Optimal. Leaf size=17 \[ \frac{\sin (x)}{b (a \sin (x)+b \cos (x))} \]
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Rubi [A] time = 0.0122878, 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} \[ \frac{\sin (x)}{b (a \sin (x)+b \cos (x))} \]
Antiderivative was successfully verified.
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Rule 3075
Rubi steps
\begin{align*} \int \frac{1}{(b \cos (x)+a \sin (x))^2} \, dx &=\frac{\sin (x)}{b (b \cos (x)+a \sin (x))}\\ \end{align*}
Mathematica [A] time = 0.02755, size = 17, normalized size = 1. \[ \frac{\sin (x)}{b (a \sin (x)+b \cos (x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 14, normalized size = 0.8 \begin{align*} -{\frac{1}{a \left ( a\tan \left ( x \right ) +b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952206, size = 19, normalized size = 1.12 \begin{align*} -\frac{1}{a^{2} \tan \left (x\right ) + a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.00399, size = 95, normalized size = 5.59 \begin{align*} -\frac{a \cos \left (x\right ) - b \sin \left (x\right )}{{\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) +{\left (a^{3} + a b^{2}\right )} \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 123.877, size = 228, normalized size = 13.41 \begin{align*} \begin{cases} \frac{\tilde{\infty } \tan{\left (\frac{x}{2} \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} - 1} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2 \tan{\left (\frac{x}{2} \right )}}{b^{2} \left (\tan ^{2}{\left (\frac{x}{2} \right )} - 1\right )} & \text{for}\: a = 0 \\\frac{x \tan ^{2}{\left (x \right )}}{2 a^{2} \sin ^{2}{\left (x \right )} - 4 a^{2} \sin{\left (x \right )} \cos{\left (x \right )} \tan{\left (x \right )} + 2 a^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (x \right )}} + \frac{x}{2 a^{2} \sin ^{2}{\left (x \right )} - 4 a^{2} \sin{\left (x \right )} \cos{\left (x \right )} \tan{\left (x \right )} + 2 a^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (x \right )}} + \frac{\tan{\left (x \right )}}{2 a^{2} \sin ^{2}{\left (x \right )} - 4 a^{2} \sin{\left (x \right )} \cos{\left (x \right )} \tan{\left (x \right )} + 2 a^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (x \right )}} & \text{for}\: b = - a \tan{\left (x \right )} \\\frac{\tan ^{2}{\left (\frac{x}{2} \right )}}{2 a^{2} \tan{\left (\frac{x}{2} \right )} - a b \tan ^{2}{\left (\frac{x}{2} \right )} + a b} - \frac{1}{2 a^{2} \tan{\left (\frac{x}{2} \right )} - a b \tan ^{2}{\left (\frac{x}{2} \right )} + a b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10501, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{{\left (a \tan \left (x\right ) + b\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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