Optimal. Leaf size=16 \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+a x \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2735, 3770} \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+a x \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x)) \sec (c+d x) \, dx &=a x+a \int \sec (c+d x) \, dx\\ &=a x+\frac {a \tanh ^{-1}(\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+a x \]
Antiderivative was successfully verified.
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fricas [B] time = 1.68, size = 36, normalized size = 2.25 \[ \frac {2 \, a d x + a \log \left (\sin \left (d x + c\right ) + 1\right ) - a \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 43, normalized size = 2.69 \[ \frac {{\left (d x + c\right )} a + a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 30, normalized size = 1.88 \[ a x +\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {c a}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 28, normalized size = 1.75 \[ \frac {{\left (d x + c\right )} a + a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 20, normalized size = 1.25 \[ a\,x+\frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.92, size = 49, normalized size = 3.06 \[ a x + a \left (\begin {cases} \frac {x \tan {\relax (c )} \sec {\relax (c )}}{\tan {\relax (c )} + \sec {\relax (c )}} + \frac {x \sec ^{2}{\relax (c )}}{\tan {\relax (c )} + \sec {\relax (c )}} & \text {for}\: d = 0 \\\frac {\log {\left (\tan {\left (c + d x \right )} + \sec {\left (c + d x \right )} \right )}}{d} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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