Optimal. Leaf size=114 \[ \frac{(a-a \cos (c+d x))^9}{9 a^{11} d}-\frac{3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac{13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac{2 (a-a \cos (c+d x))^6}{a^8 d}+\frac{4 (a-a \cos (c+d x))^5}{5 a^7 d} \]
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Rubi [A] time = 0.180202, antiderivative size = 114, 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} \[ \frac{(a-a \cos (c+d x))^9}{9 a^{11} d}-\frac{3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac{13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac{2 (a-a \cos (c+d x))^6}{a^8 d}+\frac{4 (a-a \cos (c+d x))^5}{5 a^7 d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) \sin ^9(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-a-x)^4 x^2 (-a+x)^2}{a^2} \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac{\operatorname{Subst}\left (\int (-a-x)^4 x^2 (-a+x)^2 \, dx,x,-a \cos (c+d x)\right )}{a^{11} d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^4 (-a-x)^4+12 a^3 (-a-x)^5+13 a^2 (-a-x)^6+6 a (-a-x)^7+(-a-x)^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{11} d}\\ &=\frac{4 (a-a \cos (c+d x))^5}{5 a^7 d}-\frac{2 (a-a \cos (c+d x))^6}{a^8 d}+\frac{13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac{3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac{(a-a \cos (c+d x))^9}{9 a^{11} d}\\ \end{align*}
Mathematica [A] time = 3.44935, size = 62, normalized size = 0.54 \[ \frac{2 \sin ^{10}\left (\frac{1}{2} (c+d x)\right ) (1615 \cos (c+d x)+970 \cos (2 (c+d x))+385 \cos (3 (c+d x))+70 \cos (4 (c+d x))+992)}{315 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 79, normalized size = 0.7 \begin{align*}{\frac{1}{d{a}^{2}} \left ( -{\frac{1}{3\, \left ( \sec \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{2\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{8}}}-{\frac{2}{3\, \left ( \sec \left ( dx+c \right ) \right ) ^{6}}}+{\frac{1}{5\, \left ( \sec \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{7\, \left ( \sec \left ( dx+c \right ) \right ) ^{7}}}-{\frac{1}{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 = 1.02519, size = 107, normalized size = 0.94 \begin{align*} -\frac{140 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 180 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 252 \, \cos \left (d x + c\right )^{5} - 630 \, \cos \left (d x + c\right )^{4} + 420 \, \cos \left (d x + c\right )^{3}}{1260 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7359, size = 221, normalized size = 1.94 \begin{align*} -\frac{140 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 180 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 252 \, \cos \left (d x + c\right )^{5} - 630 \, \cos \left (d x + c\right )^{4} + 420 \, \cos \left (d x + c\right )^{3}}{1260 \, 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 = 1.31472, size = 190, normalized size = 1.67 \begin{align*} -\frac{64 \,{\left (\frac{9 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{36 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{84 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{126 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{210 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 1\right )}}{315 \, a^{2} d{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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