Optimal. Leaf size=158 \[ -2 \sqrt {1+x+x^2}+\frac {1}{4} (1+2 x) \sqrt {1+x+x^2}-2 \sqrt {4+2 x+x^2}+\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {11}{2} \sinh ^{-1}\left (\frac {1+x}{\sqrt {3}}\right )+\frac {43}{8} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-2 \sqrt {7} \tanh ^{-1}\left (\frac {1+5 x}{2 \sqrt {7} \sqrt {1+x+x^2}}\right )+2 \sqrt {7} \tanh ^{-1}\left (\frac {1-2 x}{\sqrt {7} \sqrt {4+2 x+x^2}}\right ) \]
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Rubi [A]
time = 0.38, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 36, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {6874, 748,
857, 633, 221, 738, 212, 626} \begin {gather*} \frac {1}{2} \sqrt {x^2+2 x+4} (x+1)+\frac {1}{4} (2 x+1) \sqrt {x^2+x+1}-2 \sqrt {x^2+x+1}-2 \sqrt {x^2+2 x+4}-2 \sqrt {7} \tanh ^{-1}\left (\frac {5 x+1}{2 \sqrt {7} \sqrt {x^2+x+1}}\right )+2 \sqrt {7} \tanh ^{-1}\left (\frac {1-2 x}{\sqrt {7} \sqrt {x^2+2 x+4}}\right )+\frac {11}{2} \sinh ^{-1}\left (\frac {x+1}{\sqrt {3}}\right )+\frac {43}{8} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 626
Rule 633
Rule 738
Rule 748
Rule 857
Rule 6874
Rubi steps
\begin {align*} \int \frac {1+x}{-\sqrt {1+x+x^2}+\sqrt {4+2 x+x^2}} \, dx &=\int \left (-\frac {1}{\sqrt {1+x+x^2}-\sqrt {4+2 x+x^2}}-\frac {x}{\sqrt {1+x+x^2}-\sqrt {4+2 x+x^2}}\right ) \, dx\\ &=-\int \frac {1}{\sqrt {1+x+x^2}-\sqrt {4+2 x+x^2}} \, dx-\int \frac {x}{\sqrt {1+x+x^2}-\sqrt {4+2 x+x^2}} \, dx\\ &=-\int \left (-\frac {\sqrt {1+x+x^2}}{3+x}-\frac {\sqrt {4+2 x+x^2}}{3+x}\right ) \, dx-\int \left (-\sqrt {1+x+x^2}+\frac {3 \sqrt {1+x+x^2}}{3+x}-\sqrt {4+2 x+x^2}+\frac {3 \sqrt {4+2 x+x^2}}{3+x}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt {1+x+x^2}}{3+x} \, dx\right )-3 \int \frac {\sqrt {4+2 x+x^2}}{3+x} \, dx+\int \sqrt {1+x+x^2} \, dx+\int \frac {\sqrt {1+x+x^2}}{3+x} \, dx+\int \sqrt {4+2 x+x^2} \, dx+\int \frac {\sqrt {4+2 x+x^2}}{3+x} \, dx\\ &=-2 \sqrt {1+x+x^2}+\frac {1}{4} (1+2 x) \sqrt {1+x+x^2}-2 \sqrt {4+2 x+x^2}+\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {3}{8} \int \frac {1}{\sqrt {1+x+x^2}} \, dx-\frac {1}{2} \int \frac {1+5 x}{(3+x) \sqrt {1+x+x^2}} \, dx-\frac {1}{2} \int \frac {-2+4 x}{(3+x) \sqrt {4+2 x+x^2}} \, dx+\frac {3}{2} \int \frac {1+5 x}{(3+x) \sqrt {1+x+x^2}} \, dx+\frac {3}{2} \int \frac {1}{\sqrt {4+2 x+x^2}} \, dx+\frac {3}{2} \int \frac {-2+4 x}{(3+x) \sqrt {4+2 x+x^2}} \, dx\\ &=-2 \sqrt {1+x+x^2}+\frac {1}{4} (1+2 x) \sqrt {1+x+x^2}-2 \sqrt {4+2 x+x^2}+\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}-2 \int \frac {1}{\sqrt {4+2 x+x^2}} \, dx-\frac {5}{2} \int \frac {1}{\sqrt {1+x+x^2}} \, dx+6 \int \frac {1}{\sqrt {4+2 x+x^2}} \, dx+7 \int \frac {1}{(3+x) \sqrt {1+x+x^2}} \, dx+7 \int \frac {1}{(3+x) \sqrt {4+2 x+x^2}} \, dx+\frac {15}{2} \int \frac {1}{\sqrt {1+x+x^2}} \, dx-21 \int \frac {1}{(3+x) \sqrt {1+x+x^2}} \, dx-21 \int \frac {1}{(3+x) \sqrt {4+2 x+x^2}} \, dx+\frac {1}{8} \sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )+\frac {1}{4} \sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{12}}} \, dx,x,2+2 x\right )\\ &=-2 \sqrt {1+x+x^2}+\frac {1}{4} (1+2 x) \sqrt {1+x+x^2}-2 \sqrt {4+2 x+x^2}+\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {3}{2} \sinh ^{-1}\left (\frac {1+x}{\sqrt {3}}\right )+\frac {3}{8} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-14 \text {Subst}\left (\int \frac {1}{28-x^2} \, dx,x,\frac {-1-5 x}{\sqrt {1+x+x^2}}\right )-14 \text {Subst}\left (\int \frac {1}{28-x^2} \, dx,x,\frac {2-4 x}{\sqrt {4+2 x+x^2}}\right )+42 \text {Subst}\left (\int \frac {1}{28-x^2} \, dx,x,\frac {-1-5 x}{\sqrt {1+x+x^2}}\right )+42 \text {Subst}\left (\int \frac {1}{28-x^2} \, dx,x,\frac {2-4 x}{\sqrt {4+2 x+x^2}}\right )-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{12}}} \, dx,x,2+2 x\right )}{\sqrt {3}}-\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )}{2 \sqrt {3}}+\sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{12}}} \, dx,x,2+2 x\right )+\frac {1}{2} \left (5 \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )\\ &=-2 \sqrt {1+x+x^2}+\frac {1}{4} (1+2 x) \sqrt {1+x+x^2}-2 \sqrt {4+2 x+x^2}+\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {11}{2} \sinh ^{-1}\left (\frac {1+x}{\sqrt {3}}\right )+\frac {43}{8} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-2 \sqrt {7} \tanh ^{-1}\left (\frac {1+5 x}{2 \sqrt {7} \sqrt {1+x+x^2}}\right )+2 \sqrt {7} \tanh ^{-1}\left (\frac {1-2 x}{\sqrt {7} \sqrt {4+2 x+x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 5.22, size = 159, normalized size = 1.01 \begin {gather*} \frac {1}{8} \left (-14 \sqrt {1+x+x^2}+4 x \sqrt {1+x+x^2}-12 \sqrt {4+2 x+x^2}+4 x \sqrt {4+2 x+x^2}-32 \sqrt {7} \tanh ^{-1}\left (\frac {3+x-\sqrt {1+x+x^2}}{\sqrt {7}}\right )-32 \sqrt {7} \tanh ^{-1}\left (\frac {3+x-\sqrt {4+2 x+x^2}}{\sqrt {7}}\right )-43 \log \left (-1-2 x+2 \sqrt {1+x+x^2}\right )-44 \log \left (-1-x+\sqrt {4+2 x+x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 140, normalized size = 0.89
method | result | size |
default | \(-2 \sqrt {\left (3+x \right )^{2}-5 x -8}+\frac {43 \arcsinh \left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{8}+2 \sqrt {7}\, \arctanh \left (\frac {\left (-1-5 x \right ) \sqrt {7}}{14 \sqrt {\left (3+x \right )^{2}-5 x -8}}\right )-2 \sqrt {\left (3+x \right )^{2}-4 x -5}+\frac {11 \arcsinh \left (\frac {\left (1+x \right ) \sqrt {3}}{3}\right )}{2}+2 \sqrt {7}\, \arctanh \left (\frac {\left (2-4 x \right ) \sqrt {7}}{14 \sqrt {\left (3+x \right )^{2}-4 x -5}}\right )+\frac {\left (1+2 x \right ) \sqrt {x^{2}+x +1}}{4}+\frac {\left (2+2 x \right ) \sqrt {x^{2}+2 x +4}}{4}\) | \(140\) |
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 = 1.42, size = 155, normalized size = 0.98 \begin {gather*} \frac {1}{4} \, \sqrt {x^{2} + x + 1} {\left (2 \, x - 7\right )} + \frac {1}{2} \, \sqrt {x^{2} + 2 \, x + 4} {\left (x - 3\right )} + 2 \, \sqrt {7} \log \left (\frac {2 \, \sqrt {7} {\left (5 \, x + 1\right )} + 2 \, \sqrt {x^{2} + x + 1} {\left (5 \, \sqrt {7} - 14\right )} - 25 \, x - 5}{x + 3}\right ) + 2 \, \sqrt {7} \log \left (\frac {\sqrt {7} {\left (2 \, x - 1\right )} + \sqrt {x^{2} + 2 \, x + 4} {\left (2 \, \sqrt {7} - 7\right )} - 4 \, x + 2}{x + 3}\right ) - \frac {11}{2} \, \log \left (-x + \sqrt {x^{2} + 2 \, x + 4} - 1\right ) - \frac {43}{8} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 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 {x + 1}{- \sqrt {x^{2} + x + 1} + \sqrt {x^{2} + 2 x + 4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {x+1}{\sqrt {x^2+x+1}-\sqrt {x^2+2\,x+4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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