Optimal. Leaf size=25 \[ \frac {\left (-x^4-3\right ) \left (2 x^4-1\right )^{3/4}}{21 x^7} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {453, 264} \begin {gather*} -\frac {\left (2 x^4-1\right )^{3/4}}{7 x^7}-\frac {\left (2 x^4-1\right )^{3/4}}{21 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx &=-\frac {\left (-1+2 x^4\right )^{3/4}}{7 x^7}-\frac {1}{7} \int \frac {1}{x^4 \sqrt [4]{-1+2 x^4}} \, dx\\ &=-\frac {\left (-1+2 x^4\right )^{3/4}}{7 x^7}-\frac {\left (-1+2 x^4\right )^{3/4}}{21 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 23, normalized size = 0.92 \begin {gather*} -\frac {\left (x^4+3\right ) \left (2 x^4-1\right )^{3/4}}{21 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.17, size = 25, normalized size = 1.00 \begin {gather*} \frac {\left (-3-x^4\right ) \left (-1+2 x^4\right )^{3/4}}{21 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 19, normalized size = 0.76 \begin {gather*} -\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} + 3\right )}}{21 \, x^{7}} \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^{4} - 1}{{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 20, normalized size = 0.80
method | result | size |
gosper | \(-\frac {\left (x^{4}+3\right ) \left (2 x^{4}-1\right )^{\frac {3}{4}}}{21 x^{7}}\) | \(20\) |
trager | \(-\frac {\left (x^{4}+3\right ) \left (2 x^{4}-1\right )^{\frac {3}{4}}}{21 x^{7}}\) | \(20\) |
risch | \(-\frac {2 x^{8}+5 x^{4}-3}{21 x^{7} \left (2 x^{4}-1\right )^{\frac {1}{4}}}\) | \(27\) |
meijerg | \(\frac {\left (-\mathrm {signum}\left (2 x^{4}-1\right )\right )^{\frac {1}{4}} \left (1+\frac {8 x^{4}}{3}\right ) \left (-2 x^{4}+1\right )^{\frac {3}{4}}}{7 \mathrm {signum}\left (2 x^{4}-1\right )^{\frac {1}{4}} x^{7}}-\frac {\left (-\mathrm {signum}\left (2 x^{4}-1\right )\right )^{\frac {1}{4}} \left (-2 x^{4}+1\right )^{\frac {3}{4}}}{3 \mathrm {signum}\left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 29, normalized size = 1.16 \begin {gather*} -\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} + \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.20, size = 30, normalized size = 1.20 \begin {gather*} -\frac {x^4\,{\left (2\,x^4-1\right )}^{3/4}+3\,{\left (2\,x^4-1\right )}^{3/4}}{21\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 2.11, size = 233, normalized size = 9.32 \begin {gather*} \begin {cases} - \frac {2^{\frac {3}{4}} \left (-1 + \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {3}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \frac {1}{2 \left |{x^{4}}\right |} > 1 \\- \frac {2^{\frac {3}{4}} \left (1 - \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} - \begin {cases} - \frac {2^{\frac {3}{4}} \left (-1 + \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{2 \Gamma \left (\frac {1}{4}\right )} - \frac {3 \cdot 2^{\frac {3}{4}} \left (-1 + \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{16 x^{4} \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \frac {1}{2 \left |{x^{4}}\right |} > 1 \\\frac {2^{\frac {3}{4}} \left (1 - \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 \Gamma \left (\frac {1}{4}\right )} + \frac {3 \cdot 2^{\frac {3}{4}} \left (1 - \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{16 x^{4} \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________