Optimal. Leaf size=145 \[ -\frac{\csc ^{10}(c+d x)}{10 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}+\frac{\csc ^8(c+d x)}{4 a^2 d}-\frac{6 \csc ^7(c+d x)}{7 a^2 d}+\frac{6 \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^4(c+d x)}{2 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{2 a^2 d} \]
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Rubi [A] time = 0.0811955, 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 = {2707, 88} \[ -\frac{\csc ^{10}(c+d x)}{10 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}+\frac{\csc ^8(c+d x)}{4 a^2 d}-\frac{6 \csc ^7(c+d x)}{7 a^2 d}+\frac{6 \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^4(c+d x)}{2 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 88
Rubi steps
\begin{align*} \int \frac{\cot ^{11}(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^5 (a+x)^3}{x^{11}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^8}{x^{11}}-\frac{2 a^7}{x^{10}}-\frac{2 a^6}{x^9}+\frac{6 a^5}{x^8}-\frac{6 a^3}{x^6}+\frac{2 a^2}{x^5}+\frac{2 a}{x^4}-\frac{1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\csc ^2(c+d x)}{2 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^4(c+d x)}{2 a^2 d}+\frac{6 \csc ^5(c+d x)}{5 a^2 d}-\frac{6 \csc ^7(c+d x)}{7 a^2 d}+\frac{\csc ^8(c+d x)}{4 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}-\frac{\csc ^{10}(c+d x)}{10 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.205023, size = 88, normalized size = 0.61 \[ \frac{\csc ^2(c+d x) \left (-126 \csc ^8(c+d x)+280 \csc ^7(c+d x)+315 \csc ^6(c+d x)-1080 \csc ^5(c+d x)+1512 \csc ^3(c+d x)-630 \csc ^2(c+d x)-840 \csc (c+d x)+630\right )}{1260 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.139, size = 89, normalized size = 0.6 \begin{align*}{\frac{1}{d{a}^{2}} \left ( -{\frac{1}{10\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{6}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}+{\frac{6}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{2}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{2}{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 = 1.83669, size = 116, normalized size = 0.8 \begin{align*} \frac{630 \, \sin \left (d x + c\right )^{8} - 840 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 1512 \, \sin \left (d x + c\right )^{5} - 1080 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 126}{1260 \, a^{2} d \sin \left (d x + c\right )^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67582, size = 431, normalized size = 2.97 \begin{align*} -\frac{630 \, \cos \left (d x + c\right )^{8} - 1890 \, \cos \left (d x + c\right )^{6} + 1890 \, \cos \left (d x + c\right )^{4} - 945 \, \cos \left (d x + c\right )^{2} + 8 \,{\left (105 \, \cos \left (d x + c\right )^{6} - 126 \, \cos \left (d x + c\right )^{4} + 72 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 189}{1260 \,{\left (a^{2} d \cos \left (d x + c\right )^{10} - 5 \, a^{2} d \cos \left (d x + c\right )^{8} + 10 \, a^{2} d \cos \left (d x + c\right )^{6} - 10 \, a^{2} d \cos \left (d x + c\right )^{4} + 5 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.93118, size = 116, normalized size = 0.8 \begin{align*} \frac{630 \, \sin \left (d x + c\right )^{8} - 840 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 1512 \, \sin \left (d x + c\right )^{5} - 1080 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 126}{1260 \, a^{2} d \sin \left (d x + c\right )^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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