Optimal. Leaf size=47 \[ \frac {\sqrt {(x-3)^2} \left (2 \sqrt [4]{x^2-6 x+9}-2 \log \left (\sqrt [4]{x^2-6 x+9}+1\right )\right )}{x-3} \]
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Rubi [B] time = 0.20, antiderivative size = 215, normalized size of antiderivative = 4.57, number of steps used = 26, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6742, 646, 50, 63, 204, 207, 43} \begin {gather*} -\frac {2 \left (x^2-6 x+9\right )^{3/4}}{3-x}+\frac {\sqrt {x^2-6 x+9} \log (2-x)}{2 (3-x)}+\frac {\sqrt {x^2-6 x+9} \log (4-x)}{2 (3-x)}-\frac {\left (x^2-6 x+9\right )^{3/4} \tan ^{-1}\left (\sqrt {x-3}\right )}{(x-3)^{3/2}}+\frac {\sqrt [4]{x^2-6 x+9} \tan ^{-1}\left (\sqrt {x-3}\right )}{\sqrt {x-3}}-\frac {\left (x^2-6 x+9\right )^{3/4} \tanh ^{-1}\left (\sqrt {x-3}\right )}{(x-3)^{3/2}}-\frac {\sqrt [4]{x^2-6 x+9} \tanh ^{-1}\left (\sqrt {x-3}\right )}{\sqrt {x-3}}+\frac {1}{2} \log (2-x)-\frac {1}{2} \log (4-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 50
Rule 63
Rule 204
Rule 207
Rule 646
Rule 6742
Rubi steps
\begin {align*} \int \frac {1}{1+\sqrt [4]{9-6 x+x^2}} \, dx &=\int \left (-\frac {1}{2 (-4+x)}+\frac {1}{2 (-2+x)}+\frac {\sqrt [4]{9-6 x+x^2}}{2 (2-x)}+\frac {\sqrt [4]{9-6 x+x^2}}{2 (-4+x)}+\frac {\sqrt {9-6 x+x^2}}{2 (4-x)}+\frac {\sqrt {9-6 x+x^2}}{2 (-2+x)}+\frac {\left (9-6 x+x^2\right )^{3/4}}{2 (2-x)}+\frac {\left (9-6 x+x^2\right )^{3/4}}{2 (-4+x)}\right ) \, dx\\ &=\frac {1}{2} \log (2-x)-\frac {1}{2} \log (4-x)+\frac {1}{2} \int \frac {\sqrt [4]{9-6 x+x^2}}{2-x} \, dx+\frac {1}{2} \int \frac {\sqrt [4]{9-6 x+x^2}}{-4+x} \, dx+\frac {1}{2} \int \frac {\sqrt {9-6 x+x^2}}{4-x} \, dx+\frac {1}{2} \int \frac {\sqrt {9-6 x+x^2}}{-2+x} \, dx+\frac {1}{2} \int \frac {\left (9-6 x+x^2\right )^{3/4}}{2-x} \, dx+\frac {1}{2} \int \frac {\left (9-6 x+x^2\right )^{3/4}}{-4+x} \, dx\\ &=\frac {1}{2} \log (2-x)-\frac {1}{2} \log (4-x)+\frac {\sqrt [4]{9-6 x+x^2} \int \frac {\sqrt {-3+x}}{2-x} \, dx}{2 \sqrt {-3+x}}+\frac {\sqrt [4]{9-6 x+x^2} \int \frac {\sqrt {-3+x}}{-4+x} \, dx}{2 \sqrt {-3+x}}+\frac {\sqrt {9-6 x+x^2} \int \frac {-3+x}{4-x} \, dx}{2 (-3+x)}+\frac {\sqrt {9-6 x+x^2} \int \frac {-3+x}{-2+x} \, dx}{2 (-3+x)}+\frac {\left (9-6 x+x^2\right )^{3/4} \int \frac {(-3+x)^{3/2}}{2-x} \, dx}{2 (-3+x)^{3/2}}+\frac {\left (9-6 x+x^2\right )^{3/4} \int \frac {(-3+x)^{3/2}}{-4+x} \, dx}{2 (-3+x)^{3/2}}\\ &=\frac {1}{2} \log (2-x)-\frac {1}{2} \log (4-x)-\frac {\sqrt [4]{9-6 x+x^2} \int \frac {1}{(2-x) \sqrt {-3+x}} \, dx}{2 \sqrt {-3+x}}+\frac {\sqrt [4]{9-6 x+x^2} \int \frac {1}{(-4+x) \sqrt {-3+x}} \, dx}{2 \sqrt {-3+x}}+\frac {\sqrt {9-6 x+x^2} \int \left (1+\frac {1}{2-x}\right ) \, dx}{2 (-3+x)}+\frac {\sqrt {9-6 x+x^2} \int \left (-1+\frac {1}{4-x}\right ) \, dx}{2 (-3+x)}-\frac {\left (9-6 x+x^2\right )^{3/4} \int \frac {\sqrt {-3+x}}{2-x} \, dx}{2 (-3+x)^{3/2}}+\frac {\left (9-6 x+x^2\right )^{3/4} \int \frac {\sqrt {-3+x}}{-4+x} \, dx}{2 (-3+x)^{3/2}}\\ &=-\frac {2 \left (9-6 x+x^2\right )^{3/4}}{3-x}+\frac {1}{2} \log (2-x)+\frac {\sqrt {9-6 x+x^2} \log (2-x)}{2 (3-x)}-\frac {1}{2} \log (4-x)+\frac {\sqrt {9-6 x+x^2} \log (4-x)}{2 (3-x)}-\frac {\sqrt [4]{9-6 x+x^2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {-3+x}\right )}{\sqrt {-3+x}}+\frac {\sqrt [4]{9-6 x+x^2} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {-3+x}\right )}{\sqrt {-3+x}}+\frac {\left (9-6 x+x^2\right )^{3/4} \int \frac {1}{(2-x) \sqrt {-3+x}} \, dx}{2 (-3+x)^{3/2}}+\frac {\left (9-6 x+x^2\right )^{3/4} \int \frac {1}{(-4+x) \sqrt {-3+x}} \, dx}{2 (-3+x)^{3/2}}\\ &=-\frac {2 \left (9-6 x+x^2\right )^{3/4}}{3-x}+\frac {\sqrt [4]{9-6 x+x^2} \tan ^{-1}\left (\sqrt {-3+x}\right )}{\sqrt {-3+x}}-\frac {\sqrt [4]{9-6 x+x^2} \tanh ^{-1}\left (\sqrt {-3+x}\right )}{\sqrt {-3+x}}+\frac {1}{2} \log (2-x)+\frac {\sqrt {9-6 x+x^2} \log (2-x)}{2 (3-x)}-\frac {1}{2} \log (4-x)+\frac {\sqrt {9-6 x+x^2} \log (4-x)}{2 (3-x)}+\frac {\left (9-6 x+x^2\right )^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {-3+x}\right )}{(-3+x)^{3/2}}+\frac {\left (9-6 x+x^2\right )^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {-3+x}\right )}{(-3+x)^{3/2}}\\ &=-\frac {2 \left (9-6 x+x^2\right )^{3/4}}{3-x}+\frac {\sqrt [4]{9-6 x+x^2} \tan ^{-1}\left (\sqrt {-3+x}\right )}{\sqrt {-3+x}}-\frac {\left (9-6 x+x^2\right )^{3/4} \tan ^{-1}\left (\sqrt {-3+x}\right )}{(-3+x)^{3/2}}-\frac {\sqrt [4]{9-6 x+x^2} \tanh ^{-1}\left (\sqrt {-3+x}\right )}{\sqrt {-3+x}}-\frac {\left (9-6 x+x^2\right )^{3/4} \tanh ^{-1}\left (\sqrt {-3+x}\right )}{(-3+x)^{3/2}}+\frac {1}{2} \log (2-x)+\frac {\sqrt {9-6 x+x^2} \log (2-x)}{2 (3-x)}-\frac {1}{2} \log (4-x)+\frac {\sqrt {9-6 x+x^2} \log (4-x)}{2 (3-x)}\\ \end {align*}
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Mathematica [B] time = 0.24, size = 138, normalized size = 2.94 \begin {gather*} \frac {\sqrt {x-3} \left (4 \left ((x-3)^2\right )^{3/4}-\left (\left (-x+\sqrt {(x-3)^2}+3\right ) \log (2-x)\right )-\left (x+\sqrt {(x-3)^2}-3\right ) \log (4-x)\right )-2 \left (-x+\sqrt {(x-3)^2}+3\right ) \sqrt [4]{(x-3)^2} \tan ^{-1}\left (\sqrt {x-3}\right )-2 \sqrt [4]{(x-3)^2} \left (x+\sqrt {(x-3)^2}-3\right ) \tanh ^{-1}\left (\sqrt {x-3}\right )}{2 (x-3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 47, normalized size = 1.00 \begin {gather*} \frac {\sqrt {(-3+x)^2} \left (2 \sqrt [4]{9-6 x+x^2}-2 \log \left (1+\sqrt [4]{9-6 x+x^2}\right )\right )}{-3+x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 28, normalized size = 0.60 \begin {gather*} 2 \, {\left (x^{2} - 6 \, x + 9\right )}^{\frac {1}{4}} - 2 \, \log \left ({\left (x^{2} - 6 \, x + 9\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 {1}{{\left (x^{2} - 6 \, x + 9\right )}^{\frac {1}{4}} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{1+\left (x^{2}-6 x +9\right )^{\frac {1}{4}}}\, dx\]
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 {1}{{\left (x^{2} - 6 \, x + 9\right )}^{\frac {1}{4}} + 1}\,{d x} \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 = -0.02 \begin {gather*} \int \frac {1}{{\left (x^2-6\,x+9\right )}^{1/4}+1} \,d x \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 {1}{\sqrt [4]{x^{2} - 6 x + 9} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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