Optimal. Leaf size=15 \[ \frac {A \tan (c+d x)}{d}+C x \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3012, 8} \[ \frac {A \tan (c+d x)}{d}+C x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3012
Rubi steps
\begin {align*} \int \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac {A \tan (c+d x)}{d}+C \int 1 \, dx\\ &=C x+\frac {A \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {A \tan (c+d x)}{d}+C x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 31, normalized size = 2.07 \[ \frac {C d x \cos \left (d x + c\right ) + A \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 20, normalized size = 1.33 \[ \frac {{\left (d x + c\right )} C + A \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 21, normalized size = 1.40 \[ \frac {A \tan \left (d x +c \right )+C \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 20, normalized size = 1.33 \[ \frac {{\left (d x + c\right )} C + A \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 17, normalized size = 1.13 \[ \frac {A\,\mathrm {tan}\left (c+d\,x\right )+C\,d\,x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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