Optimal. Leaf size=84 \[ -\frac {\cot ^8(c+d x)}{8 a d}+\frac {\csc ^7(c+d x)}{7 a d}-\frac {3 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^3(c+d x)}{a d}-\frac {\csc (c+d x)}{a d} \]
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Rubi [A] time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2706, 2607, 30, 2606, 194} \[ -\frac {\cot ^8(c+d x)}{8 a d}+\frac {\csc ^7(c+d x)}{7 a d}-\frac {3 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^3(c+d x)}{a d}-\frac {\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 194
Rule 2606
Rule 2607
Rule 2706
Rubi steps
\begin {align*} \int \frac {\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^7(c+d x) \csc (c+d x) \, dx}{a}+\frac {\int \cot ^7(c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int x^7 \, dx,x,-\cot (c+d x)\right )}{a d}+\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac {\cot ^8(c+d x)}{8 a d}+\frac {\operatorname {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac {\cot ^8(c+d x)}{8 a d}-\frac {\csc (c+d x)}{a d}+\frac {\csc ^3(c+d x)}{a d}-\frac {3 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^7(c+d x)}{7 a d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 77, normalized size = 0.92 \[ \frac {\csc ^8(c+d x) (-513 \sin (c+d x)+371 \sin (3 (c+d x))-105 \sin (5 (c+d x))+35 \sin (7 (c+d x))-245 \cos (2 (c+d x))-35 \cos (6 (c+d x)))}{2240 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 127, normalized size = 1.51 \[ -\frac {140 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{4} + 140 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (35 \, \cos \left (d x + c\right )^{6} - 70 \, \cos \left (d x + c\right )^{4} + 56 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) - 35}{280 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 86, normalized size = 1.02 \[ -\frac {280 \, \sin \left (d x + c\right )^{7} - 140 \, \sin \left (d x + c\right )^{6} - 280 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} + 168 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 40 \, \sin \left (d x + c\right ) + 35}{280 \, a d \sin \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 87, normalized size = 1.04 \[ \frac {\frac {1}{2 \sin \left (d x +c \right )^{6}}-\frac {1}{\sin \left (d x +c \right )}-\frac {3}{5 \sin \left (d x +c \right )^{5}}+\frac {1}{7 \sin \left (d x +c \right )^{7}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{\sin \left (d x +c \right )^{3}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 86, normalized size = 1.02 \[ -\frac {280 \, \sin \left (d x + c\right )^{7} - 140 \, \sin \left (d x + c\right )^{6} - 280 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} + 168 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 40 \, \sin \left (d x + c\right ) + 35}{280 \, a d \sin \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.77, size = 83, normalized size = 0.99 \[ \frac {-{\sin \left (c+d\,x\right )}^7+\frac {{\sin \left (c+d\,x\right )}^6}{2}+{\sin \left (c+d\,x\right )}^5-\frac {3\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {3\,{\sin \left (c+d\,x\right )}^3}{5}+\frac {{\sin \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )}{7}-\frac {1}{8}}{a\,d\,{\sin \left (c+d\,x\right )}^8} \]
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 \[ \frac {\int \frac {\cot ^{9}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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