Integrand size = 27, antiderivative size = 98 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {b \cot ^5(c+d x)}{5 d}-\frac {a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d} \]
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Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2917, 2691, 3853, 3855, 2687, 30} \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {b \cot ^5(c+d x)}{5 d} \]
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Rule 30
Rule 2687
Rule 2691
Rule 2917
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = a \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+b \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx \\ & = -\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {1}{2} a \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {b \text {Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {b \cot ^5(c+d x)}{5 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{8} a \int \csc ^3(c+d x) \, dx \\ & = -\frac {b \cot ^5(c+d x)}{5 d}-\frac {a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{16} a \int \csc (c+d x) \, dx \\ & = -\frac {a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {b \cot ^5(c+d x)}{5 d}-\frac {a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.79 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \cot ^5(c+d x)}{5 d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \]
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Time = 0.44 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {b \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}}{d}\) | \(118\) |
default | \(\frac {a \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {b \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}}{d}\) | \(118\) |
parallelrisch | \(\frac {-5 a \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -12 b \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 a \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +60 b \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-60 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 a \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +120 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+120 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d}\) | \(184\) |
risch | \(\frac {-240 i b \,{\mathrm e}^{10 i \left (d x +c \right )}+15 a \,{\mathrm e}^{11 i \left (d x +c \right )}+240 i b \,{\mathrm e}^{8 i \left (d x +c \right )}+235 a \,{\mathrm e}^{9 i \left (d x +c \right )}-480 i b \,{\mathrm e}^{6 i \left (d x +c \right )}+390 a \,{\mathrm e}^{7 i \left (d x +c \right )}+480 i b \,{\mathrm e}^{4 i \left (d x +c \right )}+390 a \,{\mathrm e}^{5 i \left (d x +c \right )}-48 i b \,{\mathrm e}^{2 i \left (d x +c \right )}+235 a \,{\mathrm e}^{3 i \left (d x +c \right )}+48 i b +15 a \,{\mathrm e}^{i \left (d x +c \right )}}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}\) | \(198\) |
norman | \(\frac {-\frac {a}{384 d}+\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d}+\frac {b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d}-\frac {b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d}+\frac {b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\) | \(254\) |
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Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (88) = 176\).
Time = 0.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.91 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {96 \, b \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + 30 \, a \cos \left (d x + c\right )^{5} + 80 \, a \cos \left (d x + c\right )^{3} - 30 \, a \cos \left (d x + c\right ) - 15 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {5 \, a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {96 \, b}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (88) = 176\).
Time = 0.35 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.05 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {5 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 12 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 60 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 120 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {294 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 120 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 60 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
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Time = 11.68 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.09 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+\frac {a}{6}\right )}{64\,d}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16\,d} \]
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