Optimal. Leaf size=15 \[ \frac{\tan ^{-1}\left (\frac{b \sin (x)}{a}\right )}{a b} \]
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Rubi [A] time = 0.0260214, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3190, 205} \[ \frac{\tan ^{-1}\left (\frac{b \sin (x)}{a}\right )}{a b} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos (x)}{a^2+b^2 \sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a^2+b^2 x^2} \, dx,x,\sin (x)\right )\\ &=\frac{\tan ^{-1}\left (\frac{b \sin (x)}{a}\right )}{a b}\\ \end{align*}
Mathematica [A] time = 0.0096648, size = 15, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{b \sin (x)}{a}\right )}{a b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 16, normalized size = 1.1 \begin{align*}{\frac{1}{ab}\arctan \left ({\frac{b\sin \left ( x \right ) }{a}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4175, size = 20, normalized size = 1.33 \begin{align*} \frac{\arctan \left (\frac{b \sin \left (x\right )}{a}\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09921, size = 35, normalized size = 2.33 \begin{align*} \frac{\arctan \left (\frac{b \sin \left (x\right )}{a}\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.715625, size = 31, normalized size = 2.07 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sin{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\sin{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\- \frac{1}{b^{2} \sin{\left (x \right )}} & \text{for}\: a = 0 \\\frac{\operatorname{atan}{\left (\frac{b \sin{\left (x \right )}}{a} \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 = 1.05471, size = 20, normalized size = 1.33 \begin{align*} \frac{\arctan \left (\frac{b \sin \left (x\right )}{a}\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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