Optimal. Leaf size=145 \[ \frac{1}{4 a d (1-\cos (c+d x))}+\frac{15}{16 a d (\cos (c+d x)+1)}-\frac{1}{32 a d (1-\cos (c+d x))^2}-\frac{9}{32 a d (\cos (c+d x)+1)^2}+\frac{1}{24 a d (\cos (c+d x)+1)^3}+\frac{11 \log (1-\cos (c+d x))}{32 a d}+\frac{21 \log (\cos (c+d x)+1)}{32 a d} \]
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Rubi [A] time = 0.0980931, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac{1}{4 a d (1-\cos (c+d x))}+\frac{15}{16 a d (\cos (c+d x)+1)}-\frac{1}{32 a d (1-\cos (c+d x))^2}-\frac{9}{32 a d (\cos (c+d x)+1)^2}+\frac{1}{24 a d (\cos (c+d x)+1)^3}+\frac{11 \log (1-\cos (c+d x))}{32 a d}+\frac{21 \log (\cos (c+d x)+1)}{32 a d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \frac{\cot ^5(c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac{a^6 \operatorname{Subst}\left (\int \frac{x^6}{(a-a x)^3 (a+a x)^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^6 \operatorname{Subst}\left (\int \left (-\frac{1}{16 a^7 (-1+x)^3}-\frac{1}{4 a^7 (-1+x)^2}-\frac{11}{32 a^7 (-1+x)}+\frac{1}{8 a^7 (1+x)^4}-\frac{9}{16 a^7 (1+x)^3}+\frac{15}{16 a^7 (1+x)^2}-\frac{21}{32 a^7 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{1}{32 a d (1-\cos (c+d x))^2}+\frac{1}{4 a d (1-\cos (c+d x))}+\frac{1}{24 a d (1+\cos (c+d x))^3}-\frac{9}{32 a d (1+\cos (c+d x))^2}+\frac{15}{16 a d (1+\cos (c+d x))}+\frac{11 \log (1-\cos (c+d x))}{32 a d}+\frac{21 \log (1+\cos (c+d x))}{32 a d}\\ \end{align*}
Mathematica [A] time = 0.538169, size = 135, normalized size = 0.93 \[ -\frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (3 \csc ^4\left (\frac{1}{2} (c+d x)\right )-48 \csc ^2\left (\frac{1}{2} (c+d x)\right )-2 \sec ^6\left (\frac{1}{2} (c+d x)\right )+27 \sec ^4\left (\frac{1}{2} (c+d x)\right )-180 \sec ^2\left (\frac{1}{2} (c+d x)\right )-264 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-504 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{192 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.068, size = 126, normalized size = 0.9 \begin{align*}{\frac{1}{24\,da \left ( \cos \left ( dx+c \right ) +1 \right ) ^{3}}}-{\frac{9}{32\,da \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}+{\frac{15}{16\,da \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{21\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{32\,da}}-{\frac{1}{32\,da \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{4\,da \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{11\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{32\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17699, size = 176, normalized size = 1.21 \begin{align*} \frac{\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{4} - 39 \, \cos \left (d x + c\right )^{3} - 79 \, \cos \left (d x + c\right )^{2} + 29 \, \cos \left (d x + c\right ) + 44\right )}}{a \cos \left (d x + c\right )^{5} + a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) + a} + \frac{63 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac{33 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2057, size = 610, normalized size = 4.21 \begin{align*} \frac{66 \, \cos \left (d x + c\right )^{4} - 78 \, \cos \left (d x + c\right )^{3} - 158 \, \cos \left (d x + c\right )^{2} + 63 \,{\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 33 \,{\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 58 \, \cos \left (d x + c\right ) + 88}{96 \,{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\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 ^{5}{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4128, size = 285, normalized size = 1.97 \begin{align*} -\frac{\frac{3 \,{\left (\frac{14 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{66 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac{132 \, \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} + \frac{384 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} + \frac{\frac{132 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{21 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{2 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{3}}}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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