Integrand size = 25, antiderivative size = 46 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (1+\sin (c+d x))}{a d} \]
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Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2912, 12, 46} \[ \int \frac {\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (\sin (c+d x)+1)}{a d} \]
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Rule 12
Rule 46
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^2}{x^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a \text {Subst}\left (\int \frac {1}{x^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a \text {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {1}{a^2 x}+\frac {1}{a^2 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (1+\sin (c+d x))}{a d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (1+\sin (c+d x))}{a d} \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.59
method | result | size |
derivativedivides | \(-\frac {\csc \left (d x +c \right )-\ln \left (\csc \left (d x +c \right )+1\right )}{d a}\) | \(27\) |
default | \(-\frac {\csc \left (d x +c \right )-\ln \left (\csc \left (d x +c \right )+1\right )}{d a}\) | \(27\) |
parallelrisch | \(\frac {-\cot \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}\) | \(58\) |
risch | \(-\frac {2 i {\mathrm e}^{i \left (d x +c \right )}}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\) | \(74\) |
norman | \(\frac {-\frac {1}{2 a d}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}\) | \(93\) |
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.11 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 1}{a d \sin \left (d x + c\right )} \]
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\[ \int \frac {\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
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Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {\log \left (\sin \left (d x + c\right )\right )}{a} - \frac {1}{a \sin \left (d x + c\right )}}{d} \]
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Time = 0.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {1}{a \sin \left (d x + c\right )}}{d} \]
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Time = 9.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.20 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1}{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}}{2\,a\,d} \]
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