Optimal. Leaf size=494 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt [5]{a}-\sqrt [5]{b}} \tan \left (\frac{x}{2}\right )}{\sqrt{\sqrt [5]{a}+\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{a}-\sqrt [5]{b}} \sqrt{\sqrt [5]{a}+\sqrt [5]{b}}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac{x}{2}\right )}{\sqrt{\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt{\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac{x}{2}\right )}{\sqrt{\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt{\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac{x}{2}\right )}{\sqrt{\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt{\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac{x}{2}\right )}{\sqrt{\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt{\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]
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Rubi [A] time = 0.9185, antiderivative size = 494, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3213, 2659, 205} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt [5]{a}-\sqrt [5]{b}} \tan \left (\frac{x}{2}\right )}{\sqrt{\sqrt [5]{a}+\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{a}-\sqrt [5]{b}} \sqrt{\sqrt [5]{a}+\sqrt [5]{b}}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac{x}{2}\right )}{\sqrt{\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt{\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac{x}{2}\right )}{\sqrt{\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt{\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac{x}{2}\right )}{\sqrt{\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt{\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac{x}{2}\right )}{\sqrt{\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt{\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]
Antiderivative was successfully verified.
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Rule 3213
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{a+b \cos ^5(x)} \, dx &=\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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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*}
Mathematica [C] time = 0.203678, size = 130, normalized size = 0.26 \[ \frac{8}{5} \text{RootSum}\left [32 \text{$\#$1}^5 a+\text{$\#$1}^{10} b+5 \text{$\#$1}^8 b+10 \text{$\#$1}^6 b+10 \text{$\#$1}^4 b+5 \text{$\#$1}^2 b+b\& ,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )-i \text{$\#$1}^3 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )}{16 \text{$\#$1}^3 a+\text{$\#$1}^8 b+4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b+4 \text{$\#$1}^2 b+b}\& \right ] \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.027, size = 150, normalized size = 0.3 \begin{align*}{\frac{1}{5}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a-b \right ){{\it \_Z}}^{10}+ \left ( 5\,a+5\,b \right ){{\it \_Z}}^{8}+ \left ( 10\,a-10\,b \right ){{\it \_Z}}^{6}+ \left ( 10\,a+10\,b \right ){{\it \_Z}}^{4}+ \left ( 5\,a-5\,b \right ){{\it \_Z}}^{2}+a+b \right ) }{\frac{{{\it \_R}}^{8}+4\,{{\it \_R}}^{6}+6\,{{\it \_R}}^{4}+4\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{9}a-{{\it \_R}}^{9}b+4\,{{\it \_R}}^{7}a+4\,{{\it \_R}}^{7}b+6\,{{\it \_R}}^{5}a-6\,{{\it \_R}}^{5}b+4\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{3}b+{\it \_R}\,a-{\it \_R}\,b}\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*} \int \frac{1}{b \cos \left (x\right )^{5} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 )^{5} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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