Optimal. Leaf size=27 \[ \frac {A \tanh ^{-1}(\sin (c+d x))}{d}+B x+\frac {C \sin (c+d x)}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3023, 2735, 3770} \[ \frac {A \tanh ^{-1}(\sin (c+d x))}{d}+B x+\frac {C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3023
Rule 3770
Rubi steps
\begin {align*} \int \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac {C \sin (c+d x)}{d}+\int (A+B \cos (c+d x)) \sec (c+d x) \, dx\\ &=B x+\frac {C \sin (c+d x)}{d}+A \int \sec (c+d x) \, dx\\ &=B x+\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 = 38, normalized size = 1.41 \[ \frac {A \tanh ^{-1}(\sin (c+d x))}{d}+B x+\frac {C \sin (c) \cos (d x)}{d}+\frac {C \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 45, normalized size = 1.67 \[ \frac {2 \, B d x + A \log \left (\sin \left (d x + c\right ) + 1\right ) - A \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.63, size = 70, normalized size = 2.59 \[ \frac {{\left (d x + c\right )} B + A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 41, normalized size = 1.52 \[ B x +\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {B c}{d}+\frac {C \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 36, normalized size = 1.33 \[ \frac {{\left (d x + c\right )} B + A \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + C \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 68, normalized size = 2.52 \[ \frac {2\,A\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,\sin \left (c+d\,x\right )}{d} \]
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 \[ \int \left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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