Optimal. Leaf size=18 \[ -\frac{\log \left (a^2+b^2 \cos ^2(x)\right )}{b^2} \]
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Rubi [A] time = 0.0404996, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {12, 260} \[ -\frac{\log \left (a^2-b^2 \sin ^2(x)+b^2\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 260
Rubi steps
\begin{align*} \int \frac{\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{2 x}{a^2+b^2-b^2 x^2} \, dx,x,\sin (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x}{a^2+b^2-b^2 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac{\log \left (a^2+b^2-b^2 \sin ^2(x)\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0150171, size = 22, normalized size = 1.22 \[ -\frac{\log \left (a^2-b^2 \sin ^2(x)+b^2\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 19, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ({a}^{2}+{b}^{2} \left ( \cos \left ( x \right ) \right ) ^{2} \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.928581, size = 24, normalized size = 1.33 \begin{align*} -\frac{\log \left (b^{2} \cos \left (x\right )^{2} + a^{2}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37374, size = 41, normalized size = 2.28 \begin{align*} -\frac{\log \left (b^{2} \cos \left (x\right )^{2} + a^{2}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.0342, size = 34, normalized size = 1.89 \begin{align*} 2 \left (\begin{cases} - \frac{\cos ^{2}{\left (x \right )}}{2 a^{2}} & \text{for}\: b^{2} = 0 \\- \frac{\log{\left (a^{2} + b^{2} \cos ^{2}{\left (x \right )} \right )}}{2 b^{2}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07439, size = 130, normalized size = 7.22 \begin{align*} -\frac{\log \left (a^{2} + b^{2} - \frac{2 \, a^{2}{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac{2 \, b^{2}{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac{a^{2}{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{b^{2}{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{b^{2}} + \frac{2 \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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