Optimal. Leaf size=74 \[ \frac {1}{18} \tan ^{-1}\left (\frac {(1-x)^2}{3 \sqrt {-1+x^3}}\right )+\frac {1}{18} \tan ^{-1}\left (\frac {1}{3} \sqrt {-1+x^3}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} (1-x)}{\sqrt {-1+x^3}}\right )}{6 \sqrt {3}} \]
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Rubi [A]
time = 0.09, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {499, 455, 65,
210, 2163, 209, 2170, 212} \begin {gather*} \frac {1}{18} \text {ArcTan}\left (\frac {(1-x)^2}{3 \sqrt {x^3-1}}\right )+\frac {1}{18} \text {ArcTan}\left (\frac {\sqrt {x^3-1}}{3}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} (1-x)}{\sqrt {x^3-1}}\right )}{6 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 210
Rule 212
Rule 455
Rule 499
Rule 2163
Rule 2170
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-1+x^3} \left (8+x^3\right )} \, dx &=-\left (\frac {1}{12} \int \frac {1-x}{(2+x) \sqrt {-1+x^3}} \, dx\right )-\frac {1}{12} \int \frac {-2-2 x+x^2}{\left (4-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx-\frac {1}{4} \int \frac {x^2}{\left (-8-x^3\right ) \sqrt {-1+x^3}} \, dx\\ &=-\left (\frac {1}{12} \text {Subst}\left (\int \frac {1}{(-8-x) \sqrt {-1+x}} \, dx,x,x^3\right )\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1}{9+x^2} \, dx,x,\frac {(1-x)^2}{\sqrt {-1+x^3}}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{2-6 x^2} \, dx,x,\frac {1-x}{\sqrt {-1+x^3}}\right )\\ &=\frac {1}{18} \tan ^{-1}\left (\frac {(1-x)^2}{3 \sqrt {-1+x^3}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} (1-x)}{\sqrt {-1+x^3}}\right )}{6 \sqrt {3}}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{-9-x^2} \, dx,x,\sqrt {-1+x^3}\right )\\ &=\frac {1}{18} \tan ^{-1}\left (\frac {(1-x)^2}{3 \sqrt {-1+x^3}}\right )+\frac {1}{18} \tan ^{-1}\left (\frac {1}{3} \sqrt {-1+x^3}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} (1-x)}{\sqrt {-1+x^3}}\right )}{6 \sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 10.03, size = 48, normalized size = 0.65 \begin {gather*} \frac {x^2 \sqrt {1-x^3} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};x^3,-\frac {x^3}{8}\right )}{16 \sqrt {-1+x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 2.23, size = 421, normalized size = 5.69
method | result | size |
default | \(-\frac {\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \sqrt {x^{3}-1}}+\frac {i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1+i \sqrt {3}\right ) \sqrt {3}}{6}+\frac {i \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \sqrt {x^{3}-1}}-\frac {i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1-i \sqrt {3}\right ) \sqrt {3}}{6}-\frac {2 i \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \sqrt {x^{3}-1}}\) | \(421\) |
trager | \(\text {Expression too large to display}\) | \(511\) |
elliptic | \(\frac {\sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{6 \sqrt {x^{3}-1}}+\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{18 \sqrt {x^{3}-1}}+\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1+i \sqrt {3}\right ) \sqrt {3}}{6}+\frac {i \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{6 \sqrt {x^{3}-1}}+\frac {\sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1+i \sqrt {3}\right ) \sqrt {3}}{6}+\frac {i \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{6 \sqrt {x^{3}-1}}+\frac {\sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1-i \sqrt {3}\right ) \sqrt {3}}{6}-\frac {2 i \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}\) | \(889\) |
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. 547 vs.
\(2 (51) = 102\).
time = 1.45, size = 547, normalized size = 7.39 \begin {gather*} \frac {1}{216} \, \sqrt {3} \log \left (\frac {4 \, {\left (x^{6} + 48 \, x^{5} + 186 \, x^{4} - 56 \, x^{3} + 6 \, \sqrt {3} {\left (x^{4} + 12 \, x^{3} + 12 \, x^{2} - 16 \, x\right )} \sqrt {x^{3} - 1} - 120 \, x^{2} - 96 \, x + 64\right )}}{x^{6} - 6 \, x^{5} + 24 \, x^{4} - 56 \, x^{3} + 96 \, x^{2} - 96 \, x + 64}\right ) - \frac {1}{216} \, \sqrt {3} \log \left (\frac {4 \, {\left (x^{6} + 48 \, x^{5} + 186 \, x^{4} - 56 \, x^{3} - 6 \, \sqrt {3} {\left (x^{4} + 12 \, x^{3} + 12 \, x^{2} - 16 \, x\right )} \sqrt {x^{3} - 1} - 120 \, x^{2} - 96 \, x + 64\right )}}{x^{6} - 6 \, x^{5} + 24 \, x^{4} - 56 \, x^{3} + 96 \, x^{2} - 96 \, x + 64}\right ) + \frac {1}{54} \, \arctan \left (\frac {{\left (x^{3} - 12 \, x^{2} - 6 \, x - 10\right )} \sqrt {x^{3} - 1}}{6 \, {\left (x^{4} - x^{3} - x + 1\right )}}\right ) - \frac {1}{54} \, \arctan \left (-\frac {\sqrt {x^{3} - 1} {\left (x^{2} - 8 \, x + 10\right )} + {\left (3 \, \sqrt {3} {\left (x^{3} + x^{2} - 2 \, x\right )} - \sqrt {x^{3} - 1} {\left (x^{2} + 10 \, x - 8\right )}\right )} \sqrt {\frac {x^{6} + 48 \, x^{5} + 186 \, x^{4} - 56 \, x^{3} + 6 \, \sqrt {3} {\left (x^{4} + 12 \, x^{3} + 12 \, x^{2} - 16 \, x\right )} \sqrt {x^{3} - 1} - 120 \, x^{2} - 96 \, x + 64}{x^{6} - 6 \, x^{5} + 24 \, x^{4} - 56 \, x^{3} + 96 \, x^{2} - 96 \, x + 64}}}{3 \, {\left (x^{3} - 3 \, x^{2} + 2\right )}}\right ) - \frac {1}{54} \, \arctan \left (-\frac {\sqrt {x^{3} - 1} {\left (x^{2} - 8 \, x + 10\right )} - {\left (3 \, \sqrt {3} {\left (x^{3} + x^{2} - 2 \, x\right )} + \sqrt {x^{3} - 1} {\left (x^{2} + 10 \, x - 8\right )}\right )} \sqrt {\frac {x^{6} + 48 \, x^{5} + 186 \, x^{4} - 56 \, x^{3} - 6 \, \sqrt {3} {\left (x^{4} + 12 \, x^{3} + 12 \, x^{2} - 16 \, x\right )} \sqrt {x^{3} - 1} - 120 \, x^{2} - 96 \, x + 64}{x^{6} - 6 \, x^{5} + 24 \, x^{4} - 56 \, x^{3} + 96 \, x^{2} - 96 \, x + 64}}}{3 \, {\left (x^{3} - 3 \, x^{2} + 2\right )}}\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}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 2\right ) \left (x^{2} - 2 x + 4\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 [B]
time = 0.21, size = 533, normalized size = 7.20 \begin {gather*} \frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{9\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\,2{}\mathrm {i}}{9\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\,2{}\mathrm {i}}{9\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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