Optimal. Leaf size=234 \[ -\frac{1}{4} \sqrt{\frac{1}{5} \left (\sqrt{5}-2\right )} \log \left (\left (\frac{1}{x}+1\right )^2-\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )+\frac{1}{4} \sqrt{\frac{1}{5} \left (\sqrt{5}-2\right )} \log \left (\left (\frac{1}{x}+1\right )^2+\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )+\frac{1}{2} \tan ^{-1}\left (\frac{1}{2} \left (\left (\frac{1}{x}+1\right )^2-1\right )\right )-\frac{1}{2} \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}-\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right )-\frac{1}{2} \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}+\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.315422, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {2069, 1673, 1169, 634, 618, 204, 628, 12, 1107} \[ -\frac{1}{4} \sqrt{\frac{1}{5} \left (\sqrt{5}-2\right )} \log \left (\left (\frac{1}{x}+1\right )^2-\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )+\frac{1}{4} \sqrt{\frac{1}{5} \left (\sqrt{5}-2\right )} \log \left (\left (\frac{1}{x}+1\right )^2+\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )+\frac{1}{2} \tan ^{-1}\left (\frac{1}{2} \left (\left (\frac{1}{x}+1\right )^2-1\right )\right )-\frac{1}{2} \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}-\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right )-\frac{1}{2} \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}+\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2069
Rule 1673
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rule 12
Rule 1107
Rubi steps
\begin{align*} \int \frac{1}{1+4 x+4 x^2+4 x^4} \, dx &=-\left (16 \operatorname{Subst}\left (\int \frac{(4-4 x)^2}{1280-512 x^2+256 x^4} \, dx,x,1+\frac{1}{x}\right )\right )\\ &=-\left (16 \operatorname{Subst}\left (\int -\frac{32 x}{1280-512 x^2+256 x^4} \, dx,x,1+\frac{1}{x}\right )\right )-16 \operatorname{Subst}\left (\int \frac{16+16 x^2}{1280-512 x^2+256 x^4} \, dx,x,1+\frac{1}{x}\right )\\ &=512 \operatorname{Subst}\left (\int \frac{x}{1280-512 x^2+256 x^4} \, dx,x,1+\frac{1}{x}\right )-\frac{\operatorname{Subst}\left (\int \frac{16 \sqrt{2 \left (1+\sqrt{5}\right )}-\left (16-16 \sqrt{5}\right ) x}{\sqrt{5}-\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )}{32 \sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{\operatorname{Subst}\left (\int \frac{16 \sqrt{2 \left (1+\sqrt{5}\right )}+\left (16-16 \sqrt{5}\right ) x}{\sqrt{5}+\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )}{32 \sqrt{10 \left (1+\sqrt{5}\right )}}\\ &=256 \operatorname{Subst}\left (\int \frac{1}{1280-512 x+256 x^2} \, dx,x,\left (1+\frac{1}{x}\right )^2\right )+\frac{\left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt{2 \left (1+\sqrt{5}\right )}+2 x}{\sqrt{5}-\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )}{4 \sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{\left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2 \left (1+\sqrt{5}\right )}+2 x}{\sqrt{5}+\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )}{4 \sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{1}{20} \left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{5}-\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )-\frac{1}{20} \left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{5}+\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )\\ &=-\frac{1}{4} \sqrt{-\frac{2}{5}+\frac{1}{\sqrt{5}}} \log \left (\sqrt{5}-\sqrt{2 \left (1+\sqrt{5}\right )} \left (1+\frac{1}{x}\right )+\left (1+\frac{1}{x}\right )^2\right )+\frac{1}{4} \sqrt{-\frac{2}{5}+\frac{1}{\sqrt{5}}} \log \left (\sqrt{5}+\sqrt{2 \left (1+\sqrt{5}\right )} \left (1+\frac{1}{x}\right )+\left (1+\frac{1}{x}\right )^2\right )-512 \operatorname{Subst}\left (\int \frac{1}{-1048576-x^2} \, dx,x,-512+512 \left (1+\frac{1}{x}\right )^2\right )+\frac{1}{10} \left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (1-\sqrt{5}\right )-x^2} \, dx,x,-\sqrt{2 \left (1+\sqrt{5}\right )}+2 \left (1+\frac{1}{x}\right )\right )+\frac{1}{10} \left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (1-\sqrt{5}\right )-x^2} \, dx,x,\sqrt{2 \left (1+\sqrt{5}\right )}+2 \left (1+\frac{1}{x}\right )\right )\\ &=\frac{1}{2} \tan ^{-1}\left (\frac{1}{2} \left (-1+\left (1+\frac{1}{x}\right )^2\right )\right )-\frac{\left (1+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\frac{2-\sqrt{2 \left (1+\sqrt{5}\right )}+\frac{2}{x}}{\sqrt{2 \left (-1+\sqrt{5}\right )}}\right )}{4 \sqrt{10}}-\frac{\left (1+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\frac{2+\sqrt{2 \left (1+\sqrt{5}\right )}+\frac{2}{x}}{\sqrt{2 \left (-1+\sqrt{5}\right )}}\right )}{4 \sqrt{10}}-\frac{1}{4} \sqrt{-\frac{2}{5}+\frac{1}{\sqrt{5}}} \log \left (\sqrt{5}-\sqrt{2 \left (1+\sqrt{5}\right )} \left (1+\frac{1}{x}\right )+\left (1+\frac{1}{x}\right )^2\right )+\frac{1}{4} \sqrt{-\frac{2}{5}+\frac{1}{\sqrt{5}}} \log \left (\sqrt{5}+\sqrt{2 \left (1+\sqrt{5}\right )} \left (1+\frac{1}{x}\right )+\left (1+\frac{1}{x}\right )^2\right )\\ \end{align*}
Mathematica [C] time = 0.0145809, size = 47, normalized size = 0.2 \[ \frac{1}{4} \text{RootSum}\left [4 \text{$\#$1}^4+4 \text{$\#$1}^2+4 \text{$\#$1}+1\& ,\frac{\log (x-\text{$\#$1})}{4 \text{$\#$1}^3+2 \text{$\#$1}+1}\& \right ] \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.004, size = 41, normalized size = 0.2 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ( 4\,{{\it \_Z}}^{4}+4\,{{\it \_Z}}^{2}+4\,{\it \_Z}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{4\,{{\it \_R}}^{3}+2\,{\it \_R}+1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 8.50128, size = 2303, normalized size = 9.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.724326, size = 36, normalized size = 0.15 \begin{align*} \operatorname{RootSum}{\left (1280 t^{4} + 288 t^{2} + 32 t + 1, \left ( t \mapsto t \log{\left (- 240 t^{3} + 10 t^{2} - 54 t + x - \frac{27}{8} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15943, size = 374, normalized size = 1.6 \begin{align*} \frac{{\left (\sqrt{\sqrt{5} - 2}{\left (\frac{i}{\sqrt{5} - 2} + 1\right )} + 2 \, i + 1\right )} \log \left (2 \,{\left (7 \, i + 3\right )} x + \sqrt{29 \, \sqrt{5} + 62}{\left (\frac{19 \, i}{29 \, \sqrt{5} + 62} + 1\right )} + 3 \, i - 7\right )}{4 \,{\left (i - 2\right )}} - \frac{{\left (\sqrt{\sqrt{5} - 2}{\left (\frac{i}{\sqrt{5} - 2} + 1\right )} - 2 \, i - 1\right )} \log \left (2 \,{\left (7 \, i + 3\right )} x - \sqrt{29 \, \sqrt{5} + 62}{\left (\frac{19 \, i}{29 \, \sqrt{5} + 62} + 1\right )} + 3 \, i - 7\right )}{4 \,{\left (i - 2\right )}} - \frac{{\left (\sqrt{\sqrt{5} + 2}{\left (\frac{i}{\sqrt{5} + 2} + 1\right )} + i + 2\right )} \log \left (2 \,{\left (i + 5\right )} x + \sqrt{13 \, \sqrt{5} - 22}{\left (\frac{19 \, i}{13 \, \sqrt{5} - 22} + 1\right )} - 5 \, i + 1\right )}{4 \,{\left (2 \, i - 1\right )}} + \frac{{\left (\sqrt{\sqrt{5} + 2}{\left (\frac{i}{\sqrt{5} + 2} + 1\right )} - i - 2\right )} \log \left (2 \,{\left (i + 5\right )} x - \sqrt{13 \, \sqrt{5} - 22}{\left (\frac{19 \, i}{13 \, \sqrt{5} - 22} + 1\right )} - 5 \, i + 1\right )}{4 \,{\left (2 \, i - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]