Optimal. Leaf size=68 \[ \frac {1}{288} \sqrt [4]{-x+x^4} \left (-7 x^2-4 x^5+32 x^8\right )+\frac {7}{192} \text {ArcTan}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right )-\frac {7}{192} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(145\) vs. \(2(68)=136\).
time = 0.10, antiderivative size = 145, normalized size of antiderivative = 2.13, number of steps
used = 10, number of rules used = 9, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2046, 2049,
2057, 335, 281, 338, 304, 209, 212} \begin {gather*} \frac {7 \left (x^3-1\right )^{3/4} x^{3/4} \text {ArcTan}\left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{192 \left (x^4-x\right )^{3/4}}+\frac {1}{9} \sqrt [4]{x^4-x} x^8-\frac {1}{72} \sqrt [4]{x^4-x} x^5-\frac {7}{288} \sqrt [4]{x^4-x} x^2-\frac {7 \left (x^3-1\right )^{3/4} x^{3/4} \tanh ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{192 \left (x^4-x\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 281
Rule 304
Rule 335
Rule 338
Rule 2046
Rule 2049
Rule 2057
Rubi steps
\begin {align*} \int x^7 \sqrt [4]{-x+x^4} \, dx &=\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {1}{12} \int \frac {x^8}{\left (-x+x^4\right )^{3/4}} \, dx\\ &=-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {7}{96} \int \frac {x^5}{\left (-x+x^4\right )^{3/4}} \, dx\\ &=-\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {7}{128} \int \frac {x^2}{\left (-x+x^4\right )^{3/4}} \, dx\\ &=-\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \int \frac {x^{5/4}}{\left (-1+x^3\right )^{3/4}} \, dx}{128 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (-1+x^{12}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,x^{3/4}\right )}{96 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{96 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{192 \left (-x+x^4\right )^{3/4}}+\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{192 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}+\frac {7 x^{3/4} \left (-1+x^3\right )^{3/4} \tan ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{192 \left (-x+x^4\right )^{3/4}}-\frac {7 x^{3/4} \left (-1+x^3\right )^{3/4} \tanh ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{192 \left (-x+x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.03, size = 69, normalized size = 1.01 \begin {gather*} \frac {x^2 \sqrt [4]{x \left (-1+x^3\right )} \left (\sqrt [4]{1-x^3} \left (-7-x^3+8 x^6\right )+7 \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};x^3\right )\right )}{72 \sqrt [4]{1-x^3}} \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 = 4.36, size = 33, normalized size = 0.49
method | result | size |
meijerg | \(\frac {4 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {33}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {11}{4}\right ], \left [\frac {15}{4}\right ], x^{3}\right )}{33 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{4}}}\) | \(33\) |
trager | \(\frac {x^{2} \left (32 x^{6}-4 x^{3}-7\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{288}-\frac {7 \ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}-2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}-2 x^{3}+1\right )}{384}+\frac {7 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{384}\) | \(145\) |
risch | \(\frac {x^{2} \left (32 x^{6}-4 x^{3}-7\right ) \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}}}{288}+\frac {\left (-\frac {7 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 x^{9}-2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+5 x^{6}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}-4 x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+1}{\left (x^{2}+x +1\right )^{2} \left (-1+x \right )^{2}}\right )}{384}-\frac {7 \ln \left (\frac {2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (x^{2}+x +1\right )^{2} \left (-1+x \right )^{2}}\right )}{384}\right ) \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\) | \(454\) |
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 [A]
time = 1.09, size = 104, normalized size = 1.53 \begin {gather*} \frac {1}{288} \, {\left (32 \, x^{8} - 4 \, x^{5} - 7 \, x^{2}\right )} {\left (x^{4} - x\right )}^{\frac {1}{4}} - \frac {7}{384} \, \arctan \left (2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}\right ) + \frac {7}{384} \, \log \left (2 \, x^{3} - 2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x} x - 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} - 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 x^{7} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 88, normalized size = 1.29 \begin {gather*} \frac {1}{288} \, {\left (7 \, {\left (\frac {1}{x^{3}} - 1\right )}^{2} {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 18 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {5}{4}} - 21 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right )} x^{9} + \frac {7}{192} \, \arctan \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {7}{384} \, \log \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {7}{384} \, \log \left ({\left | {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \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 x^7\,{\left (x^4-x\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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