Optimal. Leaf size=29 \[ \tan ^{-1}\left (\sqrt [4]{x^2-2 x-2}\right )-\tanh ^{-1}\left (\sqrt [4]{x^2-2 x-2}\right ) \]
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Rubi [B] time = 1.12, antiderivative size = 262, normalized size of antiderivative = 9.03, number of steps used = 42, number of rules used = 16, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {6742, 749, 748, 746, 399, 490, 1213, 537, 444, 63, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\sqrt [4]{-x^2+2 x+2} \log \left (\sqrt {-x^2+2 x+2}+\sqrt {2} \sqrt [4]{-x^2+2 x+2}+1\right )}{2 \sqrt {2} \sqrt [4]{x^2-2 x-2}}+\frac {\sqrt [4]{-x^2+2 x+2} \log \left (3 \sqrt {-x^2+2 x+2}-3 \sqrt {2} \sqrt [4]{-x^2+2 x+2}+3\right )}{2 \sqrt {2} \sqrt [4]{x^2-2 x-2}}-\frac {\sqrt [4]{-x^2+2 x+2} \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{3-(1-x)^2}\right )}{\sqrt {2} \sqrt [4]{x^2-2 x-2}}+\frac {\sqrt [4]{-x^2+2 x+2} \tan ^{-1}\left (\sqrt {2} \sqrt [4]{3-(1-x)^2}+1\right )}{\sqrt {2} \sqrt [4]{x^2-2 x-2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 204
Rule 297
Rule 399
Rule 444
Rule 490
Rule 537
Rule 617
Rule 628
Rule 746
Rule 748
Rule 749
Rule 1162
Rule 1165
Rule 1213
Rule 6742
Rubi steps
\begin {align*} \int \frac {-1+x}{(-3+x) (1+x) \sqrt [4]{-2-2 x+x^2}} \, dx &=\int \left (\frac {1}{2 (-3+x) \sqrt [4]{-2-2 x+x^2}}+\frac {1}{2 (1+x) \sqrt [4]{-2-2 x+x^2}}\right ) \, dx\\ &=\frac {1}{2} \int \frac {1}{(-3+x) \sqrt [4]{-2-2 x+x^2}} \, dx+\frac {1}{2} \int \frac {1}{(1+x) \sqrt [4]{-2-2 x+x^2}} \, dx\\ &=\frac {\sqrt [4]{2+2 x-x^2} \int \frac {1}{(-3+x) \sqrt [4]{\frac {1}{6}+\frac {x}{6}-\frac {x^2}{12}}} \, dx}{2 \sqrt {2} \sqrt [4]{3} \sqrt [4]{-2-2 x+x^2}}+\frac {\sqrt [4]{2+2 x-x^2} \int \frac {1}{(1+x) \sqrt [4]{\frac {1}{6}+\frac {x}{6}-\frac {x^2}{12}}} \, dx}{2 \sqrt {2} \sqrt [4]{3} \sqrt [4]{-2-2 x+x^2}}\\ &=\frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {1}{3}+x\right ) \sqrt [4]{1-12 x^2}} \, dx,x,\frac {1}{6}-\frac {x}{6}\right )}{2 \sqrt [4]{3} \sqrt [4]{-2-2 x+x^2}}+\frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{3}+x\right ) \sqrt [4]{1-12 x^2}} \, dx,x,\frac {1}{6}-\frac {x}{6}\right )}{2 \sqrt [4]{3} \sqrt [4]{-2-2 x+x^2}}\\ &=-2 \frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {x}{\sqrt [4]{1-12 x^2} \left (\frac {1}{9}-x^2\right )} \, dx,x,\frac {1}{6}-\frac {x}{6}\right )}{2 \sqrt [4]{3} \sqrt [4]{-2-2 x+x^2}}\\ &=-2 \frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-12 x} \left (\frac {1}{9}-x\right )} \, dx,x,\left (\frac {1}{6}-\frac {x}{6}\right )^2\right )}{4 \sqrt [4]{3} \sqrt [4]{-2-2 x+x^2}}\\ &=2 \frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {x^2}{\frac {1}{36}+\frac {x^4}{12}} \, dx,x,\frac {\sqrt [4]{3-(-1+x)^2}}{\sqrt [4]{3}}\right )}{12 \sqrt [4]{3} \sqrt [4]{-2-2 x+x^2}}\\ &=2 \left (-\frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {1-\sqrt {3} x^2}{\frac {1}{36}+\frac {x^4}{12}} \, dx,x,\frac {\sqrt [4]{3-(-1+x)^2}}{\sqrt [4]{3}}\right )}{24\ 3^{3/4} \sqrt [4]{-2-2 x+x^2}}+\frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {1+\sqrt {3} x^2}{\frac {1}{36}+\frac {x^4}{12}} \, dx,x,\frac {\sqrt [4]{3-(-1+x)^2}}{\sqrt [4]{3}}\right )}{24\ 3^{3/4} \sqrt [4]{-2-2 x+x^2}}\right )\\ &=2 \left (\frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{3}}+2 x}{-\frac {1}{\sqrt {3}}-\frac {\sqrt {2} x}{\sqrt [4]{3}}-x^2} \, dx,x,\frac {\sqrt [4]{3-(-1+x)^2}}{\sqrt [4]{3}}\right )}{4 \sqrt {2} \sqrt [4]{-2-2 x+x^2}}+\frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{3}}-2 x}{-\frac {1}{\sqrt {3}}+\frac {\sqrt {2} x}{\sqrt [4]{3}}-x^2} \, dx,x,\frac {\sqrt [4]{3-(-1+x)^2}}{\sqrt [4]{3}}\right )}{4 \sqrt {2} \sqrt [4]{-2-2 x+x^2}}+\frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {3}}-\frac {\sqrt {2} x}{\sqrt [4]{3}}+x^2} \, dx,x,\frac {\sqrt [4]{3-(-1+x)^2}}{\sqrt [4]{3}}\right )}{4 \sqrt [4]{3} \sqrt [4]{-2-2 x+x^2}}+\frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {3}}+\frac {\sqrt {2} x}{\sqrt [4]{3}}+x^2} \, dx,x,\frac {\sqrt [4]{3-(-1+x)^2}}{\sqrt [4]{3}}\right )}{4 \sqrt [4]{3} \sqrt [4]{-2-2 x+x^2}}\right )\\ &=2 \left (\frac {\sqrt [4]{2+2 x-x^2} \log \left (\sqrt {3}-\sqrt {6} \sqrt [4]{3-(1-x)^2}+\sqrt {3} \sqrt {3-(1-x)^2}\right )}{4 \sqrt {2} \sqrt [4]{-2-2 x+x^2}}-\frac {\sqrt [4]{2+2 x-x^2} \log \left (\sqrt {3}+\sqrt {6} \sqrt [4]{3-(1-x)^2}+\sqrt {3} \sqrt {3-(1-x)^2}\right )}{4 \sqrt {2} \sqrt [4]{-2-2 x+x^2}}+\frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{3-(-1+x)^2}\right )}{2 \sqrt {2} \sqrt [4]{-2-2 x+x^2}}-\frac {\sqrt [4]{2+2 x-x^2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{3-(-1+x)^2}\right )}{2 \sqrt {2} \sqrt [4]{-2-2 x+x^2}}\right )\\ &=2 \left (-\frac {\sqrt [4]{2+2 x-x^2} \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{3-(1-x)^2}\right )}{2 \sqrt {2} \sqrt [4]{-2-2 x+x^2}}+\frac {\sqrt [4]{2+2 x-x^2} \tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{3-(1-x)^2}\right )}{2 \sqrt {2} \sqrt [4]{-2-2 x+x^2}}+\frac {\sqrt [4]{2+2 x-x^2} \log \left (\sqrt {3}-\sqrt {6} \sqrt [4]{3-(1-x)^2}+\sqrt {3} \sqrt {3-(1-x)^2}\right )}{4 \sqrt {2} \sqrt [4]{-2-2 x+x^2}}-\frac {\sqrt [4]{2+2 x-x^2} \log \left (\sqrt {3}+\sqrt {6} \sqrt [4]{3-(1-x)^2}+\sqrt {3} \sqrt {3-(1-x)^2}\right )}{4 \sqrt {2} \sqrt [4]{-2-2 x+x^2}}\right )\\ \end {align*}
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Mathematica [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x}{(-3+x) (1+x) \sqrt [4]{-2-2 x+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.06, size = 29, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\sqrt [4]{x^2-2 x-2}\right )-\tanh ^{-1}\left (\sqrt [4]{x^2-2 x-2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 42, normalized size = 1.45 \begin {gather*} \arctan \left ({\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{4}} - 1\right ) \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*} \int \frac {x - 1}{{\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{4}} {\left (x + 1\right )} {\left (x - 3\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.56, size = 149, normalized size = 5.14 \begin {gather*} -\frac {\ln \left (\frac {2 \left (x^{2}-2 x -2\right )^{\frac {3}{4}}+2 \sqrt {x^{2}-2 x -2}+x^{2}+2 \left (x^{2}-2 x -2\right )^{\frac {1}{4}}-2 x -1}{\left (-3+x \right ) \left (1+x \right )}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \left (x^{2}-2 x -2\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{2}-2 x -2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \left (x^{2}-2 x -2\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-3+x \right ) \left (1+x \right )}\right )}{2} \end {gather*}
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*} \int \frac {x - 1}{{\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{4}} {\left (x + 1\right )} {\left (x - 3\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 25, normalized size = 0.86 \begin {gather*} \mathrm {atan}\left ({\left (x^2-2\,x-2\right )}^{1/4}\right )-\mathrm {atanh}\left ({\left (x^2-2\,x-2\right )}^{1/4}\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 - 1}{\left (x - 3\right ) \left (x + 1\right ) \sqrt [4]{x^{2} - 2 x - 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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