Optimal. Leaf size=37 \[ \frac{\csc ^3(c+d x)}{3 a d}-\frac{\csc ^4(c+d x)}{4 a d} \]
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Rubi [A] time = 0.0976363, antiderivative size = 37, 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, 43} \[ \frac{\csc ^3(c+d x)}{3 a d}-\frac{\csc ^4(c+d x)}{4 a d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^5 (a-x)}{x^5} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \frac{a-x}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{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 d}-\frac{\csc ^4(c+d x)}{4 a d}\\ \end{align*}
Mathematica [A] time = 0.0439687, size = 28, normalized size = 0.76 \[ \frac{(4 \sin (c+d x)-3) \csc ^4(c+d x)}{12 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 29, normalized size = 0.8 \begin{align*}{\frac{1}{da} \left ( -{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11174, size = 35, normalized size = 0.95 \begin{align*} \frac{4 \, \sin \left (d x + c\right ) - 3}{12 \, a 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 = 1.32628, size = 104, normalized size = 2.81 \begin{align*} \frac{4 \, \sin \left (d x + c\right ) - 3}{12 \,{\left (a d \cos \left (d x + c\right )^{4} - 2 \, 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 = 1.43395, size = 35, normalized size = 0.95 \begin{align*} \frac{4 \, \sin \left (d x + c\right ) - 3}{12 \, a d \sin \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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