Optimal. Leaf size=83 \[ -48 \sqrt {2+\sqrt {1+\sqrt {x}}}+\frac {88}{3} \left (2+\sqrt {1+\sqrt {x}}\right )^{3/2}-\frac {48}{5} \left (2+\sqrt {1+\sqrt {x}}\right )^{5/2}+\frac {8}{7} \left (2+\sqrt {1+\sqrt {x}}\right )^{7/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {378, 1412, 786}
\begin {gather*} \frac {8}{7} \left (\sqrt {\sqrt {x}+1}+2\right )^{7/2}-\frac {48}{5} \left (\sqrt {\sqrt {x}+1}+2\right )^{5/2}+\frac {88}{3} \left (\sqrt {\sqrt {x}+1}+2\right )^{3/2}-48 \sqrt {\sqrt {\sqrt {x}+1}+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 378
Rule 786
Rule 1412
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+\sqrt {1+\sqrt {x}}}} \, dx &=2 \text {Subst}\left (\int \frac {x}{\sqrt {2+\sqrt {1+x}}} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \frac {-1+x}{\sqrt {2+\sqrt {x}}} \, dx,x,1+\sqrt {x}\right )\\ &=4 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )}{\sqrt {2+x}} \, dx,x,\sqrt {1+\sqrt {x}}\right )\\ &=4 \text {Subst}\left (\int \left (-\frac {6}{\sqrt {2+x}}+11 \sqrt {2+x}-6 (2+x)^{3/2}+(2+x)^{5/2}\right ) \, dx,x,\sqrt {1+\sqrt {x}}\right )\\ &=-48 \sqrt {2+\sqrt {1+\sqrt {x}}}+\frac {88}{3} \left (2+\sqrt {1+\sqrt {x}}\right )^{3/2}-\frac {48}{5} \left (2+\sqrt {1+\sqrt {x}}\right )^{5/2}+\frac {8}{7} \left (2+\sqrt {1+\sqrt {x}}\right )^{7/2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 58, normalized size = 0.70 \begin {gather*} \frac {8}{105} \sqrt {2+\sqrt {1+\sqrt {x}}} \left (-280+76 \sqrt {1+\sqrt {x}}+3 \left (-12+5 \sqrt {1+\sqrt {x}}\right ) \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 54, normalized size = 0.65
method | result | size |
derivativedivides | \(\frac {88 \left (2+\sqrt {1+\sqrt {x}}\right )^{\frac {3}{2}}}{3}-\frac {48 \left (2+\sqrt {1+\sqrt {x}}\right )^{\frac {5}{2}}}{5}+\frac {8 \left (2+\sqrt {1+\sqrt {x}}\right )^{\frac {7}{2}}}{7}-48 \sqrt {2+\sqrt {1+\sqrt {x}}}\) | \(54\) |
default | \(\frac {88 \left (2+\sqrt {1+\sqrt {x}}\right )^{\frac {3}{2}}}{3}-\frac {48 \left (2+\sqrt {1+\sqrt {x}}\right )^{\frac {5}{2}}}{5}+\frac {8 \left (2+\sqrt {1+\sqrt {x}}\right )^{\frac {7}{2}}}{7}-48 \sqrt {2+\sqrt {1+\sqrt {x}}}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 53, normalized size = 0.64 \begin {gather*} \frac {8}{7} \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {7}{2}} - \frac {48}{5} \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {5}{2}} + \frac {88}{3} \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {3}{2}} - 48 \, \sqrt {\sqrt {\sqrt {x} + 1} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 35, normalized size = 0.42 \begin {gather*} \frac {8}{105} \, {\left ({\left (15 \, \sqrt {x} + 76\right )} \sqrt {\sqrt {x} + 1} - 36 \, \sqrt {x} - 280\right )} \sqrt {\sqrt {\sqrt {x} + 1} + 2} \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 {\sqrt {\sqrt {x} + 1} + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.52, size = 82, normalized size = 0.99 \begin {gather*} \frac {8 \, {\left (15 \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {7}{2}} - 126 \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {5}{2}} + 385 \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {3}{2}} - 630 \, \sqrt {\sqrt {\sqrt {x} + 1} + 2}\right )}}{105 \, \mathrm {sgn}\left (4 \, {\left (\sqrt {x} + 1\right )}^{2} - 8 \, \sqrt {x} - 7\right ) \mathrm {sgn}\left (4 \, x - 3\right )} \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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {\sqrt {x}+1}+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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