Optimal. Leaf size=137 \[ -\frac{(a-a \cos (c+d x))^{11}}{11 a^{13} d}+\frac{4 (a-a \cos (c+d x))^{10}}{5 a^{12} d}-\frac{25 (a-a \cos (c+d x))^9}{9 a^{11} d}+\frac{19 (a-a \cos (c+d x))^8}{4 a^{10} d}-\frac{4 (a-a \cos (c+d x))^7}{a^9 d}+\frac{4 (a-a \cos (c+d x))^6}{3 a^8 d} \]
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Rubi [A] time = 0.185802, antiderivative size = 137, 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 = {3872, 2836, 12, 88} \[ -\frac{(a-a \cos (c+d x))^{11}}{11 a^{13} d}+\frac{4 (a-a \cos (c+d x))^{10}}{5 a^{12} d}-\frac{25 (a-a \cos (c+d x))^9}{9 a^{11} d}+\frac{19 (a-a \cos (c+d x))^8}{4 a^{10} d}-\frac{4 (a-a \cos (c+d x))^7}{a^9 d}+\frac{4 (a-a \cos (c+d x))^6}{3 a^8 d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) \sin ^{11}(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-a-x)^5 x^2 (-a+x)^3}{a^2} \, dx,x,-a \cos (c+d x)\right )}{a^{11} d}\\ &=\frac{\operatorname{Subst}\left (\int (-a-x)^5 x^2 (-a+x)^3 \, dx,x,-a \cos (c+d x)\right )}{a^{13} d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-8 a^5 (-a-x)^5-28 a^4 (-a-x)^6-38 a^3 (-a-x)^7-25 a^2 (-a-x)^8-8 a (-a-x)^9-(-a-x)^{10}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{13} d}\\ &=\frac{4 (a-a \cos (c+d x))^6}{3 a^8 d}-\frac{4 (a-a \cos (c+d x))^7}{a^9 d}+\frac{19 (a-a \cos (c+d x))^8}{4 a^{10} d}-\frac{25 (a-a \cos (c+d x))^9}{9 a^{11} d}+\frac{4 (a-a \cos (c+d x))^{10}}{5 a^{12} d}-\frac{(a-a \cos (c+d x))^{11}}{11 a^{13} d}\\ \end{align*}
Mathematica [A] time = 4.82492, size = 72, normalized size = 0.53 \[ \frac{4 \sin ^{12}\left (\frac{1}{2} (c+d x)\right ) (4038 \cos (c+d x)+2586 \cos (2 (c+d x))+1189 \cos (3 (c+d x))+342 \cos (4 (c+d x))+45 \cos (5 (c+d x))+2360)}{495 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.112, size = 88, normalized size = 0.6 \begin{align*} -{\frac{1}{d{a}^{2}} \left ({\frac{1}{3\, \left ( \sec \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{5\, \left ( \sec \left ( dx+c \right ) \right ) ^{10}}}-{\frac{3}{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{8}}}+ \left ( \sec \left ( dx+c \right ) \right ) ^{-6}-{\frac{1}{11\, \left ( \sec \left ( dx+c \right ) \right ) ^{11}}}-{\frac{1}{2\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}}-{\frac{2}{5\, \left ( \sec \left ( dx+c \right ) \right ) ^{5}}}+{\frac{2}{9\, \left ( \sec \left ( dx+c \right ) \right ) ^{9}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01579, size = 120, normalized size = 0.88 \begin{align*} \frac{180 \, \cos \left (d x + c\right )^{11} - 396 \, \cos \left (d x + c\right )^{10} - 440 \, \cos \left (d x + c\right )^{9} + 1485 \, \cos \left (d x + c\right )^{8} - 1980 \, \cos \left (d x + c\right )^{6} + 792 \, \cos \left (d x + c\right )^{5} + 990 \, \cos \left (d x + c\right )^{4} - 660 \, \cos \left (d x + c\right )^{3}}{1980 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80994, size = 254, normalized size = 1.85 \begin{align*} \frac{180 \, \cos \left (d x + c\right )^{11} - 396 \, \cos \left (d x + c\right )^{10} - 440 \, \cos \left (d x + c\right )^{9} + 1485 \, \cos \left (d x + c\right )^{8} - 1980 \, \cos \left (d x + c\right )^{6} + 792 \, \cos \left (d x + c\right )^{5} + 990 \, \cos \left (d x + c\right )^{4} - 660 \, \cos \left (d x + c\right )^{3}}{1980 \, a^{2} d} \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 = 1.34112, size = 250, normalized size = 1.82 \begin{align*} -\frac{64 \,{\left (\frac{11 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{55 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{165 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{330 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{462 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{198 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{990 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 1\right )}}{495 \, a^{2} d{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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