Optimal. Leaf size=32 \[ \frac{\csc (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d} \]
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Rubi [A] time = 0.06456, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2706, 2606, 30, 8} \[ \frac{\csc (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2606
Rule 30
Rule 8
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot (c+d x) \csc (c+d x) \, dx}{a}+\frac{\int \cot (c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}(\int 1 \, dx,x,\csc (c+d x))}{a d}-\frac{\operatorname{Subst}(\int x \, dx,x,\csc (c+d x))}{a d}\\ &=\frac{\csc (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.0301619, size = 24, normalized size = 0.75 \[ -\frac{(\csc (c+d x)-2) \csc (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 25, normalized size = 0.8 \begin{align*}{\frac{1}{da} \left ( -{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{2}}+\csc \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18528, size = 35, normalized size = 1.09 \begin{align*} \frac{2 \, \sin \left (d x + c\right ) - 1}{2 \, a d \sin \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25667, size = 73, normalized size = 2.28 \begin{align*} -\frac{2 \, \sin \left (d x + c\right ) - 1}{2 \,{\left (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 = 1.34424, size = 35, normalized size = 1.09 \begin{align*} \frac{2 \, \sin \left (d x + c\right ) - 1}{2 \, a d \sin \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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