Optimal. Leaf size=51 \[ x-4 \sqrt {\sqrt {x+1}+1}+\frac {8 \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}}{\sqrt {5}}+\frac {1}{\sqrt {5}}\right )}{\sqrt {5}} \]
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Rubi [A] time = 0.31, antiderivative size = 67, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1628, 618, 206} \begin {gather*} \left (\sqrt {x+1}+1\right )^2-4 \sqrt {\sqrt {x+1}+1}-2 \sqrt {x+1}+\frac {8 \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}+1}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1628
Rubi steps
\begin {align*} \int \frac {x}{x+\sqrt {1+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \left (-1+x^2\right )}{-1+x^2+\sqrt {1+x}} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x^2 (1+x) \left (-2+x^2\right )}{-1+x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (-1-x+x^3-\frac {1}{-1+x+x^2}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-2 \sqrt {1+x}-4 \sqrt {1+\sqrt {1+x}}+\left (1+\sqrt {1+x}\right )^2-4 \operatorname {Subst}\left (\int \frac {1}{-1+x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-2 \sqrt {1+x}-4 \sqrt {1+\sqrt {1+x}}+\left (1+\sqrt {1+x}\right )^2+8 \operatorname {Subst}\left (\int \frac {1}{5-x^2} \, dx,x,1+2 \sqrt {1+\sqrt {1+x}}\right )\\ &=-2 \sqrt {1+x}-4 \sqrt {1+\sqrt {1+x}}+\left (1+\sqrt {1+x}\right )^2+\frac {8 \tanh ^{-1}\left (\frac {1+2 \sqrt {1+\sqrt {1+x}}}{\sqrt {5}}\right )}{\sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 48, normalized size = 0.94 \begin {gather*} x-4 \sqrt {\sqrt {x+1}+1}+\frac {8 \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}+1}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 52, normalized size = 1.02 \begin {gather*} 1+x-4 \sqrt {1+\sqrt {1+x}}+\frac {8 \tanh ^{-1}\left (\frac {1}{\sqrt {5}}+\frac {2 \sqrt {1+\sqrt {1+x}}}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 106, normalized size = 2.08 \begin {gather*} \frac {4}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (3 \, x + 1\right )} - {\left (\sqrt {5} {\left (x + 2\right )} - 5 \, x\right )} \sqrt {x + 1} + {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} - 5\right )} \sqrt {x + 1} - 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + x - 4 \, \sqrt {\sqrt {x + 1} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 71, normalized size = 1.39 \begin {gather*} {\left (\sqrt {x + 1} + 1\right )}^{2} - \frac {4}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) - 2 \, \sqrt {x + 1} - 4 \, \sqrt {\sqrt {x + 1} + 1} - 2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 54, normalized size = 1.06
method | result | size |
derivativedivides | \(\left (1+\sqrt {1+x}\right )^{2}-2-2 \sqrt {1+x}-4 \sqrt {1+\sqrt {1+x}}+\frac {8 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}\) | \(54\) |
default | \(\left (1+\sqrt {1+x}\right )^{2}-2-2 \sqrt {1+x}-4 \sqrt {1+\sqrt {1+x}}+\frac {8 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 71, normalized size = 1.39 \begin {gather*} {\left (\sqrt {x + 1} + 1\right )}^{2} - \frac {4}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) - 2 \, \sqrt {x + 1} - 4 \, \sqrt {\sqrt {x + 1} + 1} - 2 \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 = -0.02 \begin {gather*} \int \frac {x}{x+\sqrt {\sqrt {x+1}+1}} \,d x \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 {x}{x + \sqrt {\sqrt {x + 1} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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