Optimal. Leaf size=106 \[ -\frac{\csc ^6(c+d x)}{6 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac{\cot (c+d x) \csc (c+d x)}{16 a d} \]
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Rubi [A] time = 0.172693, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3872, 2835, 2606, 30, 2611, 3768, 3770} \[ -\frac{\csc ^6(c+d x)}{6 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac{\cot (c+d x) \csc (c+d x)}{16 a d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2835
Rule 2606
Rule 30
Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\csc ^5(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac{\cot (c+d x) \csc ^4(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=-\frac{\int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{a}+\frac{\int \cot (c+d x) \csc ^6(c+d x) \, dx}{a}\\ &=\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac{\int \csc ^5(c+d x) \, dx}{6 a}-\frac{\operatorname{Subst}\left (\int x^5 \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac{\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\csc ^6(c+d x)}{6 a d}+\frac{\int \csc ^3(c+d x) \, dx}{8 a}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{16 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\csc ^6(c+d x)}{6 a d}+\frac{\int \csc (c+d x) \, dx}{16 a}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac{\cot (c+d x) \csc (c+d x)}{16 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\csc ^6(c+d x)}{6 a d}\\ \end{align*}
Mathematica [A] time = 0.474233, size = 122, normalized size = 1.15 \[ -\frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (3 \csc ^4\left (\frac{1}{2} (c+d x)\right )+12 \csc ^2\left (\frac{1}{2} (c+d x)\right )+2 \sec ^6\left (\frac{1}{2} (c+d x)\right )+3 \sec ^4\left (\frac{1}{2} (c+d x)\right )+24 \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{192 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 108, normalized size = 1. \begin{align*} -{\frac{1}{24\,da \left ( \cos \left ( dx+c \right ) +1 \right ) ^{3}}}-{\frac{1}{32\,da \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}-{\frac{\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}{16\,da \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{\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 = 1.01937, size = 176, normalized size = 1.66 \begin{align*} \frac{\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )^{2} - 5 \, \cos \left (d x + c\right ) - 8\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{3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac{3 \, \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 [B] time = 1.78485, size = 603, normalized size = 5.69 \begin{align*} \frac{6 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} - 10 \, \cos \left (d x + c\right )^{2} - 3 \,{\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 ) + 3 \,{\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 ) - 10 \, \cos \left (d x + c\right ) - 16}{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{\csc ^{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 = 1.38418, size = 246, normalized size = 2.32 \begin{align*} \frac{\frac{3 \,{\left (\frac{6 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{6 \,{\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{12 \, \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} + \frac{\frac{12 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{9 \, 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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