Optimal. Leaf size=223 \[ \frac{2 \tan ^{-1}\left (\sqrt{\frac{1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tan \left (\frac{x}{2}\right )\right )}{5 \sqrt{1-(-1)^{4/5}}}+\frac{2 \tan ^{-1}\left (\sqrt{\frac{1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tan \left (\frac{x}{2}\right )\right )}{5 \sqrt{1+(-1)^{3/5}}}+\frac{\sin (x)}{5 (\cos (x)+1)}-\frac{2 \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )}{\sqrt{-\frac{1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt{(-1)^{2/5}-1}}-\frac{2 \sqrt{-\frac{1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh ^{-1}\left (\sqrt{-\frac{1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan \left (\frac{x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )} \]
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Rubi [A] time = 0.562301, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {3213, 2648, 2659, 208, 205} \[ \frac{2 \tan ^{-1}\left (\sqrt{\frac{1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tan \left (\frac{x}{2}\right )\right )}{5 \sqrt{1-(-1)^{4/5}}}+\frac{2 \tan ^{-1}\left (\sqrt{\frac{1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tan \left (\frac{x}{2}\right )\right )}{5 \sqrt{1+(-1)^{3/5}}}+\frac{\sin (x)}{5 (\cos (x)+1)}-\frac{2 \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )}{\sqrt{-\frac{1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt{(-1)^{2/5}-1}}-\frac{2 \sqrt{-\frac{1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh ^{-1}\left (\sqrt{-\frac{1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan \left (\frac{x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )} \]
Antiderivative was successfully verified.
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Rule 3213
Rule 2648
Rule 2659
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{1+\cos ^5(x)} \, dx &=\int \left (-\frac{1}{5 (-1-\cos (x))}-\frac{1}{5 \left (-1+\sqrt [5]{-1} \cos (x)\right )}-\frac{1}{5 \left (-1-(-1)^{2/5} \cos (x)\right )}-\frac{1}{5 \left (-1+(-1)^{3/5} \cos (x)\right )}-\frac{1}{5 \left (-1-(-1)^{4/5} \cos (x)\right )}\right ) \, dx\\ &=-\left (\frac{1}{5} \int \frac{1}{-1-\cos (x)} \, dx\right )-\frac{1}{5} \int \frac{1}{-1+\sqrt [5]{-1} \cos (x)} \, dx-\frac{1}{5} \int \frac{1}{-1-(-1)^{2/5} \cos (x)} \, dx-\frac{1}{5} \int \frac{1}{-1+(-1)^{3/5} \cos (x)} \, dx-\frac{1}{5} \int \frac{1}{-1-(-1)^{4/5} \cos (x)} \, dx\\ &=\frac{\sin (x)}{5 (1+\cos (x))}-\frac{2}{5} \operatorname{Subst}\left (\int \frac{1}{-1+\sqrt [5]{-1}+\left (-1-\sqrt [5]{-1}\right ) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{2}{5} \operatorname{Subst}\left (\int \frac{1}{-1-(-1)^{2/5}+\left (-1+(-1)^{2/5}\right ) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{2}{5} \operatorname{Subst}\left (\int \frac{1}{-1+(-1)^{3/5}+\left (-1-(-1)^{3/5}\right ) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{2}{5} \operatorname{Subst}\left (\int \frac{1}{-1-(-1)^{4/5}+\left (-1+(-1)^{4/5}\right ) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{2 \tan ^{-1}\left (\sqrt{\frac{1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tan \left (\frac{x}{2}\right )\right )}{5 \sqrt{1-(-1)^{4/5}}}+\frac{2 \tan ^{-1}\left (\sqrt{\frac{1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tan \left (\frac{x}{2}\right )\right )}{5 \sqrt{1+(-1)^{3/5}}}-\frac{2 \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )}{\sqrt{-\frac{1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt{-1+(-1)^{2/5}}}-\frac{2 \sqrt{-\frac{1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh ^{-1}\left (\sqrt{-\frac{1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan \left (\frac{x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )}+\frac{\sin (x)}{5 (1+\cos (x))}\\ \end{align*}
Mathematica [C] time = 0.122576, size = 378, normalized size = 1.7 \[ \frac{1}{5} \tan \left (\frac{x}{2}\right )-\frac{1}{10} \text{RootSum}\left [\text{$\#$1}^8-2 \text{$\#$1}^7+8 \text{$\#$1}^6-14 \text{$\#$1}^5+30 \text{$\#$1}^4-14 \text{$\#$1}^3+8 \text{$\#$1}^2-2 \text{$\#$1}+1\& ,\frac{-i \text{$\#$1}^6 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )+4 i \text{$\#$1}^5 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )-15 i \text{$\#$1}^4 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )+40 i \text{$\#$1}^3 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )-15 i \text{$\#$1}^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )+4 i \text{$\#$1} \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )-i \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )+2 \text{$\#$1}^6 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )-8 \text{$\#$1}^5 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )+30 \text{$\#$1}^4 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )-80 \text{$\#$1}^3 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )+30 \text{$\#$1}^2 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )-8 \text{$\#$1} \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )+2 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )}{4 \text{$\#$1}^7-7 \text{$\#$1}^6+24 \text{$\#$1}^5-35 \text{$\#$1}^4+60 \text{$\#$1}^3-21 \text{$\#$1}^2+8 \text{$\#$1}-1}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 62, normalized size = 0.3 \begin{align*}{\frac{1}{5}\tan \left ({\frac{x}{2}} \right ) }+{\frac{1}{50}\sum _{{\it \_R}={\it RootOf} \left ( 5\,{{\it \_Z}}^{8}+10\,{{\it \_Z}}^{4}+1 \right ) }{\frac{5\,{{\it \_R}}^{6}+5\,{{\it \_R}}^{4}+5\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{7}+{{\it \_R}}^{3}}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cos \left (x\right )^{5} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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