Integrand size = 27, antiderivative size = 29 \[ \int \frac {\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d} \]
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Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2915, 12, 45} \[ \int \frac {\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d} \]
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Rule 12
Rule 45
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a (a-x)}{x} \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \frac {a-x}{x} \, dx,x,a \sin (c+d x)\right )}{a^2 d} \\ & = \frac {\text {Subst}\left (\int \left (-1+\frac {a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d} \\ & = \frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log (\sin (c+d x))-\sin (c+d x)}{a d} \]
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Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {-\ln \left (\csc \left (d x +c \right )\right )-\frac {1}{\csc \left (d x +c \right )}}{d a}\) | \(28\) |
default | \(\frac {-\ln \left (\csc \left (d x +c \right )\right )-\frac {1}{\csc \left (d x +c \right )}}{d a}\) | \(28\) |
parallelrisch | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sin \left (d x +c \right )}{a d}\) | \(41\) |
risch | \(-\frac {i x}{a}-\frac {2 i c}{a d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}-\frac {\sin \left (d x +c \right )}{a d}\) | \(52\) |
norman | \(\frac {\frac {2}{a d}+\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\) | \(136\) |
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Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - \sin \left (d x + c\right )}{a d} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\log \left (\sin \left (d x + c\right )\right )}{a} - \frac {\sin \left (d x + c\right )}{a}}{d} \]
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Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {\sin \left (d x + c\right )}{a}}{d} \]
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Time = 9.45 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]
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