Integrand size = 16, antiderivative size = 47 \[ \int \frac {1}{1+\sqrt [4]{9-6 x+x^2}} \, dx=\frac {\sqrt {(-3+x)^2} \left (2 \sqrt [4]{9-6 x+x^2}-2 \log \left (1+\sqrt [4]{9-6 x+x^2}\right )\right )}{-3+x} \]
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Leaf count is larger than twice the leaf count of optimal. \(215\) vs. \(2(47)=94\).
Time = 0.15 (sec) , antiderivative size = 215, normalized size of antiderivative = 4.57, number of steps used = 26, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6874, 660, 52, 65, 210, 213, 45} \[ \int \frac {1}{1+\sqrt [4]{9-6 x+x^2}} \, dx=-\frac {\left (x^2-6 x+9\right )^{3/4} \arctan \left (\sqrt {x-3}\right )}{(x-3)^{3/2}}+\frac {\sqrt [4]{x^2-6 x+9} \arctan \left (\sqrt {x-3}\right )}{\sqrt {x-3}}-\frac {\left (x^2-6 x+9\right )^{3/4} \text {arctanh}\left (\sqrt {x-3}\right )}{(x-3)^{3/2}}-\frac {\sqrt [4]{x^2-6 x+9} \text {arctanh}\left (\sqrt {x-3}\right )}{\sqrt {x-3}}-\frac {2 \left (x^2-6 x+9\right )^{3/4}}{3-x}+\frac {\sqrt {x^2-6 x+9} \log (2-x)}{2 (3-x)}+\frac {\sqrt {x^2-6 x+9} \log (4-x)}{2 (3-x)}+\frac {1}{2} \log (2-x)-\frac {1}{2} \log (4-x) \]
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Rule 45
Rule 52
Rule 65
Rule 210
Rule 213
Rule 660
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \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} \text {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} \text {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} \arctan \left (\sqrt {-3+x}\right )}{\sqrt {-3+x}}-\frac {\sqrt [4]{9-6 x+x^2} \text {arctanh}\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} \text {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} \text {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} \arctan \left (\sqrt {-3+x}\right )}{\sqrt {-3+x}}-\frac {\left (9-6 x+x^2\right )^{3/4} \arctan \left (\sqrt {-3+x}\right )}{(-3+x)^{3/2}}-\frac {\sqrt [4]{9-6 x+x^2} \text {arctanh}\left (\sqrt {-3+x}\right )}{\sqrt {-3+x}}-\frac {\left (9-6 x+x^2\right )^{3/4} \text {arctanh}\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*}
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \frac {1}{1+\sqrt [4]{9-6 x+x^2}} \, dx=\frac {\sqrt {(-3+x)^2} \left (2 \sqrt [4]{(-3+x)^2}-2 \log \left (1+\sqrt [4]{(-3+x)^2}\right )\right )}{-3+x} \]
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\[\int \frac {1}{1+\left (x^{2}-6 x +9\right )^{\frac {1}{4}}}d x\]
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none
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.60 \[ \int \frac {1}{1+\sqrt [4]{9-6 x+x^2}} \, dx=2 \, {\left (x^{2} - 6 \, x + 9\right )}^{\frac {1}{4}} - 2 \, \log \left ({\left (x^{2} - 6 \, x + 9\right )}^{\frac {1}{4}} + 1\right ) \]
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\[ \int \frac {1}{1+\sqrt [4]{9-6 x+x^2}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} - 6 x + 9} + 1}\, dx \]
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\[ \int \frac {1}{1+\sqrt [4]{9-6 x+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} - 6 \, x + 9\right )}^{\frac {1}{4}} + 1} \,d x } \]
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\[ \int \frac {1}{1+\sqrt [4]{9-6 x+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} - 6 \, x + 9\right )}^{\frac {1}{4}} + 1} \,d x } \]
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Timed out. \[ \int \frac {1}{1+\sqrt [4]{9-6 x+x^2}} \, dx=\int \frac {1}{{\left (x^2-6\,x+9\right )}^{1/4}+1} \,d x \]
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