Optimal. Leaf size=145 \[ -\frac {\csc ^{12}(c+d x)}{12 a^3 d}+\frac {3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {8 \csc ^9(c+d x)}{9 a^3 d}+\frac {3 \csc ^8(c+d x)}{4 a^3 d}+\frac {6 \csc ^7(c+d x)}{7 a^3 d}-\frac {4 \csc ^6(c+d x)}{3 a^3 d}+\frac {3 \csc ^4(c+d x)}{4 a^3 d}-\frac {\csc ^3(c+d x)}{3 a^3 d} \]
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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} \[ -\frac {\csc ^{12}(c+d x)}{12 a^3 d}+\frac {3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {8 \csc ^9(c+d x)}{9 a^3 d}+\frac {3 \csc ^8(c+d x)}{4 a^3 d}+\frac {6 \csc ^7(c+d x)}{7 a^3 d}-\frac {4 \csc ^6(c+d x)}{3 a^3 d}+\frac {3 \csc ^4(c+d x)}{4 a^3 d}-\frac {\csc ^3(c+d x)}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^6 (a+x)^3}{x^{13}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^9}{x^{13}}-\frac {3 a^8}{x^{12}}+\frac {8 a^6}{x^{10}}-\frac {6 a^5}{x^9}-\frac {6 a^4}{x^8}+\frac {8 a^3}{x^7}-\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 {4 \csc ^6(c+d x)}{3 a^3 d}+\frac {6 \csc ^7(c+d x)}{7 a^3 d}+\frac {3 \csc ^8(c+d x)}{4 a^3 d}-\frac {8 \csc ^9(c+d x)}{9 a^3 d}+\frac {3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {\csc ^{12}(c+d x)}{12 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 88, normalized size = 0.61 \[ \frac {\csc ^3(c+d x) \left (-231 \csc ^9(c+d x)+756 \csc ^8(c+d x)-2464 \csc ^6(c+d x)+2079 \csc ^5(c+d x)+2376 \csc ^4(c+d x)-3696 \csc ^3(c+d x)+2079 \csc (c+d x)-924\right )}{2772 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 185, normalized size = 1.28 \[ \frac {2079 \, \cos \left (d x + c\right )^{8} - 4620 \, \cos \left (d x + c\right )^{6} + 3465 \, \cos \left (d x + c\right )^{4} - 1386 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (231 \, \cos \left (d x + c\right )^{8} - 924 \, \cos \left (d x + c\right )^{6} + 792 \, \cos \left (d x + c\right )^{4} - 352 \, \cos \left (d x + c\right )^{2} + 64\right )} \sin \left (d x + c\right ) + 231}{2772 \, {\left (a^{3} d \cos \left (d x + c\right )^{12} - 6 \, a^{3} d \cos \left (d x + c\right )^{10} + 15 \, a^{3} d \cos \left (d x + c\right )^{8} - 20 \, a^{3} d \cos \left (d x + c\right )^{6} + 15 \, a^{3} d \cos \left (d x + c\right )^{4} - 6 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 6.36, size = 86, normalized size = 0.59 \[ -\frac {924 \, \sin \left (d x + c\right )^{9} - 2079 \, \sin \left (d x + c\right )^{8} + 3696 \, \sin \left (d x + c\right )^{6} - 2376 \, \sin \left (d x + c\right )^{5} - 2079 \, \sin \left (d x + c\right )^{4} + 2464 \, \sin \left (d x + c\right )^{3} - 756 \, \sin \left (d x + c\right ) + 231}{2772 \, a^{3} d \sin \left (d x + c\right )^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 89, normalized size = 0.61 \[ \frac {-\frac {4}{3 \sin \left (d x +c \right )^{6}}-\frac {1}{12 \sin \left (d x +c \right )^{12}}+\frac {6}{7 \sin \left (d x +c \right )^{7}}-\frac {8}{9 \sin \left (d x +c \right )^{9}}+\frac {3}{4 \sin \left (d x +c \right )^{8}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}+\frac {3}{11 \sin \left (d x +c \right )^{11}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 86, normalized size = 0.59 \[ -\frac {924 \, \sin \left (d x + c\right )^{9} - 2079 \, \sin \left (d x + c\right )^{8} + 3696 \, \sin \left (d x + c\right )^{6} - 2376 \, \sin \left (d x + c\right )^{5} - 2079 \, \sin \left (d x + c\right )^{4} + 2464 \, \sin \left (d x + c\right )^{3} - 756 \, \sin \left (d x + c\right ) + 231}{2772 \, a^{3} d \sin \left (d x + c\right )^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.82, size = 85, normalized size = 0.59 \[ \frac {-\frac {{\sin \left (c+d\,x\right )}^9}{3}+\frac {3\,{\sin \left (c+d\,x\right )}^8}{4}-\frac {4\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {6\,{\sin \left (c+d\,x\right )}^5}{7}+\frac {3\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {8\,{\sin \left (c+d\,x\right )}^3}{9}+\frac {3\,\sin \left (c+d\,x\right )}{11}-\frac {1}{12}}{a^3\,d\,{\sin \left (c+d\,x\right )}^{12}} \]
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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