Optimal. Leaf size=26 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
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Rubi [A] time = 0.0272752, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3190, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 205
Rubi steps
\begin{align*} \int \frac{\sin (x)}{a+b \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0192716, size = 26, normalized size = 1. \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 18, normalized size = 0.7 \begin{align*} -{\arctan \left ({b\cos \left ( x \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67127, size = 181, normalized size = 6.96 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (-\frac{b \cos \left (x\right )^{2} + 2 \, \sqrt{-a b} \cos \left (x\right ) - a}{b \cos \left (x\right )^{2} + a}\right )}{2 \, a b}, -\frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b} \cos \left (x\right )}{a}\right )}{a b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.94691, size = 87, normalized size = 3.35 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\cos{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{1}{b \cos{\left (x \right )}} & \text{for}\: a = 0 \\- \frac{\cos{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{i \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \cos{\left (x \right )} \right )}}{2 \sqrt{a} b \sqrt{\frac{1}{b}}} - \frac{i \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \cos{\left (x \right )} \right )}}{2 \sqrt{a} b \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12251, size = 23, normalized size = 0.88 \begin{align*} -\frac{\arctan \left (\frac{b \cos \left (x\right )}{\sqrt{a b}}\right )}{\sqrt{a b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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