Optimal. Leaf size=30 \[ -\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^4}{2 a^2 d} \]
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Rubi [A] time = 0.0392531, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 74} \[ -\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^4}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 74
Rubi steps
\begin{align*} \int \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x) (a+x)^3}{x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{\csc ^2(c+d x) (a+a \sin (c+d x))^4}{2 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0412311, size = 28, normalized size = 0.93 \[ -\frac{a^2 (\sin (c+d x)+1)^4 \csc ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 94, normalized size = 3.1 \begin{align*}{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-2\,{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d\sin \left ( dx+c \right ) }}-2\,{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) }{d}}-4\,{\frac{{a}^{2}\sin \left ( dx+c \right ) }{d}}-{\frac{{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11304, size = 72, normalized size = 2.4 \begin{align*} -\frac{a^{2} \sin \left (d x + c\right )^{2} + 4 \, a^{2} \sin \left (d x + c\right ) + \frac{4 \, a^{2} \sin \left (d x + c\right ) + a^{2}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44029, size = 173, normalized size = 5.77 \begin{align*} \frac{2 \, a^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 8 \,{\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2}\right )} \sin \left (d x + c\right )}{4 \,{\left (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] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int 2 \sin{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \cot ^{3}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37063, size = 63, normalized size = 2.1 \begin{align*} -\frac{a^{2}{\left (\frac{1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right )\right )}^{2} + 4 \, a^{2}{\left (\frac{1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right )\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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