Optimal. Leaf size=53 \[ -\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.057603, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2836, 12, 75} \[ -\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 75
Rubi steps
\begin{align*} \int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (a-x) (a+x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x) (a+x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a+\frac{a^3}{x^2}+\frac{a^2}{x}-x\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}-\frac{a \sin (c+d x)}{d}-\frac{a \sin ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0335284, size = 53, normalized size = 1. \[ -\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 82, normalized size = 1.6 \begin{align*}{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987907, size = 62, normalized size = 1.17 \begin{align*} -\frac{a \sin \left (d x + c\right )^{2} - 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 )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87556, size = 176, normalized size = 3.32 \begin{align*} \frac{4 \, a \cos \left (d x + c\right )^{2} + 4 \, a \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) +{\left (2 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 8 \, a}{4 \, d \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 = 1.22565, size = 63, normalized size = 1.19 \begin{align*} -\frac{a \sin \left (d x + c\right )^{2} - 2 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 2 \, a \sin \left (d x + c\right ) + \frac{2 \, a}{\sin \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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