Optimal. Leaf size=91 \[ \frac{\sin ^8(c+d x)}{8 a d}-\frac{\cos ^9(c+d x)}{9 a d}+\frac{3 \cos ^7(c+d x)}{7 a d}-\frac{3 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.161202, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3872, 2835, 2564, 30, 2565, 270} \[ \frac{\sin ^8(c+d x)}{8 a d}-\frac{\cos ^9(c+d x)}{9 a d}+\frac{3 \cos ^7(c+d x)}{7 a d}-\frac{3 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2835
Rule 2564
Rule 30
Rule 2565
Rule 270
Rubi steps
\begin{align*} \int \frac{\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac{\cos (c+d x) \sin ^9(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=\frac{\int \cos (c+d x) \sin ^7(c+d x) \, dx}{a}-\frac{\int \cos ^2(c+d x) \sin ^7(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int x^7 \, dx,x,\sin (c+d x)\right )}{a d}+\frac{\operatorname{Subst}\left (\int x^2 \left (1-x^2\right )^3 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac{\sin ^8(c+d x)}{8 a d}+\frac{\operatorname{Subst}\left (\int \left (x^2-3 x^4+3 x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac{\cos ^3(c+d x)}{3 a d}-\frac{3 \cos ^5(c+d x)}{5 a d}+\frac{3 \cos ^7(c+d x)}{7 a d}-\frac{\cos ^9(c+d x)}{9 a d}+\frac{\sin ^8(c+d x)}{8 a d}\\ \end{align*}
Mathematica [A] time = 4.23403, size = 62, normalized size = 0.68 \[ \frac{\sin ^{10}\left (\frac{1}{2} (c+d x)\right ) (6995 \cos (c+d x)+3650 \cos (2 (c+d x))+1085 \cos (3 (c+d x))+140 \cos (4 (c+d x))+4258)}{315 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.098, size = 89, normalized size = 1. \begin{align*}{\frac{1}{da} \left ({\frac{1}{3\, \left ( \sec \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{2\, \left ( \sec \left ( dx+c \right ) \right ) ^{6}}}-{\frac{1}{2\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{8\, \left ( \sec \left ( dx+c \right ) \right ) ^{8}}}+{\frac{3}{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3}{5\, \left ( \sec \left ( dx+c \right ) \right ) ^{5}}}+{\frac{3}{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 = 0.988516, size = 120, normalized size = 1.32 \begin{align*} -\frac{280 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 1080 \, \cos \left (d x + c\right )^{7} + 1260 \, \cos \left (d x + c\right )^{6} + 1512 \, \cos \left (d x + c\right )^{5} - 1890 \, \cos \left (d x + c\right )^{4} - 840 \, \cos \left (d x + c\right )^{3} + 1260 \, \cos \left (d x + c\right )^{2}}{2520 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74285, size = 254, normalized size = 2.79 \begin{align*} -\frac{280 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 1080 \, \cos \left (d x + c\right )^{7} + 1260 \, \cos \left (d x + c\right )^{6} + 1512 \, \cos \left (d x + c\right )^{5} - 1890 \, \cos \left (d x + c\right )^{4} - 840 \, \cos \left (d x + c\right )^{3} + 1260 \, \cos \left (d x + c\right )^{2}}{2520 \, a 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 = 1.30621, size = 190, normalized size = 2.09 \begin{align*} \frac{32 \,{\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{630 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1\right )}}{315 \, a 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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