Optimal. Leaf size=25 \[ \frac{a \log (\sin (c+d x))}{d}-\frac{a \csc (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0360492, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2833, 12, 43} \[ \frac{a \log (\sin (c+d x))}{d}-\frac{a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (a+x)}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{a+x}{x^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (\frac{a}{x^2}+\frac{1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0392552, size = 33, normalized size = 1.32 \[ \frac{a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}-\frac{a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.03, size = 28, normalized size = 1.1 \begin{align*} -{\frac{a}{d\sin \left ( dx+c \right ) }}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.25986, size = 34, normalized size = 1.36 \begin{align*} \frac{a \log \left (\sin \left (d x + c\right )\right ) - \frac{a}{\sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.63557, size = 82, normalized size = 3.28 \begin{align*} \frac{a \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - a}{d \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \cos{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int \sin{\left (c + d x \right )} \cos{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.28595, size = 35, normalized size = 1.4 \begin{align*} \frac{a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac{a}{\sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]