Optimal. Leaf size=134 \[ -\frac {9869 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )}{4096}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {9869 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )}{4096}-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {\sqrt [4]{x^4-x^3} \left (6144 x^4-8064 x^3+10400 x^2-1060 x-32575\right )}{30720} \]
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Rubi [C] time = 0.78, antiderivative size = 223, normalized size of antiderivative = 1.66, number of steps used = 35, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2056, 6733, 6725, 279, 331, 298, 203, 206, 321, 511, 510} \begin {gather*} \frac {4 \sqrt [4]{x^4-x^3} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}+\frac {1}{5} \sqrt [4]{x^4-x^3} x^4-\frac {21}{80} \sqrt [4]{x^4-x^3} x^3-\frac {53 \sqrt [4]{x^4-x^3} x}{1536}-\frac {6515 \sqrt [4]{x^4-x^3}}{6144}-\frac {1677 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{4096 \sqrt [4]{x-1} x^{3/4}}+\frac {1677 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{4096 \sqrt [4]{x-1} x^{3/4}}+\frac {65}{192} \sqrt [4]{x^4-x^3} x^2 \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 279
Rule 298
Rule 321
Rule 331
Rule 510
Rule 511
Rule 2056
Rule 6725
Rule 6733
Rubi steps
\begin {align*} \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (-1+x+x^4\right )}{1+x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{-1+x^4} \left (-1+x^4+x^{16}\right )}{1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-x^2 \sqrt [4]{-1+x^4}+x^{10} \sqrt [4]{-1+x^4}-x^{14} \sqrt [4]{-1+x^4}+x^{18} \sqrt [4]{-1+x^4}+\frac {x^2 \sqrt [4]{-1+x^4}}{1+x^4}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int x^2 \sqrt [4]{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int x^{10} \sqrt [4]{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int x^{14} \sqrt [4]{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int x^{18} \sqrt [4]{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{1-x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {x^{18}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{5 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {x^{14}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\sqrt [4]{-x^3+x^4}-\frac {1}{24} x \sqrt [4]{-x^3+x^4}+\frac {17}{48} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{16 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (11 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{48 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (7 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{24 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {103}{96} \sqrt [4]{-x^3+x^4}-\frac {5}{384} x \sqrt [4]{-x^3+x^4}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {\left (11 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{384 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (7 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1571 \sqrt [4]{-x^3+x^4}}{1536}-\frac {53 x \sqrt [4]{-x^3+x^4}}{1536}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {\sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{512 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{512 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (7 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {6515 \sqrt [4]{-x^3+x^4}}{6144}-\frac {53 x \sqrt [4]{-x^3+x^4}}{1536}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {\sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (7 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (231 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{2048 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{512 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {6515 \sqrt [4]{-x^3+x^4}}{6144}-\frac {53 x \sqrt [4]{-x^3+x^4}}{1536}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {25 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {25 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{1024 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{1024 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (231 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2048 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {6515 \sqrt [4]{-x^3+x^4}}{6144}-\frac {53 x \sqrt [4]{-x^3+x^4}}{1536}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {477 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{1024 \sqrt [4]{-1+x} x^{3/4}}+\frac {477 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{1024 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (231 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{4096 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (231 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{4096 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {6515 \sqrt [4]{-x^3+x^4}}{6144}-\frac {53 x \sqrt [4]{-x^3+x^4}}{1536}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {1677 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{4096 \sqrt [4]{-1+x} x^{3/4}}+\frac {1677 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{4096 \sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 118, normalized size = 0.88 \begin {gather*} \frac {x^3 \left (148035 x \left (1-x^2\right )^{3/4} F_1\left (\frac {7}{4};\frac {3}{4},1;\frac {11}{4};x,-x\right )-228025 (1-x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {2 x}{x+1}\right )+7 (x+1)^{3/4} \left (6144 x^5-14208 x^4+18464 x^3-11460 x^2-31515 x+32575\right )\right )}{215040 \left ((x-1) x^3\right )^{3/4} (x+1)^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.76, size = 134, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-x^3+x^4} \left (-32575-1060 x+10400 x^2-8064 x^3+6144 x^4\right )}{30720}-\frac {9869 \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{4096}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {9869 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{4096}-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 209, normalized size = 1.56 \begin {gather*} \frac {1}{30720} \, {\left (6144 \, x^{4} - 8064 \, x^{3} + 10400 \, x^{2} - 1060 \, x - 32575\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} + 4 \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 ) - 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {9869}{4096} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {9869}{8192} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {9869}{8192} \, \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 = 0.22, size = 184, normalized size = 1.37 \begin {gather*} \frac {1}{30720} \, {\left (32575 \, {\left (\frac {1}{x} - 1\right )}^{4} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 131360 \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 188230 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 120744 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 25155 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{5} + 2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {9869}{4096} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {9869}{8192} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {9869}{8192} \, \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 = 3.58, size = 480, normalized size = 3.58
method | result | size |
trager | \(\left (\frac {1}{5} x^{4}-\frac {21}{80} x^{3}+\frac {65}{192} x^{2}-\frac {53}{1536} x -\frac {6515}{6144}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\frac {9869 \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 )}{8192}-\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} 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 )+\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 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} 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 )+\frac {9869 \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}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} 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 )}{16384}\) | \(480\) |
risch | \(\frac {\left (6144 x^{4}-8064 x^{3}+10400 x^{2}-1060 x -32575\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{30720}+\frac {\left (-\frac {9869 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right )^{2}}\right )}{8192}+\frac {9869 \ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+2 x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+4 x -1}{\left (-1+x \right )^{2}}\right )}{8192}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}+7 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-5 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right )^{2} \left (1+x \right )}\right )-\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x -2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}-7 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+5 \RootOf \left (\textit {\_Z}^{4}-2\right ) x -\RootOf \left (\textit {\_Z}^{4}-2\right )}{\left (-1+x \right )^{2} \left (1+x \right )}\right )\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\) | \(934\) |
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^{4} + x - 1\right )}}{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^4+x-1\right )}{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^{4} + x - 1\right )}{x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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