Integrand size = 19, antiderivative size = 34 \[ \int \frac {\sin (c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\cos (c+d x)}{a d}+\frac {b \log (b+a \cos (c+d x))}{a^2 d} \]
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Time = 0.10 (sec) , antiderivative size = 34, 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} \[ \int \frac {\sin (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {b \log (a \cos (c+d x)+b)}{a^2 d}-\frac {\cos (c+d x)}{a d} \]
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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)}{-b-a \cos (c+d x)} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x}{a (-b+x)} \, dx,x,-a \cos (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \frac {x}{-b+x} \, dx,x,-a \cos (c+d x)\right )}{a^2 d} \\ & = \frac {\text {Subst}\left (\int \left (1-\frac {b}{b-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d} \\ & = -\frac {\cos (c+d x)}{a d}+\frac {b \log (b+a \cos (c+d x))}{a^2 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {\sin (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {-a \cos (c+d x)+b \log (b+a \cos (c+d x))}{a^2 d} \]
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Time = 0.47 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41
method | result | size |
derivativedivides | \(\frac {-\frac {1}{a \sec \left (d x +c \right )}-\frac {b \ln \left (\sec \left (d x +c \right )\right )}{a^{2}}+\frac {b \ln \left (a +b \sec \left (d x +c \right )\right )}{a^{2}}}{d}\) | \(48\) |
default | \(\frac {-\frac {1}{a \sec \left (d x +c \right )}-\frac {b \ln \left (\sec \left (d x +c \right )\right )}{a^{2}}+\frac {b \ln \left (a +b \sec \left (d x +c \right )\right )}{a^{2}}}{d}\) | \(48\) |
parallelrisch | \(\frac {b \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-b \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-a \left (\cos \left (d x +c \right )+1\right )}{a^{2} d}\) | \(59\) |
risch | \(-\frac {i x b}{a^{2}}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 d a}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 a d}-\frac {2 i b c}{a^{2} d}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{2} d}\) | \(90\) |
norman | \(-\frac {2}{d a \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{a^{2} d}-\frac {b \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a^{2} d}\) | \(91\) |
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\sin (c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {a \cos \left (d x + c\right ) - b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2} d} \]
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\[ \int \frac {\sin (c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\sin {\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {\sin (c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\frac {\cos \left (d x + c\right )}{a} - \frac {b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2}}}{d} \]
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Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \frac {\sin (c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\cos \left (d x + c\right )}{a d} + \frac {b \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{2} d} \]
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Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {\sin (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {b\,\ln \left (b+a\,\cos \left (c+d\,x\right )\right )-a\,\cos \left (c+d\,x\right )}{a^2\,d} \]
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