Integrand size = 19, antiderivative size = 62 \[ \int \frac {1+x^2}{\sqrt {1+\sqrt {1+x}}} \, dx=-\frac {8 \left (1187-4 x+175 x^2\right ) \sqrt {1+\sqrt {1+x}}}{3465}+\frac {4 \sqrt {1+x} \left (1091+40 x+315 x^2\right ) \sqrt {1+\sqrt {1+x}}}{3465} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.63, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1914, 1634} \[ \int \frac {1+x^2}{\sqrt {1+\sqrt {1+x}}} \, dx=\frac {4}{11} \left (\sqrt {x+1}+1\right )^{11/2}-\frac {20}{9} \left (\sqrt {x+1}+1\right )^{9/2}+\frac {32}{7} \left (\sqrt {x+1}+1\right )^{7/2}-\frac {16}{5} \left (\sqrt {x+1}+1\right )^{5/2}+\frac {4}{3} \left (\sqrt {x+1}+1\right )^{3/2}-4 \sqrt {\sqrt {x+1}+1} \]
[In]
[Out]
Rule 1634
Rule 1914
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x \left (2-2 x^2+x^4\right )}{\sqrt {1+x}} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}-4 (1+x)^{3/2}+8 (1+x)^{5/2}-5 (1+x)^{7/2}+(1+x)^{9/2}\right ) \, dx,x,\sqrt {1+x}\right ) \\ & = -4 \sqrt {1+\sqrt {1+x}}+\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-\frac {16}{5} \left (1+\sqrt {1+x}\right )^{5/2}+\frac {32}{7} \left (1+\sqrt {1+x}\right )^{7/2}-\frac {20}{9} \left (1+\sqrt {1+x}\right )^{9/2}+\frac {4}{11} \left (1+\sqrt {1+x}\right )^{11/2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94 \[ \int \frac {1+x^2}{\sqrt {1+\sqrt {1+x}}} \, dx=\frac {4 \sqrt {1+\sqrt {1+x}} \left (-2732+1366 \sqrt {1+x}+708 (1+x)-590 (1+x)^{3/2}-350 (1+x)^2+315 (1+x)^{5/2}\right )}{3465} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 5 vs. order 2.
Time = 0.39 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.61
method | result | size |
meijerg | \(\frac {\sqrt {2}\, x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {3}{4}, 1\right ], \left [\frac {3}{2}, 2\right ], -x \right )}{2}+\frac {\sqrt {2}\, x^{3} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {3}{4}, 3\right ], \left [\frac {3}{2}, 4\right ], -x \right )}{6}\) | \(38\) |
derivativedivides | \(\frac {4 \left (\sqrt {1+x}+1\right )^{\frac {11}{2}}}{11}-\frac {20 \left (\sqrt {1+x}+1\right )^{\frac {9}{2}}}{9}+\frac {32 \left (\sqrt {1+x}+1\right )^{\frac {7}{2}}}{7}-\frac {16 \left (\sqrt {1+x}+1\right )^{\frac {5}{2}}}{5}+\frac {4 \left (\sqrt {1+x}+1\right )^{\frac {3}{2}}}{3}-4 \sqrt {\sqrt {1+x}+1}\) | \(68\) |
default | \(\frac {4 \left (\sqrt {1+x}+1\right )^{\frac {11}{2}}}{11}-\frac {20 \left (\sqrt {1+x}+1\right )^{\frac {9}{2}}}{9}+\frac {32 \left (\sqrt {1+x}+1\right )^{\frac {7}{2}}}{7}-\frac {16 \left (\sqrt {1+x}+1\right )^{\frac {5}{2}}}{5}+\frac {4 \left (\sqrt {1+x}+1\right )^{\frac {3}{2}}}{3}-4 \sqrt {\sqrt {1+x}+1}\) | \(68\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.61 \[ \int \frac {1+x^2}{\sqrt {1+\sqrt {1+x}}} \, dx=-\frac {4}{3465} \, {\left (350 \, x^{2} - {\left (315 \, x^{2} + 40 \, x + 1091\right )} \sqrt {x + 1} - 8 \, x + 2374\right )} \sqrt {\sqrt {x + 1} + 1} \]
[In]
[Out]
Time = 0.53 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.42 \[ \int \frac {1+x^2}{\sqrt {1+\sqrt {1+x}}} \, dx=\frac {4 \left (\sqrt {x + 1} + 1\right )^{\frac {11}{2}}}{11} - \frac {20 \left (\sqrt {x + 1} + 1\right )^{\frac {9}{2}}}{9} + \frac {32 \left (\sqrt {x + 1} + 1\right )^{\frac {7}{2}}}{7} - \frac {16 \left (\sqrt {x + 1} + 1\right )^{\frac {5}{2}}}{5} + \frac {4 \left (\sqrt {x + 1} + 1\right )^{\frac {3}{2}}}{3} - 4 \sqrt {\sqrt {x + 1} + 1} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.08 \[ \int \frac {1+x^2}{\sqrt {1+\sqrt {1+x}}} \, dx=\frac {4}{11} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {11}{2}} - \frac {20}{9} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {9}{2}} + \frac {32}{7} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {7}{2}} - \frac {16}{5} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {\sqrt {x + 1} + 1} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.08 \[ \int \frac {1+x^2}{\sqrt {1+\sqrt {1+x}}} \, dx=\frac {4}{11} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {11}{2}} - \frac {20}{9} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {9}{2}} + \frac {32}{7} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {7}{2}} - \frac {16}{5} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {\sqrt {x + 1} + 1} \]
[In]
[Out]
Timed out. \[ \int \frac {1+x^2}{\sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {x^2+1}{\sqrt {\sqrt {x+1}+1}} \,d x \]
[In]
[Out]