Optimal. Leaf size=145 \[ -\frac {\csc ^{10}(c+d x)}{10 a^3 d}+\frac {\csc ^9(c+d x)}{3 a^3 d}-\frac {\csc ^8(c+d x)}{8 a^3 d}-\frac {5 \csc ^7(c+d x)}{7 a^3 d}+\frac {5 \csc ^6(c+d x)}{6 a^3 d}+\frac {\csc ^5(c+d x)}{5 a^3 d}-\frac {3 \csc ^4(c+d x)}{4 a^3 d}+\frac {\csc ^3(c+d x)}{3 a^3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 145, 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} \[ -\frac {\csc ^{10}(c+d x)}{10 a^3 d}+\frac {\csc ^9(c+d x)}{3 a^3 d}-\frac {\csc ^8(c+d x)}{8 a^3 d}-\frac {5 \csc ^7(c+d x)}{7 a^3 d}+\frac {5 \csc ^6(c+d x)}{6 a^3 d}+\frac {\csc ^5(c+d x)}{5 a^3 d}-\frac {3 \csc ^4(c+d x)}{4 a^3 d}+\frac {\csc ^3(c+d x)}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot ^{11}(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^5 (a+x)^2}{x^{11}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^7}{x^{11}}-\frac {3 a^6}{x^{10}}+\frac {a^5}{x^9}+\frac {5 a^4}{x^8}-\frac {5 a^3}{x^7}-\frac {a^2}{x^6}+\frac {3 a}{x^5}-\frac {1}{x^4}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\csc ^3(c+d x)}{3 a^3 d}-\frac {3 \csc ^4(c+d x)}{4 a^3 d}+\frac {\csc ^5(c+d x)}{5 a^3 d}+\frac {5 \csc ^6(c+d x)}{6 a^3 d}-\frac {5 \csc ^7(c+d x)}{7 a^3 d}-\frac {\csc ^8(c+d x)}{8 a^3 d}+\frac {\csc ^9(c+d x)}{3 a^3 d}-\frac {\csc ^{10}(c+d x)}{10 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 88, normalized size = 0.61 \[ \frac {\csc ^3(c+d x) \left (-84 \csc ^7(c+d x)+280 \csc ^6(c+d x)-105 \csc ^5(c+d x)-600 \csc ^4(c+d x)+700 \csc ^3(c+d x)+168 \csc ^2(c+d x)-630 \csc (c+d x)+280\right )}{840 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 152, normalized size = 1.05 \[ -\frac {630 \, \cos \left (d x + c\right )^{6} - 1190 \, \cos \left (d x + c\right )^{4} + 595 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (35 \, \cos \left (d x + c\right )^{6} - 126 \, \cos \left (d x + c\right )^{4} + 72 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) - 119}{840 \, {\left (a^{3} d \cos \left (d x + c\right )^{10} - 5 \, a^{3} d \cos \left (d x + c\right )^{8} + 10 \, a^{3} d \cos \left (d x + c\right )^{6} - 10 \, a^{3} d \cos \left (d x + c\right )^{4} + 5 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.48, size = 86, normalized size = 0.59 \[ \frac {280 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 168 \, \sin \left (d x + c\right )^{5} + 700 \, \sin \left (d x + c\right )^{4} - 600 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 84}{840 \, a^{3} d \sin \left (d x + c\right )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 89, normalized size = 0.61 \[ \frac {\frac {5}{6 \sin \left (d x +c \right )^{6}}+\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {5}{7 \sin \left (d x +c \right )^{7}}+\frac {1}{3 \sin \left (d x +c \right )^{9}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{10 \sin \left (d x +c \right )^{10}}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 86, normalized size = 0.59 \[ \frac {280 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 168 \, \sin \left (d x + c\right )^{5} + 700 \, \sin \left (d x + c\right )^{4} - 600 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 84}{840 \, a^{3} d \sin \left (d x + c\right )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.81, size = 86, normalized size = 0.59 \[ \frac {280\,{\sin \left (c+d\,x\right )}^7-630\,{\sin \left (c+d\,x\right )}^6+168\,{\sin \left (c+d\,x\right )}^5+700\,{\sin \left (c+d\,x\right )}^4-600\,{\sin \left (c+d\,x\right )}^3-105\,{\sin \left (c+d\,x\right )}^2+280\,\sin \left (c+d\,x\right )-84}{840\,a^3\,d\,{\sin \left (c+d\,x\right )}^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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