Optimal. Leaf size=213 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{a}-\sqrt [4]{b}} \cot (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt{\sqrt [4]{a}-\sqrt [4]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{a}-i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt{\sqrt [4]{a}-i \sqrt [4]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{a}+i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt{\sqrt [4]{a}+i \sqrt [4]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{a}+\sqrt [4]{b}} \cot (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt{\sqrt [4]{a}+\sqrt [4]{b}}} \]
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Rubi [A] time = 0.215615, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3211, 3181, 203} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{a}-\sqrt [4]{b}} \cot (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt{\sqrt [4]{a}-\sqrt [4]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{a}-i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt{\sqrt [4]{a}-i \sqrt [4]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{a}+i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt{\sqrt [4]{a}+i \sqrt [4]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{a}+\sqrt [4]{b}} \cot (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt{\sqrt [4]{a}+\sqrt [4]{b}}} \]
Antiderivative was successfully verified.
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Rule 3211
Rule 3181
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{a-b \cos ^8(x)} \, dx &=\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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\tan ^{-1}\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{\tan ^{-1}\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{\tan ^{-1}\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{\tan ^{-1}\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}}}\\ \end{align*}
Mathematica [C] time = 0.208781, size = 172, normalized size = 0.81 \[ -8 \text{RootSum}\left [-256 \text{$\#$1}^4 a+\text{$\#$1}^8 b+8 \text{$\#$1}^7 b+28 \text{$\#$1}^6 b+56 \text{$\#$1}^5 b+70 \text{$\#$1}^4 b+56 \text{$\#$1}^3 b+28 \text{$\#$1}^2 b+8 \text{$\#$1} b+b\& ,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left (\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right )-i \text{$\#$1}^3 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right )}{-128 \text{$\#$1}^3 a+\text{$\#$1}^7 b+7 \text{$\#$1}^6 b+21 \text{$\#$1}^5 b+35 \text{$\#$1}^4 b+35 \text{$\#$1}^3 b+21 \text{$\#$1}^2 b+7 \text{$\#$1} b+b}\& \right ] \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.089, size = 78, normalized size = 0.4 \begin{align*}{\frac{1}{8\,a}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{8}+4\,a{{\it \_Z}}^{6}+6\,a{{\it \_Z}}^{4}+4\,a{{\it \_Z}}^{2}+a-b \right ) }{\frac{ \left ({{\it \_R}}^{6}+3\,{{\it \_R}}^{4}+3\,{{\it \_R}}^{2}+1 \right ) \ln \left ( \tan \left ( x \right ) -{\it \_R} \right ) }{{{\it \_R}}^{7}+3\,{{\it \_R}}^{5}+3\,{{\it \_R}}^{3}+{\it \_R}}}} \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*} -\int \frac{1}{b \cos \left (x\right )^{8} - a}\,{d x} \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}{b \cos \left (x\right )^{8} - a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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