Integrand size = 11, antiderivative size = 494 \[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]
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Time = 0.64 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3292, 2738, 211} \[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]
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Rule 211
Rule 2738
Rule 3292
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-\sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \cos (x)\right )}\right ) \, dx \\ & = \frac {\int \frac {1}{\sqrt [5]{a}-\sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [5]{a}-\sqrt [5]{b}+\left (\sqrt [5]{a}+\sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}+\left (\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}+\left (\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}+\left (\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}+\left (\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}} \\ & = \frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.15 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.26 \[ \int \frac {1}{a-b \cos ^5(x)} \, dx=-\frac {8}{5} \text {RootSum}\left [b+5 b \text {$\#$1}^2+10 b \text {$\#$1}^4-32 a \text {$\#$1}^5+10 b \text {$\#$1}^6+5 b \text {$\#$1}^8+b \text {$\#$1}^{10}\&,\frac {2 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{b+4 b \text {$\#$1}^2-16 a \text {$\#$1}^3+6 b \text {$\#$1}^4+4 b \text {$\#$1}^6+b \text {$\#$1}^8}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.62 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.30
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a +b \right ) \textit {\_Z}^{10}+\left (5 a -5 b \right ) \textit {\_Z}^{8}+\left (10 a +10 b \right ) \textit {\_Z}^{6}+\left (10 a -10 b \right ) \textit {\_Z}^{4}+\left (5 a +5 b \right ) \textit {\_Z}^{2}+a -b \right )}{\sum }\frac {\left (\textit {\_R}^{8}+4 \textit {\_R}^{6}+6 \textit {\_R}^{4}+4 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a +\textit {\_R}^{9} b +4 \textit {\_R}^{7} a -4 \textit {\_R}^{7} b +6 \textit {\_R}^{5} a +6 \textit {\_R}^{5} b +4 \textit {\_R}^{3} a -4 \textit {\_R}^{3} b +\textit {\_R} a +\textit {\_R} b}\right )}{5}\) | \(148\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (9765625 a^{10}-9765625 a^{8} b^{2}\right ) \textit {\_Z}^{10}+1953125 a^{8} \textit {\_Z}^{8}+156250 a^{6} \textit {\_Z}^{6}+6250 a^{4} \textit {\_Z}^{4}+125 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (\frac {11718750 i a^{10}}{b}-11718750 i b \,a^{8}\right ) \textit {\_R}^{9}+\left (-\frac {1171875 a^{9}}{b}+1171875 b \,a^{7}\right ) \textit {\_R}^{8}+\left (\frac {2109375 i a^{8}}{b}+234375 i a^{6} b \right ) \textit {\_R}^{7}+\left (-\frac {218750 a^{7}}{b}-15625 a^{5} b \right ) \textit {\_R}^{6}+\left (\frac {143750 i a^{6}}{b}-3125 i a^{4} b \right ) \textit {\_R}^{5}-\frac {15625 a^{5} \textit {\_R}^{4}}{b}+\frac {4375 i a^{4} \textit {\_R}^{3}}{b}-\frac {500 a^{3} \textit {\_R}^{2}}{b}+\frac {50 i a^{2} \textit {\_R}}{b}-\frac {6 a}{b}\right )\) | \(217\) |
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Exception generated. \[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\int \frac {1}{a - b \cos ^{5}{\left (x \right )}}\, dx \]
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\[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\int { -\frac {1}{b \cos \left (x\right )^{5} - a} \,d x } \]
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\[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\int { -\frac {1}{b \cos \left (x\right )^{5} - a} \,d x } \]
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Time = 8.98 (sec) , antiderivative size = 1518, normalized size of antiderivative = 3.07 \[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\text {Too large to display} \]
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