Optimal. Leaf size=114 \[ -\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{3 x^3}-\frac {5 (1-x)^{3/4} \sqrt [4]{1+x}}{12 x^2}-\frac {11 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x}-\frac {3}{8} \tan ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {3}{8} \tanh ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {101, 156, 12,
95, 218, 212, 209} \begin {gather*} -\frac {3}{8} \text {ArcTan}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}-\frac {5 (1-x)^{3/4} \sqrt [4]{x+1}}{12 x^2}-\frac {11 (1-x)^{3/4} \sqrt [4]{x+1}}{24 x}-\frac {3}{8} \tanh ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 209
Rule 212
Rule 218
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^4} \, dx &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{3 x^3}+\frac {1}{3} \int \frac {\frac {5}{2}+2 x}{\sqrt [4]{1-x} x^3 (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{3 x^3}-\frac {5 (1-x)^{3/4} \sqrt [4]{1+x}}{12 x^2}-\frac {1}{6} \int \frac {-\frac {11}{4}-\frac {5 x}{2}}{\sqrt [4]{1-x} x^2 (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{3 x^3}-\frac {5 (1-x)^{3/4} \sqrt [4]{1+x}}{12 x^2}-\frac {11 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x}+\frac {1}{6} \int \frac {9}{8 \sqrt [4]{1-x} x (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{3 x^3}-\frac {5 (1-x)^{3/4} \sqrt [4]{1+x}}{12 x^2}-\frac {11 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x}+\frac {3}{16} \int \frac {1}{\sqrt [4]{1-x} x (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{3 x^3}-\frac {5 (1-x)^{3/4} \sqrt [4]{1+x}}{12 x^2}-\frac {11 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x}+\frac {3}{4} \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{3 x^3}-\frac {5 (1-x)^{3/4} \sqrt [4]{1+x}}{12 x^2}-\frac {11 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x}-\frac {3}{8} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {3}{8} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{3 x^3}-\frac {5 (1-x)^{3/4} \sqrt [4]{1+x}}{12 x^2}-\frac {11 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x}-\frac {3}{8} \tan ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {3}{8} \tanh ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 76, normalized size = 0.67 \begin {gather*} \frac {1}{24} \left (-\frac {(1-x)^{3/4} \sqrt [4]{1+x} \left (8+10 x+11 x^2\right )}{x^3}-9 \tan ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-9 \tanh ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\right ) \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 = 0.72, size = 389, normalized size = 3.41
method | result | size |
risch | \(\frac {\left (1+x \right )^{\frac {1}{4}} \left (-1+x \right ) \left (11 x^{2}+10 x +8\right ) \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{24 x^{3} \left (-\left (-1+x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}} \left (1-x \right )^{\frac {1}{4}}}+\frac {\left (-\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-2 x^{3}+2 x +1}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x -\RootOf \left (\textit {\_Z}^{2}+1\right )+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}}{x \left (1+x \right )^{2}}\right )}{16}-\frac {3 \ln \left (\frac {\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}+\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}+\sqrt {-x^{4}-2 x^{3}+2 x +1}+2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x +x^{2}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}+2 x +1}{x \left (1+x \right )^{2}}\right )}{16}\right ) \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{\left (1+x \right )^{\frac {3}{4}} \left (1-x \right )^{\frac {1}{4}}}\) | \(389\) |
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 = 0.50, size = 112, normalized size = 0.98 \begin {gather*} \frac {18 \, x^{3} \arctan \left (\frac {{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + 9 \, x^{3} \log \left (\frac {x + {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 9 \, x^{3} \log \left (-\frac {x - {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 2 \, {\left (11 \, x^{2} + 10 \, x + 8\right )} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{48 \, x^{3}} \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 {\sqrt [4]{x + 1}}{x^{4} \sqrt [4]{1 - x}}\, 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 {{\left (x+1\right )}^{1/4}}{x^4\,{\left (1-x\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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