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 = 0.08, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 30
Rule 194
Rule 2606
Rule 2607
Rule 2706
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*}
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Mathematica [A] time = 0.08, 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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fricas [A] time = 2.40, size = 53, normalized size = 1.32 \[ -\frac {3 \, \cos \relax (x)^{4} - 12 \, \cos \relax (x)^{3} - 12 \, \cos \relax (x)^{2} + 8 \, \cos \relax (x) + 8}{15 \, {\left (a \cos \relax (x)^{3} + a \cos \relax (x)^{2} - a \cos \relax (x) - a\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 59, normalized size = 1.48 \[ \frac {12 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}{48 \, a \tan \left (\frac {1}{2} \, x\right )^{3}} + \frac {3 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{5} - 20 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{3} + 90 \, a^{4} \tan \left (\frac {1}{2} \, x\right )}{240 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 45, normalized size = 1.12 \[ \frac {\frac {\left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}+6 \tan \left (\frac {x}{2}\right )-\frac {1}{3 \tan \left (\frac {x}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {x}{2}\right )}}{16 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.72, size = 70, normalized size = 1.75 \[ \frac {\frac {90 \, \sin \relax (x)}{\cos \relax (x) + 1} - \frac {20 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {3 \, \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}}}{240 \, a} + \frac {{\left (\frac {12 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - 1\right )} {\left (\cos \relax (x) + 1\right )}^{3}}{48 \, a \sin \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 45, normalized size = 1.12 \[ \frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+90\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+60\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2-5}{240\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{4}{\relax (x )}}{\cos {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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