Integrand size = 19, antiderivative size = 128 \[ \int \frac {1}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {32}{105} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {32}{105} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\sqrt {1+x} \left (-\frac {48}{35} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{7} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {378, 1412, 786} \[ \int \frac {1}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8}{7} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{7/2}-\frac {24}{5} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2}+\frac {16}{3} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2} \]
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Rule 378
Rule 786
Rule 1412
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x}{\sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {Subst}\left (\int \frac {-1+x}{\sqrt {1+\sqrt {x}}} \, dx,x,1+\sqrt {1+x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )}{\sqrt {1+x}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \text {Subst}\left (\int \left (2 \sqrt {1+x}-3 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = \frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8}{105} \sqrt {1+\sqrt {1+\sqrt {1+x}}} \left (4-18 \sqrt {1+x}+4 \sqrt {1+\sqrt {1+x}}+15 \sqrt {1+x} \sqrt {1+\sqrt {1+x}}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.37
method | result | size |
derivativedivides | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}\) | \(47\) |
default | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}\) | \(47\) |
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none
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.34 \[ \int \frac {1}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8}{105} \, {\left ({\left (15 \, \sqrt {x + 1} + 4\right )} \sqrt {\sqrt {x + 1} + 1} - 18 \, \sqrt {x + 1} + 4\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (110) = 220\).
Time = 0.98 (sec) , antiderivative size = 445, normalized size of antiderivative = 3.48 \[ \int \frac {1}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=- \frac {336 \left (x + 1\right )^{\frac {13}{4}} \sqrt [4]{\sqrt {x + 1} + 1} \sin {\left (\frac {5 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} - \frac {112 \left (x + 1\right )^{\frac {11}{4}} \sqrt [4]{\sqrt {x + 1} + 1} \sin {\left (\frac {5 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} + \frac {224 \left (x + 1\right )^{\frac {9}{4}} \sqrt [4]{\sqrt {x + 1} + 1} \sin {\left (\frac {5 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} - \frac {152 \left (x + 1\right )^{\frac {5}{2}} \left (\sqrt {x + 1} + 1\right )^{\frac {3}{4}} \cos {\left (\frac {7 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} - \frac {64 \left (x + 1\right )^{\frac {5}{2}} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} - \frac {216 \left (x + 1\right )^{3} \left (\sqrt {x + 1} + 1\right )^{\frac {3}{4}} \cos {\left (\frac {7 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} + \frac {64 \left (x + 1\right )^{2} \left (\sqrt {x + 1} + 1\right )^{\frac {3}{4}} \cos {\left (\frac {7 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} - \frac {64 \left (x + 1\right )^{2} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} \]
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Result contains higher order function than in optimal. Order 3 vs. order 2.
Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.36 \[ \int \frac {1}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8}{7} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} - \frac {24}{5} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} + \frac {16}{3} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} \]
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Result contains higher order function than in optimal. Order 3 vs. order 2.
Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8 \, {\left (15 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} - 63 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} + 70 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}}\right )}}{105 \, \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) \mathrm {sgn}\left (4 \, x + 1\right )} \]
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Timed out. \[ \int \frac {1}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}} \,d x \]
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