Optimal. Leaf size=223 \[ \frac {2 \text {ArcTan}\left (\sqrt {\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \text {ArcTan}\left (\sqrt {\frac {1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+(-1)^{3/5}}}-\frac {2 \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt {-1+(-1)^{2/5}}}-\frac {2 \sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh ^{-1}\left (\sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )}+\frac {\sin (x)}{5 (1+\cos (x))} \]
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Rubi [A]
time = 0.39, antiderivative size = 223, normalized size of antiderivative = 1.00, 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 = {3292, 2727,
2738, 214, 211} \begin {gather*} \frac {2 \text {ArcTan}\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 \text {ArcTan}\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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 2727
Rule 2738
Rule 3292
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} \text {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} \text {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} \text {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} \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.13, size = 378, normalized size = 1.70 \begin {gather*} -\frac {1}{10} \text {RootSum}\left [1-2 \text {$\#$1}+8 \text {$\#$1}^2-14 \text {$\#$1}^3+30 \text {$\#$1}^4-14 \text {$\#$1}^5+8 \text {$\#$1}^6-2 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {2 \text {ArcTan}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )-8 \text {ArcTan}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}+4 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+30 \text {ArcTan}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2-15 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-80 \text {ArcTan}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3+40 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+30 \text {ArcTan}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4-15 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-8 \text {ArcTan}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^5+4 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^5+2 \text {ArcTan}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-1+8 \text {$\#$1}-21 \text {$\#$1}^2+60 \text {$\#$1}^3-35 \text {$\#$1}^4+24 \text {$\#$1}^5-7 \text {$\#$1}^6+4 \text {$\#$1}^7}\&\right ]+\frac {1}{5} \tan \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.16, size = 62, normalized size = 0.28
method | result | size |
default | \(\frac {\tan \left (\frac {x}{2}\right )}{5}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (5 \textit {\_Z}^{8}+10 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (5 \textit {\_R}^{6}+5 \textit {\_R}^{4}+5 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7}+\textit {\_R}^{3}}\right )}{50}\) | \(62\) |
risch | \(\frac {2 i}{5 \left ({\mathrm e}^{i x}+1\right )}+\left (\munderset {\textit {\_R} =\RootOf \left (1953125 \textit {\_Z}^{8}+156250 \textit {\_Z}^{6}+6250 \textit {\_Z}^{4}+125 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}-2343750 i \textit {\_R}^{7}+234375 \textit {\_R}^{6}-140625 i \textit {\_R}^{5}+15625 \textit {\_R}^{4}-4375 i \textit {\_R}^{3}+500 \textit {\_R}^{2}-50 i \textit {\_R} +6\right )\right )\) | \(87\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [B]
time = 2.78, size = 535, normalized size = 2.40 \begin {gather*} \frac {\mathrm {tan}\left (\frac {x}{2}\right )}{5}+2\,\mathrm {atanh}\left (\frac {603979776\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{244140625\,\left (\frac {33554432\,\sqrt {5}\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{1220703125}-\frac {134217728\,\sqrt {5}}{1220703125}+\frac {67108864\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{1220703125}-\frac {301989888}{1220703125}\right )}+\frac {268435456\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{244140625\,\left (\frac {33554432\,\sqrt {5}\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{1220703125}-\frac {134217728\,\sqrt {5}}{1220703125}+\frac {67108864\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{1220703125}-\frac {301989888}{1220703125}\right )}\right )\,\sqrt {-\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}-2\,\mathrm {atanh}\left (\frac {603979776\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{244140625\,\left (\frac {33554432\,\sqrt {5}\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{1220703125}+\frac {134217728\,\sqrt {5}}{1220703125}+\frac {67108864\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{1220703125}+\frac {301989888}{1220703125}\right )}+\frac {268435456\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{244140625\,\left (\frac {33554432\,\sqrt {5}\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{1220703125}+\frac {134217728\,\sqrt {5}}{1220703125}+\frac {67108864\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{1220703125}+\frac {301989888}{1220703125}\right )}\right )\,\sqrt {\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}-2\,\mathrm {atanh}\left (\frac {603979776\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{244140625\,\left (\frac {33554432\,\sqrt {5}\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{1220703125}-\frac {134217728\,\sqrt {5}}{1220703125}-\frac {67108864\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{1220703125}+\frac {301989888}{1220703125}\right )}-\frac {268435456\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{244140625\,\left (\frac {33554432\,\sqrt {5}\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{1220703125}-\frac {134217728\,\sqrt {5}}{1220703125}-\frac {67108864\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{1220703125}+\frac {301989888}{1220703125}\right )}\right )\,\sqrt {-\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}+2\,\mathrm {atanh}\left (\frac {603979776\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{244140625\,\left (\frac {33554432\,\sqrt {5}\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{1220703125}+\frac {134217728\,\sqrt {5}}{1220703125}-\frac {67108864\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{1220703125}-\frac {301989888}{1220703125}\right )}-\frac {268435456\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{244140625\,\left (\frac {33554432\,\sqrt {5}\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{1220703125}+\frac {134217728\,\sqrt {5}}{1220703125}-\frac {67108864\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{1220703125}-\frac {301989888}{1220703125}\right )}\right )\,\sqrt {\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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