Optimal. Leaf size=19 \[ \frac {\left (x^8+x^4+1\right )^{3/4}}{3 x^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1590} \begin {gather*} \frac {\left (x^8+x^4+1\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1590
Rubi steps
\begin {align*} \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx &=\frac {\left (1+x^4+x^8\right )^{3/4}}{3 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 19, normalized size = 1.00 \begin {gather*} \frac {\left (x^8+x^4+1\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.39, size = 19, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^4+x^8\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\left (x^{8} + x^{4} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \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*} \int \frac {x^{8} - 1}{{\left (x^{8} + x^{4} + 1\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 16, normalized size = 0.84
method | result | size |
trager | \(\frac {\left (x^{8}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
risch | \(\frac {\left (x^{8}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
gosper | \(\frac {\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{4}-x^{2}+1\right )}{3 \left (x^{8}+x^{4}+1\right )^{\frac {1}{4}} x^{3}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.56, size = 43, normalized size = 2.26 \begin {gather*} \frac {x^{8} + x^{4} + 1}{3 \, {\left (x^{4} - x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.15, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\left (x^8+x^4+1\right )}^{3/4}}{3\,x^3} \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 {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{x^{4} \sqrt [4]{\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]
________________________________________________________________________________________