Optimal. Leaf size=73 \[ -\frac{\csc ^6(c+d x)}{6 a^3 d}+\frac{3 \csc ^5(c+d x)}{5 a^3 d}-\frac{3 \csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{3 a^3 d} \]
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Rubi [A] time = 0.0574368, antiderivative size = 73, normalized size of antiderivative = 1., 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, 43} \[ -\frac{\csc ^6(c+d x)}{6 a^3 d}+\frac{3 \csc ^5(c+d x)}{5 a^3 d}-\frac{3 \csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 43
Rubi steps
\begin{align*} \int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^7}-\frac{3 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{3 \csc ^5(c+d x)}{5 a^3 d}-\frac{\csc ^6(c+d x)}{6 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.098613, size = 48, normalized size = 0.66 \[ \frac{\csc ^3(c+d x) \left (-10 \csc ^3(c+d x)+36 \csc ^2(c+d x)-45 \csc (c+d x)+20\right )}{60 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.119, size = 49, normalized size = 0.7 \begin{align*}{\frac{1}{d{a}^{3}} \left ({\frac{3}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{1}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53887, size = 62, normalized size = 0.85 \begin{align*} \frac{20 \, \sin \left (d x + c\right )^{3} - 45 \, \sin \left (d x + c\right )^{2} + 36 \, \sin \left (d x + c\right ) - 10}{60 \, a^{3} d \sin \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46143, size = 208, normalized size = 2.85 \begin{align*} -\frac{45 \, \cos \left (d x + c\right )^{2} - 4 \,{\left (5 \, \cos \left (d x + c\right )^{2} - 14\right )} \sin \left (d x + c\right ) - 55}{60 \,{\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} 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 = 1.64026, size = 62, normalized size = 0.85 \begin{align*} \frac{20 \, \sin \left (d x + c\right )^{3} - 45 \, \sin \left (d x + c\right )^{2} + 36 \, \sin \left (d x + c\right ) - 10}{60 \, a^{3} d \sin \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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