Optimal. Leaf size=27 \[ \frac{a A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a A \cos (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0522252, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3962, 2592, 321, 206} \[ \frac{a A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a A \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3962
Rule 2592
Rule 321
Rule 206
Rubi steps
\begin{align*} \int (a+a \csc (c+d x)) (A-A \csc (c+d x)) \sin (c+d x) \, dx &=-((a A) \int \cos (c+d x) \cot (c+d x) \, dx)\\ &=\frac{(a A) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a A \cos (c+d x)}{d}+\frac{(a A) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a A \cos (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.022104, size = 46, normalized size = 1.7 \[ -a A \left (\frac{\cos (c+d x)}{d}+\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d}-\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 38, normalized size = 1.4 \begin{align*} -{\frac{Aa\cos \left ( dx+c \right ) }{d}}-{\frac{Aa\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01463, size = 45, normalized size = 1.67 \begin{align*} -\frac{A a \cos \left (d x + c\right ) - A a \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.506677, size = 132, normalized size = 4.89 \begin{align*} -\frac{2 \, A a \cos \left (d x + c\right ) - A a \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + A a \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 11.4959, size = 75, normalized size = 2.78 \begin{align*} A a \left (\begin{cases} - \frac{\cot{\left (c + d x \right )}}{d \csc{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x}{\csc{\left (c \right )}} & \text{otherwise} \end{cases}\right ) - A a \left (\begin{cases} \frac{x \cot{\left (c \right )} \csc{\left (c \right )}}{\cot{\left (c \right )} + \csc{\left (c \right )}} + \frac{x \csc ^{2}{\left (c \right )}}{\cot{\left (c \right )} + \csc{\left (c \right )}} & \text{for}\: d = 0 \\- \frac{\log{\left (\cot{\left (c + d x \right )} + \csc{\left (c + d x \right )} \right )}}{d} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.39327, size = 81, normalized size = 3. \begin{align*} -\frac{A a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - \frac{4 \, A a}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]