Integrand size = 25, antiderivative size = 33 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d} \]
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Time = 0.03 (sec) , antiderivative size = 33, 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, 45} \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^3(c+d x)}{3 d} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^5 (a+x)}{x^5} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a^4 \text {Subst}\left (\int \frac {a+x}{x^5} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^4 \text {Subst}\left (\int \left (\frac {a}{x^5}+\frac {1}{x^4}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d} \]
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Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(28\) |
default | \(-\frac {a \left (\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(28\) |
risch | \(\frac {4 i a \left (3 i {\mathrm e}^{4 i \left (d x +c \right )}+2 \,{\mathrm e}^{5 i \left (d x +c \right )}-2 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}\) | \(56\) |
parallelrisch | \(-\frac {a \left (165+36 \cos \left (2 d x +2 c \right )+256 \sin \left (d x +c \right )-9 \cos \left (4 d x +4 c \right )\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12288 d}\) | \(61\) |
norman | \(\frac {-\frac {a}{64 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d}-\frac {5 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {5 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(169\) |
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Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {4 \, a \sin \left (d x + c\right ) + 3 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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\[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=a \left (\int \cos {\left (c + d x \right )} \csc ^{5}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{5}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {4 \, a \sin \left (d x + c\right ) + 3 \, a}{12 \, d \sin \left (d x + c\right )^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {4 \, a \sin \left (d x + c\right ) + 3 \, a}{12 \, d \sin \left (d x + c\right )^{4}} \]
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Time = 9.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {a}{4}+\frac {a\,\sin \left (c+d\,x\right )}{3}}{d\,{\sin \left (c+d\,x\right )}^4} \]
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