Optimal. Leaf size=54 \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.037076, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 75} \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 75
Rubi steps
\begin{align*} \int \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x) (a+x)^2}{x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+\frac{a^3}{x^3}+\frac{a^2}{x^2}-\frac{a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a \csc (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \log (\sin (c+d x))}{d}-\frac{a \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.117517, size = 60, normalized size = 1.11 \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}-\frac{a \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 83, normalized size = 1.5 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d\sin \left ( dx+c \right ) }}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{d}}-2\,{\frac{a\sin \left ( dx+c \right ) }{d}}-{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08976, size = 61, normalized size = 1.13 \begin{align*} -\frac{2 \, a \log \left (\sin \left (d x + c\right )\right ) + 2 \, a \sin \left (d x + c\right ) + \frac{2 \, a \sin \left (d x + c\right ) + a}{\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 [A] time = 1.49611, size = 167, normalized size = 3.09 \begin{align*} -\frac{2 \,{\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 2 \,{\left (a \cos \left (d x + c\right )^{2} - 2 \, a\right )} \sin \left (d x + c\right ) - a}{2 \,{\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 \left (\int \sin{\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 = 1.32525, size = 81, normalized size = 1.5 \begin{align*} -\frac{2 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 2 \, a \sin \left (d x + c\right ) - \frac{3 \, a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right ) - a}{\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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