Optimal. Leaf size=61 \[ -\tan ^{-1}\left (\frac {3+\sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )+3 \tanh ^{-1}\left (\frac {1-3 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right ) \]
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Rubi [A]
time = 0.36, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1047, 738, 212,
210} \begin {gather*} 3 \tanh ^{-1}\left (\frac {1-3 \sqrt {x+1}}{2 \sqrt {x+\sqrt {x+1}}}\right )-\text {ArcTan}\left (\frac {\sqrt {x+1}+3}{2 \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 738
Rule 1047
Rubi steps
\begin {align*} \int \frac {1+2 \sqrt {1+x}}{x \sqrt {1+x} \sqrt {x+\sqrt {1+x}}} \, dx &=2 \text {Subst}\left (\int \frac {1+2 x}{\left (-1+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3-\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\right )-6 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+3 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=-\tan ^{-1}\left (\frac {3+\sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )+3 \tanh ^{-1}\left (\frac {1-3 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 55, normalized size = 0.90 \begin {gather*} -2 \tan ^{-1}\left (1+\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}\right )-6 \tanh ^{-1}\left (1-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 68, normalized size = 1.11
method | result | size |
derivativedivides | \(-3 \arctanh \left (\frac {-1+3 \sqrt {1+x}}{2 \sqrt {\left (\sqrt {1+x}-1\right )^{2}+3 \sqrt {1+x}-2}}\right )+\arctan \left (\frac {-3-\sqrt {1+x}}{2 \sqrt {\left (\sqrt {1+x}+1\right )^{2}-\sqrt {1+x}-2}}\right )\) | \(68\) |
default | \(-3 \arctanh \left (\frac {-1+3 \sqrt {1+x}}{2 \sqrt {\left (\sqrt {1+x}-1\right )^{2}+3 \sqrt {1+x}-2}}\right )+\arctan \left (\frac {-3-\sqrt {1+x}}{2 \sqrt {\left (\sqrt {1+x}+1\right )^{2}-\sqrt {1+x}-2}}\right )\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.56, size = 62, normalized size = 1.02 \begin {gather*} \arctan \left (\frac {2 \, \sqrt {x + \sqrt {x + 1}} {\left (\sqrt {x + 1} - 3\right )}}{x - 8}\right ) + 3 \, \log \left (\frac {2 \, \sqrt {x + \sqrt {x + 1}} {\left (\sqrt {x + 1} + 1\right )} - 3 \, x - 2 \, \sqrt {x + 1} - 2}{x}\right ) \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 {2 \sqrt {x + 1} + 1}{x \sqrt {x + 1} \sqrt {x + \sqrt {x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.07, size = 65, normalized size = 1.07 \begin {gather*} 2 \, \arctan \left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1} - 1\right ) - 3 \, \log \left ({\left | \sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1} + 2 \right |}\right ) + 3 \, \log \left ({\left | \sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1} \right |}\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 = -0.02 \begin {gather*} \int \frac {2\,\sqrt {x+1}+1}{x\,\sqrt {x+\sqrt {x+1}}\,\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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