Integrand size = 43, antiderivative size = 184 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {88}{15} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{5} \sqrt {1+x} \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\frac {32}{15} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}-2 \sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {-1+\sqrt {2}}}\right )-2 \sqrt {2 \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {1+\sqrt {2}}}\right ) \]
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Time = 1.18 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.88, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {1600, 912, 1301, 1107, 213, 209} \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=-2 \sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {\sqrt {2}-1}}\right )-2 \sqrt {2 \left (\sqrt {2}-1\right )} \text {arctanh}\left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {1+\sqrt {2}}}\right )+\frac {8}{5} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2}-\frac {16}{3} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}+8 \sqrt {\sqrt {\sqrt {x+1}+1}+1} \]
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Rule 209
Rule 213
Rule 912
Rule 1107
Rule 1301
Rule 1600
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^2 \sqrt {1+x}}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {Subst}\left (\int \frac {x^2}{(-1+x) \sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{\sqrt {1+x} \left (-2+x^2\right )} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \text {Subst}\left (\int \frac {(-1+x)^2 (1+x)^{3/2}}{-2+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 8 \text {Subst}\left (\int \frac {x^4 \left (-2+x^2\right )^2}{-1-2 x^2+x^4} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = 8 \text {Subst}\left (\int \left (1-2 x^2+x^4+\frac {1}{-1-2 x^2+x^4}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = 8 \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}+\frac {8}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+8 \text {Subst}\left (\int \frac {1}{-1-2 x^2+x^4} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = 8 \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}+\frac {8}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\left (2 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-1-\sqrt {2}+x^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )-\left (2 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-1+\sqrt {2}+x^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = 8 \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}+\frac {8}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}-2 \sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {-1+\sqrt {2}}}\right )-2 \sqrt {2 \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {1+\sqrt {2}}}\right ) \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8}{15} \left (11+3 \sqrt {1+x}-4 \sqrt {1+\sqrt {1+x}}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}-2 \sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )-2 \sqrt {2 \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.62
method | result | size |
derivativedivides | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}-\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+8 \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {\sqrt {2}-1}}\right )}{\sqrt {\sqrt {2}-1}}-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\) | \(115\) |
default | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}-\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+8 \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {\sqrt {2}-1}}\right )}{\sqrt {\sqrt {2}-1}}-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\) | \(115\) |
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none
Time = 0.26 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=-\sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (2 \, \sqrt {2} {\left (\sqrt {2} + 2\right )} \sqrt {\sqrt {2} - 1} + 4 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right ) + \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} + 2\right )} \sqrt {\sqrt {2} - 1} + 4 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right ) + \frac {8}{15} \, {\left (3 \, \sqrt {x + 1} - 4 \, \sqrt {\sqrt {x + 1} + 1} + 11\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} + \frac {1}{2} \, \sqrt {-8 \, \sqrt {2} - 8} \log \left ({\left (\sqrt {2} - 2\right )} \sqrt {-8 \, \sqrt {2} - 8} + 4 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right ) - \frac {1}{2} \, \sqrt {-8 \, \sqrt {2} - 8} \log \left (-{\left (\sqrt {2} - 2\right )} \sqrt {-8 \, \sqrt {2} - 8} + 4 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right ) \]
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Timed out. \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int { \frac {\sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}}{x \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (126) = 252\).
Time = 8.39 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=-\frac {\frac {30 \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}{\sqrt {\sqrt {2} - 1}}\right )}{\mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )} + \frac {15 \, \sqrt {2 \, \sqrt {2} - 2} \log \left (\sqrt {\sqrt {2} + 1} + \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )}{\mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )} - \frac {15 \, \sqrt {2 \, \sqrt {2} - 2} \log \left ({\left | -\sqrt {\sqrt {2} + 1} + \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \right |}\right )}{\mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )} - \frac {8 \, {\left (3 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} - 10 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )}}{\mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )}}{15 \, \mathrm {sgn}\left (4 \, x + 1\right )} \]
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Timed out. \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}}{x\,\sqrt {\sqrt {\sqrt {x+1}+1}+1}} \,d x \]
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