Optimal. Leaf size=14 \[ \frac{a \tan (x)}{c}+\frac{b x}{c} \]
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Rubi [A] time = 0.0576156, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3175, 3012, 8} \[ \frac{a \tan (x)}{c}+\frac{b x}{c} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3012
Rule 8
Rubi steps
\begin{align*} \int \frac{a+b \cos ^2(x)}{c-c \sin ^2(x)} \, dx &=\frac{\int \left (a+b \cos ^2(x)\right ) \sec ^2(x) \, dx}{c}\\ &=\frac{a \tan (x)}{c}+\frac{b \int 1 \, dx}{c}\\ &=\frac{b x}{c}+\frac{a \tan (x)}{c}\\ \end{align*}
Mathematica [A] time = 0.0121171, size = 14, normalized size = 1. \[ \frac{a \tan (x)}{c}+\frac{b x}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 17, normalized size = 1.2 \begin{align*}{\frac{a\tan \left ( x \right ) }{c}}+{\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 = 1.52015, size = 19, normalized size = 1.36 \begin{align*} \frac{b x}{c} + \frac{a \tan \left (x\right )}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44106, size = 49, normalized size = 3.5 \begin{align*} \frac{b x \cos \left (x\right ) + a \sin \left (x\right )}{c \cos \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.20449, size = 51, normalized size = 3.64 \begin{align*} - \frac{2 a \tan{\left (\frac{x}{2} \right )}}{c \tan ^{2}{\left (\frac{x}{2} \right )} - c} + \frac{b x \tan ^{2}{\left (\frac{x}{2} \right )}}{c \tan ^{2}{\left (\frac{x}{2} \right )} - c} - \frac{b x}{c \tan ^{2}{\left (\frac{x}{2} \right )} - c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13976, size = 31, normalized size = 2.21 \begin{align*} \frac{b \arctan \left (\frac{{\left | c \right |} \tan \left (x\right )}{c}\right )}{{\left | c \right |}} + \frac{a \tan \left (x\right )}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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