Optimal. Leaf size=30 \[ -\frac{\csc ^3(c+d x) (a \sin (c+d x)+a)^3}{3 a d} \]
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Rubi [A] time = 0.0581122, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 37} \[ -\frac{\csc ^3(c+d x) (a \sin (c+d x)+a)^3}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 37
Rubi steps
\begin{align*} \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^4 (a+x)^2}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{(a+x)^2}{x^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{\csc ^3(c+d x) (a+a \sin (c+d x))^3}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.0233936, size = 20, normalized size = 0.67 \[ -\frac{a^2 (\csc (c+d x)+1)^3}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 39, normalized size = 1.3 \begin{align*}{\frac{{a}^{2}}{d} \left ( - \left ( \sin \left ( dx+c \right ) \right ) ^{-1}-{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}- \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 = 1.14513, size = 55, normalized size = 1.83 \begin{align*} -\frac{3 \, a^{2} \sin \left (d x + c\right )^{2} + 3 \, a^{2} \sin \left (d x + c\right ) + a^{2}}{3 \, d \sin \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55655, size = 130, normalized size = 4.33 \begin{align*} -\frac{3 \, a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} \sin \left (d x + c\right ) - 4 \, a^{2}}{3 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\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 = 1.16261, size = 55, normalized size = 1.83 \begin{align*} -\frac{3 \, a^{2} \sin \left (d x + c\right )^{2} + 3 \, a^{2} \sin \left (d x + c\right ) + a^{2}}{3 \, d \sin \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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