Optimal. Leaf size=187 \[ \frac {\left (x^2+x^6\right )^{3/4}}{2 x \left (1+x^4\right )}-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{8 \sqrt [4]{2}}+\frac {\text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{8\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{8 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{8\ 2^{3/4}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 0.19, antiderivative size = 46, normalized size of antiderivative = 0.25, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2081, 1493,
477, 524} \begin {gather*} -\frac {2 x^5 \sqrt [4]{x^4+1} F_1\left (\frac {9}{8};1,\frac {5}{4};\frac {17}{8};x^4,-x^4\right )}{9 \sqrt [4]{x^6+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 477
Rule 524
Rule 1493
Rule 2081
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/2}}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/2}}{\left (-1+x^4\right ) \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (-1+x^8\right ) \left (1+x^8\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {2 x^5 \sqrt [4]{1+x^4} F_1\left (\frac {9}{8};1,\frac {5}{4};\frac {17}{8};x^4,-x^4\right )}{9 \sqrt [4]{x^2+x^6}}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 218, normalized size = 1.17 \begin {gather*} \frac {\sqrt {x} \left (8 \sqrt {x}-2^{3/4} \sqrt [4]{1+x^4} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt [4]{2} \sqrt [4]{1+x^4} \text {ArcTan}\left (\frac {2^{3/4} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt {2} x-\sqrt {1+x^4}}\right )-2^{3/4} \sqrt [4]{1+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \sqrt [4]{1+x^4} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+\sqrt {2} \sqrt {1+x^4}}\right )\right )}{16 \sqrt [4]{x^2+x^6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 44.78, size = 651, normalized size = 3.48
method | result | size |
risch | \(\text {Expression too large to display}\) | \(651\) |
trager | \(\text {Expression too large to display}\) | \(655\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 765 vs.
\(2 (144) = 288\).
time = 3.98, size = 765, normalized size = 4.09 \begin {gather*} \frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{5} + x\right )} \arctan \left (\frac {2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{2 \, {\left (x^{5} + x\right )}}\right ) - 2^{\frac {3}{4}} {\left (x^{5} + x\right )} \log \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) + 2^{\frac {3}{4}} {\left (x^{5} + x\right )} \log \left (-\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) - 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} + x\right )} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {x^{5} + 2 \, x^{3} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + 2 \, x^{3} + x}} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{2 \, {\left (x^{5} - 2 \, x^{3} + x\right )}}\right ) - 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} + x\right )} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {x^{5} + 2 \, x^{3} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x - 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + 2 \, x^{3} + x}} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{2 \, {\left (x^{5} - 2 \, x^{3} + x\right )}}\right ) - 2^{\frac {1}{4}} {\left (x^{5} + x\right )} \log \left (\frac {2 \, {\left (x^{5} + 2 \, x^{3} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + 2 \, x^{3} + x}\right ) + 2^{\frac {1}{4}} {\left (x^{5} + x\right )} \log \left (\frac {2 \, {\left (x^{5} + 2 \, x^{3} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x - 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + 2 \, x^{3} + x}\right ) + 32 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{64 \, {\left (x^{5} + x\right )}} \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 {x^{4}}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^8-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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