Optimal. Leaf size=55 \[ -\frac{a^2 \csc ^4(c+d x)}{4 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.0663529, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ -\frac{a^2 \csc ^4(c+d x)}{4 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^5 (a+x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \frac{(a+x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^4 \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{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^4(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0290794, size = 55, normalized size = 1. \[ -\frac{a^2 \csc ^4(c+d x)}{4 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 39, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}}{d} \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 = 1.23087, size = 58, normalized size = 1.05 \begin{align*} -\frac{6 \, a^{2} \sin \left (d x + c\right )^{2} + 8 \, a^{2} \sin \left (d x + c\right ) + 3 \, a^{2}}{12 \, 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 = 1.62856, size = 138, normalized size = 2.51 \begin{align*} \frac{6 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a^{2} \sin \left (d x + c\right ) - 9 \, a^{2}}{12 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{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 = 1.19858, size = 58, normalized size = 1.05 \begin{align*} -\frac{6 \, a^{2} \sin \left (d x + c\right )^{2} + 8 \, a^{2} \sin \left (d x + c\right ) + 3 \, a^{2}}{12 \, d \sin \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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