Optimal. Leaf size=108 \[ -\frac {\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right ) \sqrt {\frac {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}{\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {4 x^4+4 x^2+4 x+1}} \]
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Rubi [A] time = 0.22, 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 = {2069, 6719, 1103} \[ -\frac {\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right ) \sqrt {\frac {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}{\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {4 x^4+4 x^2+4 x+1}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 2069
Rule 6719
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx &=-\left (16 \operatorname {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 ) \operatorname {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] time = 0.59, size = 249, normalized size = 2.31 \[ \frac {(2-i) \sqrt {-\frac {1}{10}+\frac {i}{5}} \sqrt {\frac {\left (2 i+\sqrt {-1-2 i}-\sqrt {-1+2 i}\right ) \left (-2 x+\sqrt {-1-2 i}-i\right )}{\left (-2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (2 x+\sqrt {-1-2 i}+i\right )}} \left (2 i x^2+2 x+1\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {\left (2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (2 x+\sqrt {-1+2 i}-i\right )}{\sqrt {-1+2 i} \left (2 x+\sqrt {-1-2 i}+i\right )}}}{\sqrt {2}}\right )|\frac {1}{2} \left (5-\sqrt {5}\right )\right )}{\sqrt {\frac {(1+2 i) \left ((-1+i)+\sqrt {-1-2 i}\right ) \left (2 i x^2+2 x+1\right )}{\left (2 x+\sqrt {-1-2 i}+i\right )^2}} \sqrt {4 x^4+4 x^2+4 x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.85, size = 961, normalized size = 8.90 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{\sqrt {4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {4\,x^4+4\,x^2+4\,x+1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {4 x^{4} + 4 x^{2} + 4 x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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