Optimal. Leaf size=55 \[ -\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d} \]
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Rubi [A] time = 0.0460195, antiderivative size = 55, 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} \[ -\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 43
Rubi steps
\begin{align*} \int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^5}-\frac{2 a}{x^4}+\frac{1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^4(c+d x)}{4 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0702285, size = 38, normalized size = 0.69 \[ \frac{\csc ^4(c+d x) (8 \sin (c+d x)+3 \cos (2 (c+d x))-6)}{12 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.139, size = 39, normalized size = 0.7 \begin{align*}{\frac{1}{d{a}^{2}} \left ( -{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{2}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05331, size = 49, normalized size = 0.89 \begin{align*} -\frac{6 \, \sin \left (d x + c\right )^{2} - 8 \, \sin \left (d x + c\right ) + 3}{12 \, a^{2} d \sin \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.9948, size = 138, normalized size = 2.51 \begin{align*} \frac{6 \, \cos \left (d x + c\right )^{2} + 8 \, \sin \left (d x + c\right ) - 9}{12 \,{\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} 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 = 1.19987, size = 49, normalized size = 0.89 \begin{align*} -\frac{6 \, \sin \left (d x + c\right )^{2} - 8 \, \sin \left (d x + c\right ) + 3}{12 \, a^{2} d \sin \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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