Optimal. Leaf size=158 \[ -9 \text {RootSum}\left [2 \text {$\#$1}^8-6 \text {$\#$1}^4+3\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )-2 \log (x)}{2 \text {$\#$1}^7-3 \text {$\#$1}^3}\& \right ]+\frac {1}{4} \sqrt [4]{x^4-x^3} (4 x+11)-\frac {177}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )+\frac {177}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right ) \]
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Rubi [B] time = 1.19, antiderivative size = 678, normalized size of antiderivative = 4.29, number of steps used = 34, number of rules used = 12, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2056, 6728, 50, 63, 240, 212, 206, 203, 101, 157, 93, 298} \begin {gather*} -\sqrt [4]{x^4-x^3} (1-x)+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{x^4-x^3}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{x^4-x^3}+\frac {3}{4} \sqrt [4]{x^4-x^3}+\frac {3 \left (15+7 \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 {3 \left (15-7 \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 {3 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{x-1} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {3 \left (15+7 \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 {3 \left (15-7 \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 {3 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{x-1} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \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 50
Rule 63
Rule 93
Rule 101
Rule 157
Rule 203
Rule 206
Rule 212
Rule 240
Rule 298
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (2-x+2 x^2\right ) \sqrt [4]{-x^3+x^4}}{-2-2 x+x^2} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (2-x+2 x^2\right )}{-2-2 x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \left (2 \sqrt [4]{-1+x} x^{3/4}+\frac {3 \sqrt [4]{-1+x} x^{3/4} (2+x)}{-2-2 x+x^2}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \int \sqrt [4]{-1+x} x^{3/4} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4} (2+x)}{-2-2 x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\left ((1-x) \sqrt [4]{-x^3+x^4}\right )+\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}} \, dx}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [4]{-1+x} x^{3/4}}{-2-2 \sqrt {3}+2 x}+\frac {\left (1-\sqrt {3}\right ) \sqrt [4]{-1+x} x^{3/4}}{-2+2 \sqrt {3}+2 x}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{16 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-2+2 \sqrt {3}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-2-2 \sqrt {3}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \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 )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{2} \left (1-\sqrt {3}\right )+\frac {1}{2} \left (-3+4 \sqrt {3}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2+2 \sqrt {3}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{2} \left (1+\sqrt {3}\right )+\frac {1}{2} \left (-3-4 \sqrt {3}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2-2 \sqrt {3}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \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 )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1-\sqrt {3}\right ) \left (3-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2+2 \sqrt {3}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2-2 \sqrt {3}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (-3+4 \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 (3 \left (1+\sqrt {3}\right ) \left (3+4 \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 {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \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 )}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \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 )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (12 \left (1-\sqrt {3}\right ) \left (3-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {3}-2 \sqrt {3} x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (12 \left (1+\sqrt {3}\right ) \left (3+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-2-2 \sqrt {3}+2 \sqrt {3} x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (-3+4 \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 (3 \left (1+\sqrt {3}\right ) \left (3+4 \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 {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {3 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {3 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1-\sqrt {3}\right ) \left (3-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {3}}-\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (3-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {3}}+\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {3}}-\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {3}}+\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (-3+4 \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 (3 \left (1+\sqrt {3}\right ) \left (3+4 \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 {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {3 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {3 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (-3+4 \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 (3 \left (1-\sqrt {3}\right ) \left (-3+4 \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 (3 \left (1+\sqrt {3}\right ) \left (3+4 \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 (3 \left (1+\sqrt {3}\right ) \left (3+4 \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 {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {3 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {3 \left (15-7 \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 {3 \left (15+7 \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 {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {3 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {3 \left (15-7 \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 {3 \left (15+7 \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 {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 158, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{(x-1) x^3} \left (15 \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};1-x\right )+2 (x-1) \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {5}{4};\frac {9}{4};1-x\right )+15 \left (3 \sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )+\left (\sqrt {3}-2\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {\left (-3+\sqrt {3}\right ) (x-1)}{3 x}\right )-\left (2+\sqrt {3}\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (3+\sqrt {3}\right ) (x-1)}{3 x}\right )\right )\right )}{5 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 158, normalized size = 1.00 \begin {gather*} \frac {1}{4} (11+4 x) \sqrt [4]{-x^3+x^4}-\frac {177}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {177}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-9 \text {RootSum}\left [3-6 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 590, normalized size = 3.73 \begin {gather*} -6 \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {71 \, \sqrt {3} + 123} {\left (19 \, \sqrt {3} - 33\right )} - {\left (19 \, \sqrt {3} x - 33 \, x\right )} \sqrt {71 \, \sqrt {3} + 123} \sqrt {-\frac {\sqrt {2} {\left (4 \, \sqrt {3} x^{2} - 7 \, x^{2}\right )} \sqrt {71 \, \sqrt {3} + 123} - 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}}\right )} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}}}{12 \, x}\right ) + \frac {3}{2} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}} \log \left (\frac {3 \, {\left (\sqrt {2} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - \frac {3}{2} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}} \log \left (-\frac {3 \, {\left (\sqrt {2} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + 2 \, \sqrt {2} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {1}{4}} \arctan \left (-\frac {3 \, \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (19 \, \sqrt {3} + 33\right )} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {3}{4}} - {\left (19 \, \sqrt {3} x + 33 \, x\right )} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {3}{4}} \sqrt {\frac {{\left (4 \, \sqrt {3} x^{2} + 7 \, x^{2}\right )} \sqrt {-11502 \, \sqrt {3} + 19926} + 18 \, \sqrt {x^{4} - x^{3}}}{x^{2}}}}{972 \, x}\right ) - \frac {1}{2} \, \sqrt {2} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (\sqrt {3} x + 2 \, x\right )} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {1}{4}} + 6 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (\sqrt {3} x + 2 \, x\right )} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {1}{4}} - 6 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x + 11\right )} + \frac {177}{8} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {177}{16} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {177}{16} \, \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 [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 30.38, size = 2789, normalized size = 17.65
method | result | size |
trager | \(\text {Expression too large to display}\) | \(2789\) |
risch | \(\text {Expression too large to display}\) | \(5736\) |
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 (2 \, x^{2} - x + 2\right )}}{x^{2} - 2 \, x - 2}\,{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 (2\,x^2-x+2\right )}{-x^2+2\,x+2} \,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 (2 x^{2} - x + 2\right )}{x^{2} - 2 x - 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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