = 695, normalized size of antiderivative = 4.73, number of steps used = 25, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {2056, 6728, 101, 157, 63, 240, 212, 206, 203, 93, 298, 205, 208} \begin {gather*} \frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3}+\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {\left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {\left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 63
Rule 93
Rule 101
Rule 157
Rule 203
Rule 205
Rule 206
Rule 208
Rule 212
Rule 240
Rule 298
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} (1+x)}{1-x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-1+x} x^{3/4}}{-1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-1+x} x^{3/4}}{-1+i \sqrt {3}+2 x}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1-i \sqrt {3}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1+i \sqrt {3}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{4} \left (1+i \sqrt {3}\right )+\frac {1}{2} \left (-1-2 i \sqrt {3}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-i \sqrt {3}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{4} \left (1-i \sqrt {3}\right )+\frac {1}{2} \left (-1+2 i \sqrt {3}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+i \sqrt {3}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-i \sqrt {3}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+i \sqrt {3}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (4 \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1-i \sqrt {3}-\left (1-i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+i \sqrt {3}-\left (1+i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {-i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {-i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}+\sqrt {-i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {-i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-i+\sqrt {3}}-\sqrt {i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-i+\sqrt {3}}+\sqrt {i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {2 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 133, normalized size = 0.90 \begin {gather*} \frac {4 \sqrt [4]{(x-1) x^3} \left (\sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};1-x\right )+\sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )-\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (-i+\sqrt {3}\right ) (x-1)}{\left (i+\sqrt {3}\right ) x}\right )-\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (i+\sqrt {3}\right ) (x-1)}{\left (-i+\sqrt {3}\right ) x}\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 147, normalized size = 1.00 \begin {gather*} \sqrt [4]{-x^3+x^4}-\frac {7}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {7}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 836, normalized size = 5.69 \begin {gather*} -\frac {1}{4} \, {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (\frac {2 \, {\left (2 \, x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \frac {1}{4} \, {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (\frac {2 \, {\left (2 \, x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) - \frac {1}{2} \, \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} \log \left (\frac {4 \, {\left (x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} \log \left (\frac {4 \, {\left (x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} \sqrt {\frac {2 \, x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} - 2 \, \sqrt {3} x - 4 \, x}{2 \, x}\right ) + \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} \sqrt {\frac {2 \, x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {3} x + 4 \, x}{2 \, x}\right ) - 2 \, \sqrt {\sqrt {3} + 2} \arctan \left (\frac {2 \, {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} \sqrt {\frac {x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} + \sqrt {3} x - 2 \, x}{x}\right ) - 2 \, \sqrt {\sqrt {3} + 2} \arctan \left (\frac {2 \, {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} \sqrt {\frac {x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} - \sqrt {3} x + 2 \, x}{x}\right ) + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} + \frac {7}{2} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {7}{4} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {7}{4} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 384, normalized size = 2.61 \begin {gather*} \frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{4} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{4} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{4} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{4} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - x {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {7}{2} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {7}{4} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {7}{4} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 48.67, size = 1746, normalized size = 11.88
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1746\) |
risch | \(\text {Expression too large to display}\) | \(5889\) |
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 \begin {gather*} \int \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{x^{2} - x + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^4-x^3\right )}^{1/4}\,\left (x+1\right )}{x^2-x+1} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x + 1\right )}{x^{2} - x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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