Optimal. Leaf size=40 \[ -\frac{\cot ^5(x)}{5 a}+\frac{\csc ^5(x)}{5 a}-\frac{2 \csc ^3(x)}{3 a}+\frac{\csc (x)}{a} \]
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Rubi [A] time = 0.0805112, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2706, 2607, 30, 2606, 194} \[ -\frac{\cot ^5(x)}{5 a}+\frac{\csc ^5(x)}{5 a}-\frac{2 \csc ^3(x)}{3 a}+\frac{\csc (x)}{a} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2607
Rule 30
Rule 2606
Rule 194
Rubi steps
\begin{align*} \int \frac{\cot ^4(x)}{a+a \cos (x)} \, dx &=-\frac{\int \cot ^5(x) \csc (x) \, dx}{a}+\frac{\int \cot ^4(x) \csc ^2(x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (x)\right )}{a}+\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (x)\right )}{a}\\ &=-\frac{\cot ^5(x)}{5 a}+\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (x)\right )}{a}\\ &=-\frac{\cot ^5(x)}{5 a}+\frac{\csc (x)}{a}-\frac{2 \csc ^3(x)}{3 a}+\frac{\csc ^5(x)}{5 a}\\ \end{align*}
Mathematica [A] time = 0.0831036, size = 41, normalized size = 1.02 \[ -\frac{(8 \cos (x)+36 \cos (2 x)+24 \cos (3 x)-3 \cos (4 x)-25) \csc ^3(x)}{120 a (\cos (x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 45, normalized size = 1.1 \begin{align*}{\frac{1}{16\,a} \left ({\frac{1}{5} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{4}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}+6\,\tan \left ( x/2 \right ) +4\, \left ( \tan \left ( x/2 \right ) \right ) ^{-1}-{\frac{1}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12601, size = 95, normalized size = 2.38 \begin{align*} \frac{\frac{90 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{20 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}}{240 \, a} + \frac{{\left (\frac{12 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}{\left (\cos \left (x\right ) + 1\right )}^{3}}{48 \, a \sin \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33074, size = 153, normalized size = 3.82 \begin{align*} -\frac{3 \, \cos \left (x\right )^{4} - 12 \, \cos \left (x\right )^{3} - 12 \, \cos \left (x\right )^{2} + 8 \, \cos \left (x\right ) + 8}{15 \,{\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )} \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot ^{4}{\left (x \right )}}{\cos{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33536, size = 80, normalized size = 2. \begin{align*} \frac{12 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 1}{48 \, a \tan \left (\frac{1}{2} \, x\right )^{3}} + \frac{3 \, a^{4} \tan \left (\frac{1}{2} \, x\right )^{5} - 20 \, a^{4} \tan \left (\frac{1}{2} \, x\right )^{3} + 90 \, a^{4} \tan \left (\frac{1}{2} \, x\right )}{240 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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