Optimal. Leaf size=36 \[ \frac{(A-B) \log (\sin (c+d x)+1)}{a d}+\frac{B \sin (c+d x)}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0817913, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2833, 43} \[ \frac{(A-B) \log (\sin (c+d x)+1)}{a d}+\frac{B \sin (c+d x)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{A+\frac{B x}{a}}{a+x} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{B}{a}+\frac{A-B}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{(A-B) \log (1+\sin (c+d x))}{a d}+\frac{B \sin (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.0316451, size = 31, normalized size = 0.86 \[ \frac{(A-B) \log (\sin (c+d x)+1)+B \sin (c+d x)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.032, size = 51, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) A}{da}}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) B}{da}}+{\frac{B\sin \left ( dx+c \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.06111, size = 46, normalized size = 1.28 \begin{align*} \frac{\frac{{\left (A - B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac{B \sin \left (d x + c\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.79853, size = 76, normalized size = 2.11 \begin{align*} \frac{{\left (A - B\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + B \sin \left (d x + c\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.677483, size = 60, normalized size = 1.67 \begin{align*} \begin{cases} \frac{A \log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a d} - \frac{B \log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a d} + \frac{B \sin{\left (c + d x \right )}}{a d} & \text{for}\: d \neq 0 \\\frac{x \left (A + B \sin{\left (c \right )}\right ) \cos{\left (c \right )}}{a \sin{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.35858, size = 47, normalized size = 1.31 \begin{align*} \frac{\frac{{\left (A - B\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac{B \sin \left (d x + c\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]