Optimal. Leaf size=26 \[ -\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3872, 2707, 43} \[ -\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2707
Rule 3872
Rubi steps
\begin {align*} \int (a+a \sec (c+d x)) \sin (c+d x) \, dx &=-\int (-a-a \cos (c+d x)) \tan (c+d x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {-a+x}{x} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {a}{x}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 1.42 \[ \frac {a \sin (c) \sin (d x)}{d}-\frac {a \cos (c) \cos (d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 25, normalized size = 0.96 \[ -\frac {a \cos \left (d x + c\right ) + a \log \left (-\cos \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.45, size = 32, normalized size = 1.23 \[ -\frac {a \cos \left (d x + c\right )}{d} - \frac {a \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 28, normalized size = 1.08 \[ \frac {a \ln \left (\sec \left (d x +c \right )\right )}{d}-\frac {a}{d \sec \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 23, normalized size = 0.88 \[ -\frac {a \cos \left (d x + c\right ) + a \log \left (\cos \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 20, normalized size = 0.77 \[ -\frac {a\,\left (\cos \left (c+d\,x\right )+\ln \left (\cos \left (c+d\,x\right )\right )\right )}{d} \]
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 \[ a \left (\int \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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