Optimal. Leaf size=31 \[ \frac {\log (\cos (c+d x)+1)}{a d}-\frac {\cos (c+d x)}{a d} \]
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Rubi [A] time = 0.07, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3872, 2833, 12, 43} \[ \frac {\log (\cos (c+d x)+1)}{a d}-\frac {\cos (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin (c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{a (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{-a+x} \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {a}{a-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\cos (c+d x)}{a d}+\frac {\log (1+\cos (c+d x))}{a d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 28, normalized size = 0.90 \[ -\frac {\cos (c+d x)-2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 28, normalized size = 0.90 \[ -\frac {\cos \left (d x + c\right ) - \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 34, normalized size = 1.10 \[ -\frac {\cos \left (d x + c\right )}{a d} + \frac {\log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 49, normalized size = 1.58 \[ -\frac {1}{d a \sec \left (d x +c \right )}-\frac {\ln \left (\sec \left (d x +c \right )\right )}{d a}+\frac {\ln \left (1+\sec \left (d x +c \right )\right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 30, normalized size = 0.97 \[ -\frac {\frac {\cos \left (d x + c\right )}{a} - \frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 25, normalized size = 0.81 \[ \frac {\ln \left (\cos \left (c+d\,x\right )+1\right )-\cos \left (c+d\,x\right )}{a\,d} \]
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 \[ \frac {\int \frac {\sin {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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