Optimal. Leaf size=32 \[ \frac{\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0164173, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3075} \[ \frac{\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3075
Rubi steps
\begin{align*} \int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx &=\frac{\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0400738, size = 32, normalized size = 1. \[ \frac{\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.128, size = 21, normalized size = 0.7 \begin{align*} -{\frac{1}{db \left ( a+b\tan \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.981473, size = 28, normalized size = 0.88 \begin{align*} -\frac{1}{{\left (b^{2} \tan \left (d x + c\right ) + a b\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.15933, size = 132, normalized size = 4.12 \begin{align*} -\frac{b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{{\left (a^{3} + a b^{2}\right )} d \cos \left (d x + c\right ) +{\left (a^{2} b + b^{3}\right )} 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*} \int \frac{1}{\left (a \cos{\left (c + d x \right )} + b \sin{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11441, size = 27, normalized size = 0.84 \begin{align*} -\frac{1}{{\left (b \tan \left (d x + c\right ) + a\right )} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]