Optimal. Leaf size=15 \[ \frac{b x}{c}-\frac{a \cot (x)}{c} \]
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Rubi [A] time = 0.0890223, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {453, 205} \[ \frac{b x}{c}-\frac{a \cot (x)}{c} \]
Antiderivative was successfully verified.
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Rule 453
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b \sin ^2(x)}{c-c \cos ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{a+(a+b) x^2}{x^2 \left (c+c x^2\right )} \, dx,x,\tan (x)\right )\\ &=-\frac{a \cot (x)}{c}+b \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\tan (x)\right )\\ &=\frac{b x}{c}-\frac{a \cot (x)}{c}\\ \end{align*}
Mathematica [A] time = 0.0129729, size = 15, normalized size = 1. \[ \frac{b x}{c}-\frac{a \cot (x)}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 20, normalized size = 1.3 \begin{align*} -{\frac{a}{c\tan \left ( x \right ) }}+{\frac{b\arctan \left ( \tan \left ( x \right ) \right ) }{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56041, size = 23, normalized size = 1.53 \begin{align*} \frac{b x}{c} - \frac{a}{c \tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34039, size = 49, normalized size = 3.27 \begin{align*} \frac{b x \sin \left (x\right ) - a \cos \left (x\right )}{c \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.87029, size = 24, normalized size = 1.6 \begin{align*} \frac{a \tan{\left (\frac{x}{2} \right )}}{2 c} - \frac{a}{2 c \tan{\left (\frac{x}{2} \right )}} + \frac{b x}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16905, size = 39, normalized size = 2.6 \begin{align*} \frac{b x}{c} + \frac{a \tan \left (\frac{1}{2} \, x\right )}{2 \, c} - \frac{a}{2 \, c \tan \left (\frac{1}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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