Optimal. Leaf size=43 \[ \frac{\sin (c+d x)}{a^2 d}-\frac{\csc (c+d x)}{a^2 d}-\frac{2 \log (\sin (c+d x))}{a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0993012, antiderivative size = 43, 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} \[ \frac{\sin (c+d x)}{a^2 d}-\frac{\csc (c+d x)}{a^2 d}-\frac{2 \log (\sin (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2836
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (a-x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{a^2}{x^2}-\frac{2 a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac{\csc (c+d x)}{a^2 d}-\frac{2 \log (\sin (c+d x))}{a^2 d}+\frac{\sin (c+d x)}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0468369, size = 32, normalized size = 0.74 \[ -\frac{-\sin (c+d x)+\csc (c+d x)+2 \log (\sin (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.116, size = 46, normalized size = 1.1 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d{a}^{2}}}-{\frac{1}{d{a}^{2}\sin \left ( dx+c \right ) }}-2\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01414, size = 55, normalized size = 1.28 \begin{align*} -\frac{\frac{2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}} - \frac{\sin \left (d x + c\right )}{a^{2}} + \frac{1}{a^{2} \sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.11527, size = 107, normalized size = 2.49 \begin{align*} -\frac{\cos \left (d x + c\right )^{2} + 2 \, \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right )}{a^{2} d \sin \left (d x + c\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 = 1.26741, size = 72, normalized size = 1.67 \begin{align*} -\frac{\frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac{\sin \left (d x + c\right )}{a^{2}} - \frac{2 \, \sin \left (d x + c\right ) - 1}{a^{2} \sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]