Optimal. Leaf size=121 \[ \frac {1}{4} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-3+2 \text {$\#$1}^4}\& \right ]+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\& \right ] \]
[Out]
________________________________________________________________________________________
Rubi [C] Result contains complex when optimal does not.
time = 0.71, antiderivative size = 333, normalized size of antiderivative = 2.75, number of steps
used = 17, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2081, 6860,
1284, 1543, 525, 524} \begin {gather*} \frac {\left (\sqrt {3}+3 i\right ) x \sqrt [4]{x^4-x^2} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,-\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )}{9 \left (\sqrt {3}+i\right ) \sqrt [4]{1-x^2}}+\frac {\left (\sqrt {3}+3 i\right ) x \sqrt [4]{x^4-x^2} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )}{9 \left (\sqrt {3}+i\right ) \sqrt [4]{1-x^2}}+\frac {\left (-\sqrt {3}+3 i\right ) x \sqrt [4]{x^4-x^2} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,-\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )}{9 \left (-\sqrt {3}+i\right ) \sqrt [4]{1-x^2}}+\frac {\left (-\sqrt {3}+3 i\right ) x \sqrt [4]{x^4-x^2} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )}{9 \left (-\sqrt {3}+i\right ) \sqrt [4]{1-x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 524
Rule 525
Rule 1284
Rule 1543
Rule 2081
Rule 6860
Rubi steps
\begin {align*} \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^2+x^4}}{1+x^4+x^8} \, dx &=\frac {\sqrt [4]{-x^2+x^4} \int \frac {\sqrt {x} \sqrt [4]{-1+x^2} \left (1+x^4\right )}{1+x^4+x^8} \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\sqrt [4]{-x^2+x^4} \int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \sqrt {x} \sqrt [4]{-1+x^2}}{1-i \sqrt {3}+2 x^4}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \sqrt {x} \sqrt [4]{-1+x^2}}{1+i \sqrt {3}+2 x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{-1+x^2}}{1-i \sqrt {3}+2 x^4} \, dx}{3 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{-1+x^2}}{1+i \sqrt {3}+2 x^4} \, dx}{3 \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\left (2 \left (3-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{1-i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \left (3+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\left (2 \left (3-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1+i \sqrt {3}\right )} \left (-\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4\right )}-\frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1+i \sqrt {3}\right )} \left (\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \left (3+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1-i \sqrt {3}\right )} \left (-\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4\right )}-\frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1-i \sqrt {3}\right )} \left (\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{-\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{3 \sqrt {\frac {1}{2} \left (-1+i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{3 \sqrt {\frac {1}{2} \left (-1+i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{-\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{3 \sqrt {\frac {1}{2} \left (-1-i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{3 \sqrt {\frac {1}{2} \left (-1-i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{-\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{3 \sqrt {\frac {1}{2} \left (-1+i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{1-x^2}}-\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{3 \sqrt {\frac {1}{2} \left (-1+i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{1-x^2}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{-\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{3 \sqrt {\frac {1}{2} \left (-1-i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{1-x^2}}-\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{3 \sqrt {\frac {1}{2} \left (-1-i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{1-x^2}}\\ &=\frac {\left (3 i+\sqrt {3}\right ) x \sqrt [4]{-x^2+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,-\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )}{9 \left (i+\sqrt {3}\right ) \sqrt [4]{1-x^2}}+\frac {\left (3 i+\sqrt {3}\right ) x \sqrt [4]{-x^2+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )}{9 \left (i+\sqrt {3}\right ) \sqrt [4]{1-x^2}}+\frac {\left (3 i-\sqrt {3}\right ) x \sqrt [4]{-x^2+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,-\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )}{9 \left (i-\sqrt {3}\right ) \sqrt [4]{1-x^2}}+\frac {\left (3 i-\sqrt {3}\right ) x \sqrt [4]{-x^2+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )}{9 \left (i-\sqrt {3}\right ) \sqrt [4]{1-x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 152, normalized size = 1.26 \begin {gather*} \frac {x^{3/2} \left (-1+x^2\right )^{3/4} \left (\text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-3+2 \text {$\#$1}^4}\&\right ]+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\right ]\right )}{4 \left (x^2 \left (-1+x^2\right )\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 100.24, size = 6392, normalized size = 52.83
method | result | size |
trager | \(\text {Expression too large to display}\) | \(6392\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x^{4} + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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+1\right )\,{\left (x^4-x^2\right )}^{1/4}}{x^8+x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________