Optimal. Leaf size=30 \[ -\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^2}{2 a d} \]
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Rubi [A] time = 0.0417927, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 37} \[ -\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^2}{2 a d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 37
Rubi steps
\begin{align*} \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^3 (a+x)}{x^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \frac{a+x}{x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{\csc ^2(c+d x) (a+a \sin (c+d x))^2}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.0191811, size = 29, normalized size = 0.97 \[ -\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 27, normalized size = 0.9 \begin{align*}{\frac{a}{d} \left ( - \left ( \sin \left ( dx+c \right ) \right ) ^{-1}-{\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 = 1.24401, size = 32, normalized size = 1.07 \begin{align*} -\frac{2 \, a \sin \left (d x + c\right ) + a}{2 \, d \sin \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5656, size = 69, normalized size = 2.3 \begin{align*} \frac{2 \, a \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(-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 = 1.29335, size = 32, normalized size = 1.07 \begin{align*} -\frac{2 \, a \sin \left (d x + c\right ) + a}{2 \, d \sin \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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