Optimal. Leaf size=22 \[ \frac{\log (a \sin (c+d x)+b \sec (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0484212, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023, Rules used = {4383} \[ \frac{\log (a \sin (c+d x)+b \sec (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4383
Rubi steps
\begin{align*} \int \frac{a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{b \sec (c+d x)+a \sin (c+d x)} \, dx &=\frac{\log (b \sec (c+d x)+a \sin (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.462226, size = 29, normalized size = 1.32 \[ \frac{\log (a \sin (2 (c+d x))+2 b)-\log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.09, size = 23, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( b\sec \left ( dx+c \right ) +a\sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.965781, size = 30, normalized size = 1.36 \begin{align*} \frac{\log \left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.76279, size = 85, normalized size = 3.86 \begin{align*} \frac{\log \left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b\right ) - \log \left (-\cos \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 7.52807, size = 63, normalized size = 2.86 \begin{align*} \begin{cases} x \tan{\left (c \right )} & \text{for}\: a = 0 \wedge d = 0 \\\frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text{for}\: a = 0 \\\frac{x \left (a \cos{\left (c \right )} + b \tan{\left (c \right )} \sec{\left (c \right )}\right )}{a \sin{\left (c \right )} + b \sec{\left (c \right )}} & \text{for}\: d = 0 \\\frac{\log{\left (\sin{\left (c + d x \right )} + \frac{b \sec{\left (c + d x \right )}}{a} \right )}}{d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30741, size = 57, normalized size = 2.59 \begin{align*} \frac{2 \, \log \left (b \tan \left (d x + c\right )^{2} + a \tan \left (d x + c\right ) + b\right ) - \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]