Optimal. Leaf size=74 \[ \frac {8}{9 (1-3 x)}-\frac {4 \sqrt {x} \sqrt {1+x}}{3 (1-3 x)}-\frac {8}{9} \sinh ^{-1}\left (\sqrt {x}\right )+\frac {10}{9} \tanh ^{-1}\left (\frac {2 \sqrt {x}}{\sqrt {1+x}}\right )+\frac {5}{9} \log (1-3 x) \]
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Rubi [A]
time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6874, 99, 163,
56, 221, 95, 213} \begin {gather*} -\frac {4 \sqrt {x} \sqrt {x+1}}{3 (1-3 x)}+\frac {8}{9 (1-3 x)}+\frac {5}{9} \log (1-3 x)-\frac {8}{9} \sinh ^{-1}\left (\sqrt {x}\right )+\frac {10}{9} \tanh ^{-1}\left (\frac {2 \sqrt {x}}{\sqrt {x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 95
Rule 99
Rule 163
Rule 213
Rule 221
Rule 6874
Rubi steps
\begin {align*} \int \frac {1}{\left (2 \sqrt {x}+\sqrt {1+x}\right )^2} \, dx &=\int \left (\frac {8}{3 (-1+3 x)^2}-\frac {4 \sqrt {x} \sqrt {1+x}}{(-1+3 x)^2}+\frac {5}{3 (-1+3 x)}\right ) \, dx\\ &=\frac {8}{9 (1-3 x)}+\frac {5}{9} \log (1-3 x)-4 \int \frac {\sqrt {x} \sqrt {1+x}}{(-1+3 x)^2} \, dx\\ &=\frac {8}{9 (1-3 x)}-\frac {4 \sqrt {x} \sqrt {1+x}}{3 (1-3 x)}+\frac {5}{9} \log (1-3 x)-\frac {4}{3} \int \frac {\frac {1}{2}+x}{\sqrt {x} \sqrt {1+x} (-1+3 x)} \, dx\\ &=\frac {8}{9 (1-3 x)}-\frac {4 \sqrt {x} \sqrt {1+x}}{3 (1-3 x)}+\frac {5}{9} \log (1-3 x)-\frac {4}{9} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx-\frac {10}{9} \int \frac {1}{\sqrt {x} \sqrt {1+x} (-1+3 x)} \, dx\\ &=\frac {8}{9 (1-3 x)}-\frac {4 \sqrt {x} \sqrt {1+x}}{3 (1-3 x)}+\frac {5}{9} \log (1-3 x)-\frac {8}{9} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right )-\frac {20}{9} \text {Subst}\left (\int \frac {1}{-1+4 x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+x}}\right )\\ &=\frac {8}{9 (1-3 x)}-\frac {4 \sqrt {x} \sqrt {1+x}}{3 (1-3 x)}-\frac {8}{9} \sinh ^{-1}\left (\sqrt {x}\right )+\frac {10}{9} \tanh ^{-1}\left (\frac {2 \sqrt {x}}{\sqrt {1+x}}\right )+\frac {5}{9} \log (1-3 x)\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 69, normalized size = 0.93 \begin {gather*} \frac {2 \left (-4+6 \sqrt {x} \sqrt {1+x}+(-1+3 x) \tanh ^{-1}\left (\sqrt {\frac {x}{1+x}}\right )+5 (-1+3 x) \log \left (1-x+\sqrt {x} \sqrt {1+x}\right )\right )}{-9+27 x} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(114\) vs.
\(2(54)=108\).
time = 0.02, size = 115, normalized size = 1.55
method | result | size |
default | \(-\frac {8}{9 \left (3 x -1\right )}+\frac {5 \ln \left (3 x -1\right )}{9}-\frac {\sqrt {x}\, \sqrt {1+x}\, \left (12 \ln \left (\frac {1}{2}+x +\sqrt {x \left (1+x \right )}\right ) x -15 \arctanh \left (\frac {5 x +1}{4 \sqrt {x \left (1+x \right )}}\right ) x -4 \ln \left (\frac {1}{2}+x +\sqrt {x \left (1+x \right )}\right )+5 \arctanh \left (\frac {5 x +1}{4 \sqrt {x \left (1+x \right )}}\right )-12 \sqrt {x \left (1+x \right )}\right )}{9 \sqrt {x \left (1+x \right )}\, \left (3 x -1\right )}\) | \(115\) |
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 = 0.32, size = 105, normalized size = 1.42 \begin {gather*} -\frac {5 \, {\left (3 \, x - 1\right )} \log \left (3 \, \sqrt {x + 1} \sqrt {x} - 3 \, x - 1\right ) - 4 \, {\left (3 \, x - 1\right )} \log \left (2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x - 1\right ) - 5 \, {\left (3 \, x - 1\right )} \log \left (\sqrt {x + 1} \sqrt {x} - x + 1\right ) - 5 \, {\left (3 \, x - 1\right )} \log \left (3 \, x - 1\right ) - 12 \, \sqrt {x + 1} \sqrt {x} - 12 \, x + 12}{9 \, {\left (3 \, x - 1\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 {1}{\left (2 \sqrt {x} + \sqrt {x + 1}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs.
\(2 (54) = 108\).
time = 0.03, size = 165, normalized size = 2.23 \begin {gather*} 2 \left (\frac {-5 x-1}{6 \left (3 x-1\right )}+\frac {5}{18} \ln \left |3 x-1\right |+2 \left (\frac {2}{9} \ln \left (\sqrt {x+1}-\sqrt {x}\right )+\frac {5}{36} \ln \left |\left (\sqrt {x+1}-\sqrt {x}\right )^{2}-3\right |-\frac {5}{36} \ln \left |3 \left (\sqrt {x+1}-\sqrt {x}\right )^{2}-1\right |-\frac {10 \left (\sqrt {x+1}-\sqrt {x}\right )^{2}-6}{9 \left (3 \left (\sqrt {x+1}-\sqrt {x}\right )^{4}-10 \left (\sqrt {x+1}-\sqrt {x}\right )^{2}+3\right )}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.71, size = 82, normalized size = 1.11 \begin {gather*} \frac {10\,\mathrm {atanh}\left (\frac {2662400\,\sqrt {x}}{81\,\left (\frac {665600\,x}{81\,{\left (\sqrt {x+1}-1\right )}^2}+\frac {665600}{81}\right )\,\left (\sqrt {x+1}-1\right )}\right )}{9}+\frac {5\,\ln \left (x-\frac {1}{3}\right )}{9}-\frac {16\,\mathrm {atanh}\left (\frac {\sqrt {x}}{\sqrt {x+1}-1}\right )}{9}-\frac {8}{27\,\left (x-\frac {1}{3}\right )}+\frac {4\,\sqrt {x}\,\sqrt {x+1}}{3\,\left (3\,x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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