Integrand size = 10, antiderivative size = 245 \[ \int \frac {1}{a+b \cos ^8(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{-a}-\sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{-a}+\sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b}}} \]
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Time = 0.65 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3290, 3260, 209} \[ \int \frac {1}{a+b \cos ^8(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{-a}+\sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {a \sqrt [4]{b}+(-a)^{5/4}} \cot (x)}{(-a)^{5/8}}\right )}{4 (-a)^{3/8} \sqrt {a \sqrt [4]{b}+(-a)^{5/4}}} \]
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Rule 209
Rule 3260
Rule 3290
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{1-\frac {\sqrt [4]{b} \cos ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac {\int \frac {1}{1-\frac {i \sqrt [4]{b} \cos ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac {\int \frac {1}{1+\frac {i \sqrt [4]{b} \cos ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac {\int \frac {1}{1+\frac {\sqrt [4]{b} \cos ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{1+\left (1-\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\cot (x)\right )}{4 a}-\frac {\text {Subst}\left (\int \frac {1}{1+\left (1-\frac {i \sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\cot (x)\right )}{4 a}-\frac {\text {Subst}\left (\int \frac {1}{1+\left (1+\frac {i \sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\cot (x)\right )}{4 a}-\frac {\text {Subst}\left (\int \frac {1}{1+\left (1+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\cot (x)\right )}{4 a} \\ & = \frac {\arctan \left (\frac {\sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{-a}+\sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {(-a)^{5/4}+a \sqrt [4]{b}} \cot (x)}{(-a)^{5/8}}\right )}{4 (-a)^{3/8} \sqrt {(-a)^{5/4}+a \sqrt [4]{b}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.70 \[ \int \frac {1}{a+b \cos ^8(x)} \, dx=8 \text {RootSum}\left [b+8 b \text {$\#$1}+28 b \text {$\#$1}^2+56 b \text {$\#$1}^3+256 a \text {$\#$1}^4+70 b \text {$\#$1}^4+56 b \text {$\#$1}^5+28 b \text {$\#$1}^6+8 b \text {$\#$1}^7+b \text {$\#$1}^8\&,\frac {2 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{b+7 b \text {$\#$1}+21 b \text {$\#$1}^2+128 a \text {$\#$1}^3+35 b \text {$\#$1}^3+35 b \text {$\#$1}^4+21 b \text {$\#$1}^5+7 b \text {$\#$1}^6+b \text {$\#$1}^7}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.37 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.31
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} a +4 \textit {\_Z}^{6} a +6 \textit {\_Z}^{4} a +4 \textit {\_Z}^{2} a +a +b \right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{4}+3 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{\textit {\_R}^{7}+3 \textit {\_R}^{5}+3 \textit {\_R}^{3}+\textit {\_R}}}{8 a}\) | \(76\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (16777216 a^{8}+16777216 b \,a^{7}\right ) \textit {\_Z}^{8}+1048576 a^{6} \textit {\_Z}^{6}+24576 a^{4} \textit {\_Z}^{4}+256 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {4194304 i a^{8}}{b}+4194304 i a^{7}\right ) \textit {\_R}^{7}+\left (-\frac {524288 a^{7}}{b}-524288 a^{6}\right ) \textit {\_R}^{6}+\left (\frac {196608 i a^{6}}{b}-65536 i a^{5}\right ) \textit {\_R}^{5}+\left (-\frac {24576 a^{5}}{b}+8192 a^{4}\right ) \textit {\_R}^{4}+\left (\frac {3072 i a^{4}}{b}+1024 i a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {384 a^{3}}{b}-128 a^{2}\right ) \textit {\_R}^{2}+\left (\frac {16 i a^{2}}{b}-16 i a \right ) \textit {\_R} -\frac {2 a}{b}+1\right )\) | \(193\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 665467 vs. \(2 (165) = 330\).
Time = 6.13 (sec) , antiderivative size = 665467, normalized size of antiderivative = 2716.19 \[ \int \frac {1}{a+b \cos ^8(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{a+b \cos ^8(x)} \, dx=\int \frac {1}{a + b \cos ^{8}{\left (x \right )}}\, dx \]
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\[ \int \frac {1}{a+b \cos ^8(x)} \, dx=\int { \frac {1}{b \cos \left (x\right )^{8} + a} \,d x } \]
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\[ \int \frac {1}{a+b \cos ^8(x)} \, dx=\int { \frac {1}{b \cos \left (x\right )^{8} + a} \,d x } \]
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Time = 4.60 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.88 \[ \int \frac {1}{a+b \cos ^8(x)} \, dx=\sum _{k=1}^8\ln \left ({\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^4\,a^5\,b^5\,\left ({\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^2\,a^2\,64+1\right )\,\left (\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )\,a\,\mathrm {tan}\left (x\right )\,8-1\right )\,4096\right )\,\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right ) \]
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