Optimal. Leaf size=24 \[ \frac {A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \sin (c+d x)}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3014, 3770} \[ \frac {A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3014
Rule 3770
Rubi steps
\begin {align*} \int \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac {C \sin (c+d x)}{d}+A \int \sec (c+d x) \, dx\\ &=\frac {A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 1.46 \[ \frac {A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \sin (c) \cos (d x)}{d}+\frac {C \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 40, normalized size = 1.67 \[ \frac {A \log \left (\sin \left (d x + c\right ) + 1\right ) - A \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 40, normalized size = 1.67 \[ \frac {A \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - A \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 32, normalized size = 1.33 \[ \frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 38, normalized size = 1.58 \[ \frac {A \log \left (\sin \left (d x + c\right ) + 1\right ) - A \log \left (\sin \left (d x + c\right ) - 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 22, normalized size = 0.92 \[ \frac {C\,\sin \left (c+d\,x\right )+A\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{d} \]
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 \[ \int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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