Integrand size = 19, antiderivative size = 82 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=a x+\frac {3 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {3 a \cos (c+d x)}{2 d}+\frac {a \cot (c+d x)}{d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d} \]
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Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2789, 2672, 294, 327, 212, 3554, 8} \[ \int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {3 a \cos (c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {a \cot (c+d x)}{d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d}+a x \]
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Rule 8
Rule 212
Rule 294
Rule 327
Rule 2672
Rule 2789
Rule 3554
Rubi steps \begin{align*} \text {integral}& = \int \left (a \cos (c+d x) \cot ^3(c+d x)+a \cot ^4(c+d x)\right ) \, dx \\ & = a \int \cos (c+d x) \cot ^3(c+d x) \, dx+a \int \cot ^4(c+d x) \, dx \\ & = -\frac {a \cot ^3(c+d x)}{3 d}-a \int \cot ^2(c+d x) \, dx-\frac {a \text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a \cot (c+d x)}{d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+a \int 1 \, dx+\frac {(3 a) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 d} \\ & = a x-\frac {3 a \cos (c+d x)}{2 d}+\frac {a \cot (c+d x)}{d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 d} \\ & = a x+\frac {3 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {3 a \cos (c+d x)}{2 d}+\frac {a \cot (c+d x)}{d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.52 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos (c+d x)}{d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d} \]
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Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(86\) |
default | \(\frac {a \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(86\) |
risch | \(a x -\frac {a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a \left (12 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}-12 i {\mathrm e}^{2 i \left (d x +c \right )}+8 i-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\) | \(139\) |
parallelrisch | \(\frac {a \left (24 d x -36 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-18 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (d x +c \right )-32 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \cot \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-75 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{24 d}\) | \(173\) |
norman | \(\frac {a x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a}{24 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {7 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {7 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {9 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(180\) |
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Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (74) = 148\).
Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.95 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {16 \, a \cos \left (d x + c\right )^{3} + 9 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 9 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 12 \, a \cos \left (d x + c\right ) + 6 \, {\left (2 \, a d x \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right )^{3} - 2 \, a d x + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.12 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {4 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a + 3 \, a {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.72 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, {\left (d x + c\right )} a - 36 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {48 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {66 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
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Time = 9.72 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.78 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+17\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {2\,a\,\mathrm {atan}\left (\frac {4\,a^2}{6\,a^2+4\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {6\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{6\,a^2+4\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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