Optimal. Leaf size=17 \[ \frac{\sin (x)}{a (a \cos (x)+b \sin (x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0127537, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3075} \[ \frac{\sin (x)}{a (a \cos (x)+b \sin (x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3075
Rubi steps
\begin{align*} \int \frac{1}{(a \cos (x)+b \sin (x))^2} \, dx &=\frac{\sin (x)}{a (a \cos (x)+b \sin (x))}\\ \end{align*}
Mathematica [A] time = 0.0205677, size = 17, normalized size = 1. \[ \frac{\sin (x)}{a (a \cos (x)+b \sin (x))} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.08, size = 14, normalized size = 0.8 \begin{align*} -{\frac{1}{b \left ( a+b\tan \left ( x \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12893, size = 19, normalized size = 1.12 \begin{align*} -\frac{1}{b^{2} \tan \left (x\right ) + a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.46397, size = 95, normalized size = 5.59 \begin{align*} -\frac{b \cos \left (x\right ) - a \sin \left (x\right )}{{\left (a^{3} + a b^{2}\right )} \cos \left (x\right ) +{\left (a^{2} b + b^{3}\right )} \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 99.0802, size = 75, normalized size = 4.41 \begin{align*} \begin{cases} - \frac{\tan ^{2}{\left (\frac{x}{2} \right )}}{a b \tan ^{2}{\left (\frac{x}{2} \right )} - a b - 2 b^{2} \tan{\left (\frac{x}{2} \right )}} + \frac{1}{a b \tan ^{2}{\left (\frac{x}{2} \right )} - a b - 2 b^{2} \tan{\left (\frac{x}{2} \right )}} & \text{for}\: b \neq 0 \\- \frac{2 \tan{\left (\frac{x}{2} \right )}}{a^{2} \left (\tan ^{2}{\left (\frac{x}{2} \right )} - 1\right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1023, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{{\left (b \tan \left (x\right ) + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]