Optimal. Leaf size=35 \[ \log \left (-3+2 x+2 x^2+2 \sqrt {-4-3 x-2 x^2+2 x^3+x^4}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.11, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1694, 12, 1121,
635, 212} \begin {gather*} -\tanh ^{-1}\left (\frac {7-4 \left (x+\frac {1}{2}\right )^2}{\sqrt {16 \left (x+\frac {1}{2}\right )^4-56 \left (x+\frac {1}{2}\right )^2-51}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 635
Rule 1121
Rule 1694
Rubi steps
\begin {align*} \int \frac {1+2 x}{\sqrt {-4-3 x-2 x^2+2 x^3+x^4}} \, dx &=\text {Subst}\left (\int \frac {8 x}{\sqrt {-51-56 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=8 \text {Subst}\left (\int \frac {x}{\sqrt {-51-56 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=4 \text {Subst}\left (\int \frac {1}{\sqrt {-51-56 x+16 x^2}} \, dx,x,\left (\frac {1}{2}+x\right )^2\right )\\ &=8 \text {Subst}\left (\int \frac {1}{64-x^2} \, dx,x,\frac {8 \left (-7+4 \left (\frac {1}{2}+x\right )^2\right )}{\sqrt {-51-56 \left (\frac {1}{2}+x\right )^2+(1+2 x)^4}}\right )\\ &=-\tanh ^{-1}\left (\frac {7-(1+2 x)^2}{\sqrt {-51-14 (1+2 x)^2+(1+2 x)^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 35, normalized size = 1.00 \begin {gather*} \log \left (-3+2 x+2 x^2+2 \sqrt {-4-3 x-2 x^2+2 x^3+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.02, size = 782, normalized size = 22.34
method | result | size |
trager | \(-\ln \left (-2 x^{2}+2 \sqrt {x^{4}+2 x^{3}-2 x^{2}-3 x -4}-2 x +3\right )\) | \(36\) |
default | \(\text {Expression too large to display}\) | \(782\) |
elliptic | \(\text {Expression too large to display}\) | \(782\) |
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 = 0.40, size = 33, normalized size = 0.94 \begin {gather*} \log \left (2 \, x^{2} + 2 \, x + 2 \, \sqrt {x^{4} + 2 \, x^{3} - 2 \, x^{2} - 3 \, x - 4} - 3\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 x + 1}{\sqrt {\left (x^{2} + x - 4\right ) \left (x^{2} + x + 1\right )}}\, 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. 67 vs.
\(2 (33) = 66\).
time = 0.40, size = 67, normalized size = 1.91 \begin {gather*} \frac {1}{4} \, \sqrt {{\left (x^{2} + x\right )}^{2} - 3 \, x^{2} - 3 \, x - 4} {\left (2 \, x^{2} + 2 \, x - 3\right )} + \frac {25}{8} \, \log \left ({\left | -2 \, x^{2} - 2 \, x + 2 \, \sqrt {{\left (x^{2} + x\right )}^{2} - 3 \, x^{2} - 3 \, x - 4} + 3 \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 = -0.03 \begin {gather*} \int \frac {2\,x+1}{\sqrt {x^4+2\,x^3-2\,x^2-3\,x-4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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