Integrand size = 27, antiderivative size = 37 \[ \int \frac {\cot ^3(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc ^3(c+d x)}{3 a d} \]
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Time = 0.08 (sec) , antiderivative size = 37, 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} \[ \int \frac {\cot ^3(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc ^3(c+d x)}{3 a d} \]
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Rule 12
Rule 45
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^4 (a-x)}{x^4} \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {a \text {Subst}\left (\int \frac {a-x}{x^4} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a \text {Subst}\left (\int \left (\frac {a}{x^4}-\frac {1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc ^3(c+d x)}{3 a d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76 \[ \int \frac {\cot ^3(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^3(c+d x) (-2+3 \sin (c+d x))}{6 a d} \]
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Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(-\frac {\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}}{d a}\) | \(30\) |
default | \(-\frac {\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}}{d a}\) | \(30\) |
risch | \(-\frac {2 \left (-4 i {\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{4 i \left (d x +c \right )}-3 \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}\) | \(57\) |
parallelrisch | \(\frac {-\left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}\) | \(74\) |
norman | \(\frac {-\frac {1}{24 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}+\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}-\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(90\) |
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int \frac {\cot ^3(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \, \sin \left (d x + c\right ) - 2}{6 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \frac {\cot ^3(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^3(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \, \sin \left (d x + c\right ) - 2}{6 \, a d \sin \left (d x + c\right )^{3}} \]
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Time = 0.41 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^3(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \, \sin \left (d x + c\right ) - 2}{6 \, a d \sin \left (d x + c\right )^{3}} \]
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Time = 9.66 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^3(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {5\,\sin \left (c+d\,x\right )}{16}+\frac {\sin \left (3\,c+3\,d\,x\right )}{16}-\frac {1}{3}}{a\,d\,{\sin \left (c+d\,x\right )}^3} \]
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