Optimal. Leaf size=109 \[ -\frac{\csc ^{12}(c+d x)}{12 a d}+\frac{\csc ^{11}(c+d x)}{11 a d}+\frac{\csc ^{10}(c+d x)}{5 a d}-\frac{2 \csc ^9(c+d x)}{9 a d}-\frac{\csc ^8(c+d x)}{8 a d}+\frac{\csc ^7(c+d x)}{7 a d} \]
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Rubi [A] time = 0.121952, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac{\csc ^{12}(c+d x)}{12 a d}+\frac{\csc ^{11}(c+d x)}{11 a d}+\frac{\csc ^{10}(c+d x)}{5 a d}-\frac{2 \csc ^9(c+d x)}{9 a d}-\frac{\csc ^8(c+d x)}{8 a d}+\frac{\csc ^7(c+d x)}{7 a d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^{13} (a-x)^3 (a+x)^2}{x^{13}} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{a^6 \operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^2}{x^{13}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^6 \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{13}}-\frac{a^4}{x^{12}}-\frac{2 a^3}{x^{11}}+\frac{2 a^2}{x^{10}}+\frac{a}{x^9}-\frac{1}{x^8}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\csc ^7(c+d x)}{7 a d}-\frac{\csc ^8(c+d x)}{8 a d}-\frac{2 \csc ^9(c+d x)}{9 a d}+\frac{\csc ^{10}(c+d x)}{5 a d}+\frac{\csc ^{11}(c+d x)}{11 a d}-\frac{\csc ^{12}(c+d x)}{12 a d}\\ \end{align*}
Mathematica [A] time = 0.108163, size = 68, normalized size = 0.62 \[ \frac{\csc ^7(c+d x) \left (-2310 \csc ^5(c+d x)+2520 \csc ^4(c+d x)+5544 \csc ^3(c+d x)-6160 \csc ^2(c+d x)-3465 \csc (c+d x)+3960\right )}{27720 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.215, size = 69, normalized size = 0.6 \begin{align*}{\frac{1}{da} \left ({\frac{1}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}+{\frac{1}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{1}{11\, \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}-{\frac{1}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{2}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{1}{12\, \left ( \sin \left ( dx+c \right ) \right ) ^{12}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0265, size = 89, normalized size = 0.82 \begin{align*} \frac{3960 \, \sin \left (d x + c\right )^{5} - 3465 \, \sin \left (d x + c\right )^{4} - 6160 \, \sin \left (d x + c\right )^{3} + 5544 \, \sin \left (d x + c\right )^{2} + 2520 \, \sin \left (d x + c\right ) - 2310}{27720 \, a d \sin \left (d x + c\right )^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24107, size = 363, normalized size = 3.33 \begin{align*} -\frac{3465 \, \cos \left (d x + c\right )^{4} - 1386 \, \cos \left (d x + c\right )^{2} - 40 \,{\left (99 \, \cos \left (d x + c\right )^{4} - 44 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 231}{27720 \,{\left (a d \cos \left (d x + c\right )^{12} - 6 \, a d \cos \left (d x + c\right )^{10} + 15 \, a d \cos \left (d x + c\right )^{8} - 20 \, a d \cos \left (d x + c\right )^{6} + 15 \, a d \cos \left (d x + c\right )^{4} - 6 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \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 = 1.32478, size = 89, normalized size = 0.82 \begin{align*} \frac{3960 \, \sin \left (d x + c\right )^{5} - 3465 \, \sin \left (d x + c\right )^{4} - 6160 \, \sin \left (d x + c\right )^{3} + 5544 \, \sin \left (d x + c\right )^{2} + 2520 \, \sin \left (d x + c\right ) - 2310}{27720 \, a d \sin \left (d x + c\right )^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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