Integrand size = 27, antiderivative size = 114 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {b \cot (c+d x) \csc (c+d x)}{16 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {b \cot ^3(c+d x) \csc ^3(c+d x)}{6 d} \]
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Time = 0.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2917, 2687, 14, 2691, 3853, 3855} \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {b \text {arctanh}(\cos (c+d x))}{16 d}-\frac {b \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {b \cot (c+d x) \csc (c+d x)}{16 d} \]
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Rule 14
Rule 2687
Rule 2691
Rule 2917
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = a \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+b \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx \\ & = -\frac {b \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {1}{2} b \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {a \text {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {b \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {b \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{8} b \int \csc ^3(c+d x) \, dx+\frac {a \text {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {b \cot (c+d x) \csc (c+d x)}{16 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {b \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{16} b \int \csc (c+d x) \, dx \\ & = -\frac {b \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {b \cot (c+d x) \csc (c+d x)}{16 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {b \cot ^3(c+d x) \csc ^3(c+d x)}{6 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(239\) vs. \(2(114)=228\).
Time = 0.12 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.10 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {2 a \cot (c+d x)}{35 d}-\frac {b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {b \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {b \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{35 d}+\frac {8 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac {a \cot (c+d x) \csc ^6(c+d x)}{7 d}-\frac {b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {b \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {b \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \]
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Time = 0.48 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.20
method | result | size |
parallelrisch | \(\frac {256 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -3 \left (a \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {7 \cos \left (3 d x +3 c \right )}{15}+\frac {\cos \left (5 d x +5 c \right )}{15}-\frac {\cos \left (7 d x +7 c \right )}{105}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {13 b \left (\cos \left (d x +c \right )+\frac {47 \cos \left (3 d x +3 c \right )}{78}+\frac {\cos \left (5 d x +5 c \right )}{26}\right )}{6}\right ) \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096 d}\) | \(137\) |
derivativedivides | \(\frac {a \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+b \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(138\) |
default | \(\frac {a \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+b \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(138\) |
risch | \(\frac {105 b \,{\mathrm e}^{13 i \left (d x +c \right )}+3360 i a \,{\mathrm e}^{10 i \left (d x +c \right )}+1540 b \,{\mathrm e}^{11 i \left (d x +c \right )}+3360 i a \,{\mathrm e}^{8 i \left (d x +c \right )}+1085 b \,{\mathrm e}^{9 i \left (d x +c \right )}+6720 i a \,{\mathrm e}^{6 i \left (d x +c \right )}+1344 i a \,{\mathrm e}^{4 i \left (d x +c \right )}-1085 b \,{\mathrm e}^{5 i \left (d x +c \right )}+672 i a \,{\mathrm e}^{2 i \left (d x +c \right )}-1540 b \,{\mathrm e}^{3 i \left (d x +c \right )}-96 i a -105 b \,{\mathrm e}^{i \left (d x +c \right )}}{840 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}\) | \(198\) |
norman | \(\frac {-\frac {a}{896 d}+\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2240 d}+\frac {3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d}-\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {3 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d}-\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2240 d}+\frac {a \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{896 d}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d}+\frac {b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {b \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\) | \(288\) |
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (102) = 204\).
Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.94 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {192 \, a \cos \left (d x + c\right )^{7} - 672 \, a \cos \left (d x + c\right )^{5} + 105 \, {\left (b \cos \left (d x + c\right )^{6} - 3 \, b \cos \left (d x + c\right )^{4} + 3 \, b \cos \left (d x + c\right )^{2} - b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 105 \, {\left (b \cos \left (d x + c\right )^{6} - 3 \, b \cos \left (d x + c\right )^{4} + 3 \, b \cos \left (d x + c\right )^{2} - b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 70 \, {\left (3 \, b \cos \left (d x + c\right )^{5} + 8 \, b \cos \left (d x + c\right )^{3} - 3 \, b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3360 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {35 \, b {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {96 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a}{\tan \left (d x + c\right )^{7}}}{3360 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (102) = 204\).
Time = 0.38 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.01 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 35 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 840 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {2178 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 105 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \]
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Time = 12.06 (sec) , antiderivative size = 399, normalized size of antiderivative = 3.50 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {15\,a\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-15\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-21\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-105\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+315\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-315\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+105\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+21\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-105\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-105\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+105\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+105\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+35\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-35\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+840\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{13440\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
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