Optimal. Leaf size=26 \[ -\frac {a \cos (c+d x)}{d}-\frac {b \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2638, 3475} \[ -\frac {a \cos (c+d x)}{d}-\frac {b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3475
Rubi steps
\begin {align*} \int (a \sin (c+d x)+b \tan (c+d x)) \, dx &=a \int \sin (c+d x) \, dx+b \int \tan (c+d x) \, dx\\ &=-\frac {a \cos (c+d x)}{d}-\frac {b \log (\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 1.42 \[ \frac {a \sin (c) \sin (d x)}{d}-\frac {a \cos (c) \cos (d x)}{d}-\frac {b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 25, normalized size = 0.96 \[ -\frac {a \cos \left (d x + c\right ) + b \log \left (-\cos \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 27, normalized size = 1.04 \[ -\frac {a \cos \left (d x + c\right )}{d} - \frac {b \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 1.19 \[ \frac {b \ln \left (\tan ^{2}\left (d x +c \right )+1\right )}{2 d}-\frac {a \cos \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 25, normalized size = 0.96 \[ -\frac {a \cos \left (d x + c\right )}{d} + \frac {b \log \left (\sec \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 40, normalized size = 1.54 \[ \frac {2\,b\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 37, normalized size = 1.42 \[ a \left (\begin {cases} - \frac {\cos {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \sin {\relax (c )} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {for}\: d \neq 0 \\x \tan {\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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