Optimal. Leaf size=34 \[ \frac{b \log (a \cos (c+d x)+b)}{a^2 d}-\frac{\cos (c+d x)}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0766032, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2833, 12, 43} \[ \frac{b \log (a \cos (c+d x)+b)}{a^2 d}-\frac{\cos (c+d x)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3872
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{a+b \sec (c+d x)} \, dx &=-\int \frac{\cos (c+d x) \sin (c+d x)}{-b-a \cos (c+d x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{a (-b+x)} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{-b+x} \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{b}{b-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=-\frac{\cos (c+d x)}{a d}+\frac{b \log (b+a \cos (c+d x))}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0180372, size = 30, normalized size = 0.88 \[ \frac{b \log (a \cos (c+d x)+b)-a \cos (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 53, normalized size = 1.6 \begin{align*}{\frac{b\ln \left ( a+b\sec \left ( dx+c \right ) \right ) }{d{a}^{2}}}-{\frac{1}{ad\sec \left ( dx+c \right ) }}-{\frac{b\ln \left ( \sec \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 = 1.02355, size = 45, normalized size = 1.32 \begin{align*} -\frac{\frac{\cos \left (d x + c\right )}{a} - \frac{b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.70301, size = 74, normalized size = 2.18 \begin{align*} -\frac{a \cos \left (d x + c\right ) - b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2} d} \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*} \int \frac{\sin{\left (c + d x \right )}}{a + b \sec{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30374, size = 51, normalized size = 1.5 \begin{align*} -\frac{\cos \left (d x + c\right )}{a d} + \frac{b \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]