Integrand size = 29, antiderivative size = 216 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {21 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d} \]
[Out]
Time = 0.28 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2687, 14, 2691, 3853, 3855, 276} \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {21 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]
[In]
[Out]
Rule 14
Rule 276
Rule 2687
Rule 2691
Rule 2952
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \cot ^4(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^6(c+d x)+a^3 \cot ^4(c+d x) \csc ^7(c+d x)\right ) \, dx \\ & = a^3 \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx \\ & = -\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {1}{10} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx-\frac {1}{8} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{80} \left (3 a^3\right ) \int \csc ^7(c+d x) \, dx+\frac {1}{16} \left (3 a^3\right ) \int \csc ^5(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{32} a^3 \int \csc ^5(c+d x) \, dx+\frac {1}{64} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {9 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{128} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac {1}{128} \left (9 a^3\right ) \int \csc (c+d x) \, dx \\ & = -\frac {9 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{256} \left (3 a^3\right ) \int \csc (c+d x) \, dx \\ & = -\frac {21 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d} \\ \end{align*}
Time = 7.52 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.69 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (-4096 \cot \left (\frac {1}{2} (c+d x)\right )-1260 \csc ^2\left (\frac {1}{2} (c+d x)\right )-5040 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+5040 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1260 \sec ^2\left (\frac {1}{2} (c+d x)\right )-180 \sec ^4\left (\frac {1}{2} (c+d x)\right )-390 \sec ^6\left (\frac {1}{2} (c+d x)\right )+75 \sec ^8\left (\frac {1}{2} (c+d x)\right )+6 \sec ^{10}\left (\frac {1}{2} (c+d x)\right )+64 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-4 \csc ^4\left (\frac {1}{2} (c+d x)\right ) (-45+\sin (c+d x))+5 \csc ^8\left (\frac {1}{2} (c+d x)\right ) (-15+4 \sin (c+d x))-2 \csc ^{10}\left (\frac {1}{2} (c+d x)\right ) (3+10 \sin (c+d x))+6 \csc ^6\left (\frac {1}{2} (c+d x)\right ) (65+42 \sin (c+d x))+4096 \tan \left (\frac {1}{2} (c+d x)\right )-504 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )-40 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+40 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{61440 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
[In]
[Out]
Time = 0.70 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.78
method | result | size |
parallelrisch | \(-\frac {3231 \left (-\frac {917504 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1077}+\left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {926 \cos \left (3 d x +3 c \right )}{3231}-\frac {3946 \cos \left (5 d x +5 c \right )}{16155}-\frac {203 \cos \left (7 d x +7 c \right )}{6462}+\frac {7 \cos \left (9 d x +9 c \right )}{2154}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {77824 \cos \left (d x +c \right )}{16155}+\frac {31744 \cos \left (3 d x +3 c \right )}{16155}+\frac {1024 \cos \left (5 d x +5 c \right )}{16155}-\frac {512 \cos \left (7 d x +7 c \right )}{5385}+\frac {512 \cos \left (9 d x +9 c \right )}{48465}\right ) \left (\sec ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{3}}{33554432 d}\) | \(168\) |
risch | \(\frac {a^{3} \left (315 \,{\mathrm e}^{19 i \left (d x +c \right )}-3045 \,{\mathrm e}^{17 i \left (d x +c \right )}-23676 \,{\mathrm e}^{15 i \left (d x +c \right )}-76800 i {\mathrm e}^{14 i \left (d x +c \right )}+27780 \,{\mathrm e}^{13 i \left (d x +c \right )}+7680 i {\mathrm e}^{16 i \left (d x +c \right )}+96930 \,{\mathrm e}^{11 i \left (d x +c \right )}+122880 i {\mathrm e}^{8 i \left (d x +c \right )}+96930 \,{\mathrm e}^{9 i \left (d x +c \right )}+15360 i {\mathrm e}^{6 i \left (d x +c \right )}+27780 \,{\mathrm e}^{7 i \left (d x +c \right )}-15360 i {\mathrm e}^{12 i \left (d x +c \right )}-23676 \,{\mathrm e}^{5 i \left (d x +c \right )}-64512 i {\mathrm e}^{10 i \left (d x +c \right )}-3045 \,{\mathrm e}^{3 i \left (d x +c \right )}+15360 i {\mathrm e}^{4 i \left (d x +c \right )}+315 \,{\mathrm e}^{i \left (d x +c \right )}-5120 i {\mathrm e}^{2 i \left (d x +c \right )}+512 i\right )}{1920 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}+\frac {21 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}-\frac {21 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}\) | \(272\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315 \sin \left (d x +c \right )^{5}}\right )+a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{32 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{256 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}+\frac {3 \cos \left (d x +c \right )}{256}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\) | \(352\) |
default | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315 \sin \left (d x +c \right )^{5}}\right )+a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{32 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{256 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}+\frac {3 \cos \left (d x +c \right )}{256}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\) | \(352\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.57 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {630 \, a^{3} \cos \left (d x + c\right )^{9} - 2940 \, a^{3} \cos \left (d x + c\right )^{7} + 768 \, a^{3} \cos \left (d x + c\right )^{5} + 2940 \, a^{3} \cos \left (d x + c\right )^{3} - 630 \, a^{3} \cos \left (d x + c\right ) - 315 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 512 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{9} - 9 \, a^{3} \cos \left (d x + c\right )^{7} + 12 \, a^{3} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{7680 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
[In]
[Out]
Timed out. \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.43 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {21 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} + 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 630 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1536 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}} - \frac {512 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{53760 \, d} \]
[In]
[Out]
none
Time = 0.63 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.65 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 384 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5040 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 3600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {14762 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 3600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 384 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{61440 \, d} \]
[In]
[Out]
Time = 10.74 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.83 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{512\,d}-\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{512\,d}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {21\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}-\frac {15\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}+\frac {15\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]
[In]
[Out]