Optimal. Leaf size=127 \[ -\frac {\csc ^8(c+d x)}{8 a^2 d}+\frac {2 \csc ^7(c+d x)}{7 a^2 d}+\frac {\csc ^6(c+d x)}{6 a^2 d}-\frac {4 \csc ^5(c+d x)}{5 a^2 d}+\frac {\csc ^4(c+d x)}{4 a^2 d}+\frac {2 \csc ^3(c+d x)}{3 a^2 d}-\frac {\csc ^2(c+d x)}{2 a^2 d} \]
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Rubi [A] time = 0.07, antiderivative size = 127, normalized size of antiderivative = 1.00, 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} \[ -\frac {\csc ^8(c+d x)}{8 a^2 d}+\frac {2 \csc ^7(c+d x)}{7 a^2 d}+\frac {\csc ^6(c+d x)}{6 a^2 d}-\frac {4 \csc ^5(c+d x)}{5 a^2 d}+\frac {\csc ^4(c+d x)}{4 a^2 d}+\frac {2 \csc ^3(c+d x)}{3 a^2 d}-\frac {\csc ^2(c+d x)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot ^9(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^4 (a+x)^2}{x^9} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^6}{x^9}-\frac {2 a^5}{x^8}-\frac {a^4}{x^7}+\frac {4 a^3}{x^6}-\frac {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)}{4 a^2 d}-\frac {4 \csc ^5(c+d x)}{5 a^2 d}+\frac {\csc ^6(c+d x)}{6 a^2 d}+\frac {2 \csc ^7(c+d x)}{7 a^2 d}-\frac {\csc ^8(c+d x)}{8 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 78, normalized size = 0.61 \[ \frac {\csc ^2(c+d x) \left (-105 \csc ^6(c+d x)+240 \csc ^5(c+d x)+140 \csc ^4(c+d x)-672 \csc ^3(c+d x)+210 \csc ^2(c+d x)+560 \csc (c+d x)-420\right )}{840 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 127, normalized size = 1.00 \[ \frac {420 \, \cos \left (d x + c\right )^{6} - 1050 \, \cos \left (d x + c\right )^{4} + 700 \, \cos \left (d x + c\right )^{2} + 16 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 28 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 175}{840 \, {\left (a^{2} d \cos \left (d x + c\right )^{8} - 4 \, a^{2} d \cos \left (d x + c\right )^{6} + 6 \, a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.77, size = 76, normalized size = 0.60 \[ -\frac {420 \, \sin \left (d x + c\right )^{6} - 560 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} + 672 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 240 \, \sin \left (d x + c\right ) + 105}{840 \, a^{2} d \sin \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 79, normalized size = 0.62 \[ \frac {\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {4}{5 \sin \left (d x +c \right )^{5}}+\frac {2}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}+\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {2}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 76, normalized size = 0.60 \[ -\frac {420 \, \sin \left (d x + c\right )^{6} - 560 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} + 672 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 240 \, \sin \left (d x + c\right ) + 105}{840 \, a^{2} d \sin \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.54, size = 76, normalized size = 0.60 \[ \frac {-420\,{\sin \left (c+d\,x\right )}^6+560\,{\sin \left (c+d\,x\right )}^5+210\,{\sin \left (c+d\,x\right )}^4-672\,{\sin \left (c+d\,x\right )}^3+140\,{\sin \left (c+d\,x\right )}^2+240\,\sin \left (c+d\,x\right )-105}{840\,a^2\,d\,{\sin \left (c+d\,x\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{9}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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