Optimal. Leaf size=156 \[ -\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )-2 \log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]-\frac {4}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {4}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right ) \]
________________________________________________________________________________________
Rubi [C] time = 0.69, antiderivative size = 746, normalized size of antiderivative = 4.78, number of steps used = 54, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2056, 6725, 105, 50, 63, 240, 212, 206, 203, 93, 298} \begin {gather*} \frac {1}{3} (-1)^{2/3} \sqrt [4]{x^4-x^3}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{x^4-x^3}+\frac {1}{3} \sqrt [4]{x^4-x^3}-\frac {(-1)^{2/3} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [3]{-1} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \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 {2^{3/4} \sqrt [4]{1-i \sqrt {3}} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{1-\sqrt [3]{-1}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{1+i \sqrt {3}} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}-\frac {(-1)^{2/3} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [3]{-1} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \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}}-\frac {2^{3/4} \sqrt [4]{1-i \sqrt {3}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1-\sqrt [3]{-1}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{1+i \sqrt {3}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 93
Rule 105
Rule 203
Rule 206
Rule 212
Rule 240
Rule 298
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {(-1+x)^{5/4}}{\sqrt [4]{x} \left (1+x^3\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \left (-\frac {(-1+x)^{5/4}}{3 (-1-x) \sqrt [4]{x}}-\frac {(-1+x)^{5/4}}{3 \sqrt [4]{x} \left (-1+\sqrt [3]{-1} x\right )}-\frac {(-1+x)^{5/4}}{3 \sqrt [4]{x} \left (-1-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {\sqrt [4]{-x^3+x^4} \int \frac {(-1+x)^{5/4}}{(-1-x) \sqrt [4]{x}} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \int \frac {(-1+x)^{5/4}}{\sqrt [4]{x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \int \frac {(-1+x)^{5/4}}{\sqrt [4]{x} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{(-1-x) \sqrt [4]{x}} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\sqrt [3]{-1} \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left ((-1)^{2/3} \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (1-\sqrt [3]{-1}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{3} \sqrt [4]{-x^3+x^4}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{-x^3+x^4}+\frac {1}{3} (-1)^{2/3} \sqrt [4]{-x^3+x^4}-\frac {\sqrt [4]{-x^3+x^4} \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{12 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1-x) (-1+x)^{3/4} \sqrt [4]{x}} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\sqrt [3]{-1} \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{12 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left ((-1)^{2/3} \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{12 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\sqrt [3]{-1} \left (1-\sqrt [3]{-1}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-\sqrt [3]{-1}\right )^2 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\right )^2 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{3} \sqrt [4]{-x^3+x^4}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{-x^3+x^4}+\frac {1}{3} (-1)^{2/3} \sqrt [4]{-x^3+x^4}-\frac {\sqrt [4]{-x^3+x^4} \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 (8 \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 (16 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\sqrt [3]{-1} \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 ((-1)^{2/3} \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 )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [3]{-1} \left (1-\sqrt [3]{-1}\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 )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \left (1-\sqrt [3]{-1}\right )^2 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1-\left (-1-(-1)^{2/3}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\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 )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\right )^2 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1-\left (-1+\sqrt [3]{-1}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{3} \sqrt [4]{-x^3+x^4}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{-x^3+x^4}+\frac {1}{3} (-1)^{2/3} \sqrt [4]{-x^3+x^4}-\frac {\sqrt [4]{-x^3+x^4} \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 (8 \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 (\sqrt [3]{-1} \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 ((-1)^{2/3} \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 )}{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 {\left (4 \sqrt [3]{-1} \left (1-\sqrt [3]{-1}\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 )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\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 )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\right )^2 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {1-\sqrt [3]{-1}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt {1-\sqrt [3]{-1}} \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\right )^2 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {1-\sqrt [3]{-1}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt {1-\sqrt [3]{-1}} \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \left (1-\sqrt [3]{-1}\right )^2 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {1+(-1)^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt {1+(-1)^{2/3}} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 \left (1-\sqrt [3]{-1}\right )^2 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {1+(-1)^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt {1+(-1)^{2/3}} \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{3} \sqrt [4]{-x^3+x^4}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{-x^3+x^4}+\frac {1}{3} (-1)^{2/3} \sqrt [4]{-x^3+x^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 {2^{3/4} \sqrt [4]{1-i \sqrt {3}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1-\sqrt [3]{-1}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {2 (-1)^{2/3} \left (1+(-1)^{2/3}\right )^{5/4} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \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 {2^{3/4} \sqrt [4]{1-i \sqrt {3}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1-\sqrt [3]{-1}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {2 (-1)^{2/3} \left (1+(-1)^{2/3}\right )^{5/4} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \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]{-1+x}}{\sqrt [4]{x}}\right )}{6 \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]{-1+x}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \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 (4 \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 (\sqrt [3]{-1} \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 )}{6 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\sqrt [3]{-1} \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 )}{6 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left ((-1)^{2/3} \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 )}{6 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left ((-1)^{2/3} \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 )}{6 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [3]{-1} \left (1-\sqrt [3]{-1}\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 )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [3]{-1} \left (1-\sqrt [3]{-1}\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 )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\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 )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\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 )}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{3} \sqrt [4]{-x^3+x^4}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{-x^3+x^4}+\frac {1}{3} (-1)^{2/3} \sqrt [4]{-x^3+x^4}-\frac {\sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [3]{-1} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{-1+x} x^{3/4}}-\frac {(-1)^{2/3} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{6 \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 {2^{3/4} \sqrt [4]{1-i \sqrt {3}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1-\sqrt [3]{-1}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {2 (-1)^{2/3} \left (1+(-1)^{2/3}\right )^{5/4} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [3]{-1} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{-1+x} x^{3/4}}-\frac {(-1)^{2/3} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{6 \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 {2^{3/4} \sqrt [4]{1-i \sqrt {3}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1-\sqrt [3]{-1}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {2 (-1)^{2/3} \left (1+(-1)^{2/3}\right )^{5/4} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.08, size = 101, normalized size = 0.65 \begin {gather*} \frac {4 \sqrt [4]{(x-1) x^3} \left (2 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {x-1}{2 x}\right )-\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {2 i (x-1)}{\left (-i+\sqrt {3}\right ) x}\right )-\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {2 i (x-1)}{\left (i+\sqrt {3}\right ) x}\right )\right )}{3 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.00, size = 156, normalized size = 1.00 \begin {gather*} -\frac {4}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} 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 {1}{3} \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.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.55, size = 881, normalized size = 5.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.31, size = 392, normalized size = 2.51 \begin {gather*} \frac {1}{6} \, {\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}{6} \, {\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}{6} \, {\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}{6} \, {\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}{12} \, {\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}{12} \, {\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}{12} \, {\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}{12} \, {\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}{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 {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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 92.96, size = 16806, normalized size = 107.73
method | result | size |
trager | \(\text {Expression too large to display}\) | \(16806\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}}{{\left (x^{3} + 1\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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\,\left (x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________