Optimal. Leaf size=121 \[ -\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x+x^5}}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{x+x^5}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{x+x^5}-\sqrt [3]{2} \left (x+x^5\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 0.54, antiderivative size = 463, normalized size of antiderivative = 3.83, number
of steps used = 17, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules
used = {2081, 6847, 6857, 251, 1452, 440, 476, 502, 2174, 206, 31, 648, 631, 210, 642}
\begin {gather*} -\frac {3 x \sqrt [3]{x^4+1} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^4,-x^4\right )}{\sqrt [3]{x^5+x}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{x^4+1} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (x^{4/3}+1\right )}{\sqrt [3]{x^4+1}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^5+x}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{x^4+1} \text {ArcTan}\left (\frac {\frac {\sqrt [3]{2} \left (x^{4/3}+1\right )}{\sqrt [3]{x^4+1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^5+x}}+\frac {3 x \sqrt [3]{x^4+1} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x^5+x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^4+1} \log \left (\left (1-x^{4/3}\right )^2 \left (x^{4/3}+1\right )\right )}{8 \sqrt [3]{2} \sqrt [3]{x^5+x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^4+1} \log \left (\frac {2^{2/3} \left (x^{4/3}+1\right )^2}{\left (x^4+1\right )^{2/3}}-\frac {\sqrt [3]{2} \left (x^{4/3}+1\right )}{\sqrt [3]{x^4+1}}+1\right )}{4 \sqrt [3]{2} \sqrt [3]{x^5+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^4+1} \log \left (\frac {\sqrt [3]{2} \left (x^{4/3}+1\right )}{\sqrt [3]{x^4+1}}+1\right )}{2 \sqrt [3]{2} \sqrt [3]{x^5+x}}-\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \log \left (x^{4/3}-2^{2/3} \sqrt [3]{x^4+1}+1\right )}{8 \sqrt [3]{2} \sqrt [3]{x^5+x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 251
Rule 440
Rule 476
Rule 502
Rule 631
Rule 642
Rule 648
Rule 1452
Rule 2081
Rule 2174
Rule 6847
Rule 6857
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x+x^5}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \int \frac {-1+x^2}{\sqrt [3]{x} \left (1+x^2\right ) \sqrt [3]{1+x^4}} \, dx}{\sqrt [3]{x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \text {Subst}\left (\int \frac {-1+x^3}{\left (1+x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^6}}-\frac {2}{\left (1+x^3\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{\left (1-x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=-\frac {3 x \sqrt [3]{1+x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^4,-x^4\right )}{\sqrt [3]{x+x^5}}+\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{4/3}\right )}{2 \sqrt [3]{x+x^5}}\\ &=-\frac {3 x \sqrt [3]{1+x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^4,-x^4\right )}{\sqrt [3]{x+x^5}}+\frac {3 x^3 \sqrt [3]{1+x^4} F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};x^4,-x^4\right )}{4 \sqrt [3]{x+x^5}}+\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}\\ \end {align*}
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Mathematica [F]
time = 20.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 9.67, size = 1135, normalized size = 9.38
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1135\) |
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. 320 vs.
\(2 (89) = 178\).
time = 2.44, size = 320, normalized size = 2.64 \begin {gather*} \frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (12 \cdot 2^{\frac {1}{6}} \left (-1\right )^{\frac {2}{3}} {\left (x^{8} - 14 \, x^{6} + 6 \, x^{4} - 14 \, x^{2} + 1\right )} {\left (x^{5} + x\right )}^{\frac {2}{3}} - 24 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (x^{9} + x^{7} + x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}} + 2^{\frac {5}{6}} {\left (x^{12} + 24 \, x^{10} - 57 \, x^{8} + 56 \, x^{6} - 57 \, x^{4} + 24 \, x^{2} + 1\right )}\right )}}{6 \, {\left (x^{12} - 48 \, x^{10} + 15 \, x^{8} - 88 \, x^{6} + 15 \, x^{4} - 48 \, x^{2} + 1\right )}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {12 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{5} - x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{8} - 14 \, x^{6} + 6 \, x^{4} - 14 \, x^{2} + 1\right )} - 6 \, {\left (x^{5} + x\right )}^{\frac {2}{3}} {\left (x^{4} - 4 \, x^{2} + 1\right )}}{x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + 2 \, x^{2} + 1\right )} - 3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} + x\right )}^{\frac {2}{3}} + 6 \, {\left (x^{5} + x\right )}^{\frac {1}{3}} x}{x^{4} + 2 \, x^{2} + 1}\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 {\left (x - 1\right ) \left (x + 1\right )}{\sqrt [3]{x \left (x^{4} + 1\right )} \left (x^{2} + 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^2-1}{\left (x^2+1\right )\,{\left (x^5+x\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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