Optimal. Leaf size=14 \[ \frac {a \tan (x)}{c}+\frac {b x}{c} \]
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Rubi [A] time = 0.06, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3175, 3012, 8} \[ \frac {a \tan (x)}{c}+\frac {b x}{c} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3012
Rule 3175
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*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ \frac {a \tan (x)}{c}+\frac {b x}{c} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 18, normalized size = 1.29 \[ \frac {b x \cos \relax (x) + a \sin \relax (x)}{c \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 23, normalized size = 1.64 \[ \frac {b \arctan \left (\frac {{\left | c \right |} \tan \relax (x)}{c}\right )}{{\left | c \right |}} + \frac {a \tan \relax (x)}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 17, normalized size = 1.21 \[ \frac {a \tan \relax (x )}{c}+\frac {b \arctan \left (\tan \relax (x )\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 14, normalized size = 1.00 \[ \frac {b x}{c} + \frac {a \tan \relax (x)}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.31, size = 12, normalized size = 0.86 \[ \frac {b\,x+a\,\mathrm {tan}\relax (x)}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.33, size = 51, normalized size = 3.64 \[ - \frac {2 a \tan {\left (\frac {x}{2} \right )}}{c \tan ^{2}{\left (\frac {x}{2} \right )} - c} + \frac {b x \tan ^{2}{\left (\frac {x}{2} \right )}}{c \tan ^{2}{\left (\frac {x}{2} \right )} - c} - \frac {b x}{c \tan ^{2}{\left (\frac {x}{2} \right )} - c} \]
Verification of antiderivative is not currently implemented for this CAS.
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