Optimal. Leaf size=47 \[ -\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.100767, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 43} \[ -\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2836
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^3 (a-x)^2}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^3}-\frac{2 a}{x^2}+\frac{1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{2 \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0451609, size = 38, normalized size = 0.81 \[ \frac{-\csc ^2(c+d x)+4 \csc (c+d x)+2 \log (\sin (c+d x))}{2 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.123, size = 48, normalized size = 1. \begin{align*} 2\,{\frac{1}{d{a}^{2}\sin \left ( dx+c \right ) }}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}-{\frac{1}{2\,d{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.09743, size = 54, normalized size = 1.15 \begin{align*} \frac{\frac{2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}} + \frac{4 \, \sin \left (d x + c\right ) - 1}{a^{2} \sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.06619, size = 140, normalized size = 2.98 \begin{align*} \frac{2 \,{\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, \sin \left (d x + c\right ) + 1}{2 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27659, size = 70, normalized size = 1.49 \begin{align*} \frac{\frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac{3 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 1}{a^{2} \sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]