Integrand size = 19, antiderivative size = 31 \[ \int \frac {\sin (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\cos (c+d x)}{a d}+\frac {\log (1+\cos (c+d x))}{a d} \]
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Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3957, 2912, 12, 45} \[ \int \frac {\sin (c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\log (\cos (c+d x)+1)}{a d}-\frac {\cos (c+d x)}{a d} \]
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Rule 12
Rule 45
Rule 2912
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) \sin (c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x}{a (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \frac {x}{-a+x} \, dx,x,-a \cos (c+d x)\right )}{a^2 d} \\ & = \frac {\text {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*}
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {\sin (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\cos (c+d x)-2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a d} \]
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Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
method | result | size |
parallelrisch | \(\frac {-\ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+1-\cos \left (d x +c \right )}{a d}\) | \(32\) |
derivativedivides | \(\frac {\ln \left (1+\sec \left (d x +c \right )\right )-\frac {1}{\sec \left (d x +c \right )}-\ln \left (\sec \left (d x +c \right )\right )}{d a}\) | \(37\) |
default | \(\frac {\ln \left (1+\sec \left (d x +c \right )\right )-\frac {1}{\sec \left (d x +c \right )}-\ln \left (\sec \left (d x +c \right )\right )}{d a}\) | \(37\) |
norman | \(\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d a \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {\ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a d}\) | \(58\) |
risch | \(-\frac {i x}{a}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 d a}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 d a}-\frac {2 i c}{d a}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}\) | \(73\) |
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Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {\sin (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\cos \left (d x + c\right ) - \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{a d} \]
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\[ \int \frac {\sin (c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sin {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
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Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {\sin (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {\cos \left (d x + c\right )}{a} - \frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a}}{d} \]
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Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {\sin (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\cos \left (d x + c\right )}{a d} + \frac {\log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{a d} \]
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Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {\sin (c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )-\cos \left (c+d\,x\right )}{a\,d} \]
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