Optimal. Leaf size=108 \[ -\frac {\left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right ) \sqrt {\frac {5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4}{\left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {1+4 x+4 x^2+4 x^4}} \]
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Rubi [A]
time = 0.14, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2094, 6851,
1117} \begin {gather*} -\frac {\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right ) \sqrt {\frac {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}{\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right )^2}} x^2 F\left (2 \text {ArcTan}\left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {4 x^4+4 x^2+4 x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1117
Rule 2094
Rule 6851
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx &=-\left (16 \text {Subst}\left (\int \frac {1}{(4-4 x)^2 \sqrt {\frac {1280-512 x^2+256 x^4}{(4-4 x)^4}}} \, dx,x,1+\frac {1}{x}\right )\right )\\ &=-\frac {\left (\sqrt {1280-512 \left (1+\frac {1}{x}\right )^2+256 \left (1+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1280-512 x^2+256 x^4}} \, dx,x,1+\frac {1}{x}\right )}{\sqrt {1+4 x+4 x^2+4 x^4}}\\ &=-\frac {\left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right ) \sqrt {\frac {5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4}{\left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {1+4 x+4 x^2+4 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.39, size = 249, normalized size = 2.31 \begin {gather*} \frac {(2-i) \sqrt {-\frac {1}{10}+\frac {i}{5}} \sqrt {\frac {\left (2 i+\sqrt {-1-2 i}-\sqrt {-1+2 i}\right ) \left (-i+\sqrt {-1-2 i}-2 x\right )}{\left (-2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (i+\sqrt {-1-2 i}+2 x\right )}} \left (1+2 x+2 i x^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {\left (2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (-i+\sqrt {-1+2 i}+2 x\right )}{\sqrt {-1+2 i} \left (i+\sqrt {-1-2 i}+2 x\right )}}}{\sqrt {2}}\right )|\frac {1}{2} \left (5-\sqrt {5}\right )\right )}{\sqrt {\frac {(1+2 i) \left ((-1+i)+\sqrt {-1-2 i}\right ) \left (1+2 x+2 i x^2\right )}{\left (i+\sqrt {-1-2 i}+2 x\right )^2}} \sqrt {1+4 x+4 x^2+4 x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.95, size = 961, normalized size = 8.90
method | result | size |
default | \(\text {Expression too large to display}\) | \(961\) |
elliptic | \(\text {Expression too large to display}\) | \(961\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {4 x^{4} + 4 x^{2} + 4 x + 1}}\, 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 {1}{\sqrt {4\,x^4+4\,x^2+4\,x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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