Optimal. Leaf size=108 \[ -\frac{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right ) \sqrt{\frac{\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5}{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{1+\frac{1}{x}}{\sqrt [4]{5}}\right )|\frac{1}{10} \left (5+\sqrt{5}\right )\right )}{2 \sqrt [4]{5} \sqrt{4 x^4+4 x^2+4 x+1}} \]
[Out]
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Rubi [A] time = 0.363689, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right ) \sqrt{\frac{\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5}{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{1+\frac{1}{x}}{\sqrt [4]{5}}\right )|\frac{1}{10} \left (5+\sqrt{5}\right )\right )}{2 \sqrt [4]{5} \sqrt{4 x^4+4 x^2+4 x+1}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[1 + 4*x + 4*x^2 + 4*x^4],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - 16 \int ^{1 + \frac{1}{x}} \frac{1}{\sqrt{\frac{256 x^{4} - 512 x^{2} + 1280}{\left (- 4 x + 4\right )^{4}}} \left (- 4 x + 4\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(4*x**4+4*x**2+4*x+1)**(1/2),x)
[Out]
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Mathematica [C] time = 0.970966, 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.
[In] Integrate[1/Sqrt[1 + 4*x + 4*x^2 + 4*x^4],x]
[Out]
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Maple [C] time = 1.227, size = 961, normalized size = 8.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(4*x^4+4*x^2+4*x+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(4*x^4 + 4*x^2 + 4*x + 1),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(4*x^4 + 4*x^2 + 4*x + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 x^{4} + 4 x^{2} + 4 x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(4*x**4+4*x**2+4*x+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(4*x^4 + 4*x^2 + 4*x + 1),x, algorithm="giac")
[Out]