Optimal. Leaf size=230 \[ \frac {1}{16} \sqrt [4]{x^4-x^3} (4 x-5)-\frac {57}{32} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )+\frac {4}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {57}{32} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )-\frac {4}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^4-x^3}}{\sqrt {x^4-x^3}-x^2}\right )}{12 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^4-x^3}}{\sqrt {2}}}{x \sqrt [4]{x^4-x^3}}\right )}{12 \sqrt {2}} \]
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Rubi [B] time = 1.05, antiderivative size = 480, normalized size of antiderivative = 2.09, number of steps used = 40, number of rules used = 18, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {2056, 6728, 50, 63, 240, 212, 206, 203, 101, 157, 93, 297, 1162, 617, 204, 1165, 628, 298} \begin {gather*} -\frac {1}{4} \sqrt [4]{x^4-x^3} (1-x)-\frac {1}{16} \sqrt [4]{x^4-x^3}+\frac {5 \sqrt [4]{x^4-x^3} \log \left (\frac {\sqrt {x}}{\sqrt {x-1}}-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{x-1}}+1\right )}{24 \sqrt {2} \sqrt [4]{x-1} x^{3/4}}-\frac {5 \sqrt [4]{x^4-x^3} \log \left (\frac {\sqrt {x}}{\sqrt {x-1}}+\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{x-1}}+1\right )}{24 \sqrt {2} \sqrt [4]{x-1} x^{3/4}}-\frac {5 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{12 \sqrt {2} \sqrt [4]{x-1} x^{3/4}}+\frac {5 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{x-1}}+1\right )}{12 \sqrt {2} \sqrt [4]{x-1} x^{3/4}}+\frac {57 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{x-1} x^{3/4}}+\frac {4 \sqrt [4]{2} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {57 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{x-1} x^{3/4}}-\frac {4 \sqrt [4]{2} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 93
Rule 101
Rule 157
Rule 203
Rule 204
Rule 206
Rule 212
Rule 240
Rule 297
Rule 298
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (1+x^2\right )}{-1+x+2 x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \left (\frac {1}{2} \sqrt [4]{-1+x} x^{3/4}+\frac {(3-x) \sqrt [4]{-1+x} x^{3/4}}{2 \left (-1+x+2 x^2\right )}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \sqrt [4]{-1+x} x^{3/4} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \int \frac {(3-x) \sqrt [4]{-1+x} x^{3/4}}{-1+x+2 x^2} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}+\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}} \, dx}{16 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \int \left (\frac {10 \sqrt [4]{-1+x} x^{3/4}}{3 (-2+4 x)}-\frac {16 \sqrt [4]{-1+x} x^{3/4}}{3 (4+4 x)}\right ) \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{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}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-2+4 x} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (8 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{4+4 x} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{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 )}{16 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{2}-x}{(-1+x)^{3/4} \sqrt [4]{x} (-2+4 x)} \, dx}{12 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \int \frac {-3+5 x}{(-1+x)^{3/4} \sqrt [4]{x} (4+4 x)} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{48 \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^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{16 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} (-2+4 x)} \, dx}{12 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{6 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (16 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} (4+4 x)} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{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 )}{32 \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 )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \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 )}{12 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-2-2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (10 \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 )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (64 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{4-8 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{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 )}{32 \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 )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \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 )}{12 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{-2-2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{6 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{-2-2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{6 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (10 \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 )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt {2} \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt {2} \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{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 )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \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 )}{32 \sqrt [4]{-1+x} x^{3/4}}-\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \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 )}{24 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \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 )}{24 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{24 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{24 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \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 )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \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 )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}+\frac {57 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {57 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}-\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {5 \sqrt [4]{-x^3+x^4} \log \left (1-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}+\frac {\sqrt {x}}{\sqrt {-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}-\frac {5 \sqrt [4]{-x^3+x^4} \log \left (1+\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}+\frac {\sqrt {x}}{\sqrt {-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{12 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{12 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}-\frac {5 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{12 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}+\frac {5 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{12 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}+\frac {57 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {57 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}-\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {5 \sqrt [4]{-x^3+x^4} \log \left (1-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}+\frac {\sqrt {x}}{\sqrt {-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}-\frac {5 \sqrt [4]{-x^3+x^4} \log \left (1+\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}+\frac {\sqrt {x}}{\sqrt {-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 126, normalized size = 0.55 \begin {gather*} \frac {\sqrt [4]{(x-1) x^3} \left (-30 \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};1-x\right )+12 (x-1) \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {5}{4};\frac {9}{4};1-x\right )+5 \left (27 \sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )-5 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1}{x}-1\right )-16 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {x-1}{2 x}\right )\right )\right )}{30 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.69, size = 230, normalized size = 1.00 \begin {gather*} \frac {1}{16} (-5+4 x) \sqrt [4]{-x^3+x^4}-\frac {57}{32} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {4}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-x^3+x^4}}{-x^2+\sqrt {-x^3+x^4}}\right )}{12 \sqrt {2}}+\frac {57}{32} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {4}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {5 \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-x^3+x^4}}{\sqrt {2}}}{x \sqrt [4]{-x^3+x^4}}\right )}{12 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 423, normalized size = 1.84 \begin {gather*} \frac {5}{12} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {\frac {x^{2} + \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x + \sqrt {x^{4} - x^{3}}}{x^{2}}} - x - \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {5}{12} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {\frac {x^{2} - \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x + \sqrt {x^{4} - x^{3}}}{x^{2}}} + x - \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {5}{48} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{2} + \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x + \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \frac {5}{48} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{2} - \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x + \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \frac {1}{16} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x - 5\right )} + \frac {8}{3} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x \sqrt {\frac {\sqrt {2} x^{2} + \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2^{\frac {3}{4}} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {57}{32} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {57}{64} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {57}{64} \, \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 [A] time = 0.48, size = 244, normalized size = 1.06 \begin {gather*} -\frac {1}{16} \, {\left (5 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} + \frac {1}{3} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {5}{24} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{24} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{48} \, \sqrt {2} \log \left (\sqrt {2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {5}{48} \, \sqrt {2} \log \left (-\sqrt {2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {57}{32} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {57}{64} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {57}{64} \, \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 [C] time = 5.30, size = 851, normalized size = 3.70
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trager | \(\left (-\frac {5}{16}+\frac {x}{4}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\frac {2 \RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )}{3}-\frac {2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )}{3}+\frac {57 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-4 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{128}-\frac {5 \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{4}-x^{3}}\, x +2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+4 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2} \left (-1+2 x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}}{48}+\frac {5 \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{4}-x^{3}}\, x +2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+4 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2} \left (-1+2 x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{48}-\frac {5 \RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{4}-x^{3}}\, x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+2 x \right )}\right )}{24}-\frac {57 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-2 x^{3}+x^{2}}{x^{2}}\right )}{64}\) | \(851\) |
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^{2} + 1\right )}}{2 \, 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 = -0.00 \begin {gather*} \int \frac {\left (x^2+1\right )\,{\left (x^4-x^3\right )}^{1/4}}{2\,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^{2} + 1\right )}{\left (x + 1\right ) \left (2 x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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