Optimal. Leaf size=51 \[ -\frac{\cot ^4(c+d x)}{4 a d}+\frac{\csc ^3(c+d x)}{3 a d}-\frac{\csc (c+d x)}{a d} \]
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Rubi [A] time = 0.0889617, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2706, 2607, 30, 2606} \[ -\frac{\cot ^4(c+d x)}{4 a d}+\frac{\csc ^3(c+d x)}{3 a d}-\frac{\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2607
Rule 30
Rule 2606
Rubi steps
\begin{align*} \int \frac{\cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^3(c+d x) \csc (c+d x) \, dx}{a}+\frac{\int \cot ^3(c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,-\cot (c+d x)\right )}{a d}+\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac{\cot ^4(c+d x)}{4 a d}-\frac{\csc (c+d x)}{a d}+\frac{\csc ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.0486413, size = 30, normalized size = 0.59 \[ -\frac{(\csc (c+d x)-1)^3 (3 \csc (c+d x)+5)}{12 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 49, normalized size = 1. \begin{align*}{\frac{1}{da} \left ( - \left ( \sin \left ( dx+c \right ) \right ) ^{-1}-{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10562, size = 62, normalized size = 1.22 \begin{align*} -\frac{12 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 3}{12 \, a d \sin \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40033, size = 162, normalized size = 3.18 \begin{align*} -\frac{6 \, \cos \left (d x + c\right )^{2} - 4 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 3}{12 \,{\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + 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 )}}{\sin{\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 = 1.32141, size = 62, normalized size = 1.22 \begin{align*} -\frac{12 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 3}{12 \, a d \sin \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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