Integrand size = 29, antiderivative size = 334 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 a \left (a^2+8 b^2\right ) \text {arctanh}(\cos (c+d x))}{128 d}-\frac {b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d} \]
[Out]
Time = 0.63 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2972, 3126, 3110, 3100, 2827, 3853, 3855, 3852, 8} \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 a \left (a^2+8 b^2\right ) \text {arctanh}(\cos (c+d x))}{128 d}-\frac {b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d} \]
[In]
[Out]
Rule 8
Rule 2827
Rule 2972
Rule 3100
Rule 3110
Rule 3126
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}-\frac {\int \csc ^7(c+d x) (a+b \sin (c+d x))^3 \left (3 \left (21 a^2-4 b^2\right )+3 a b \sin (c+d x)-8 \left (7 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{56 a^2} \\ & = \frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}-\frac {\int \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (9 b \left (23 a^2-4 b^2\right )-3 a \left (7 a^2-2 b^2\right ) \sin (c+d x)-6 b \left (35 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{336 a^2} \\ & = \frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}-\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x)) \left (-3 \left (35 a^4-148 a^2 b^2+24 b^4\right )-3 a b \left (109 a^2-2 b^2\right ) \sin (c+d x)-12 b^2 \left (53 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{1680 a^2} \\ & = -\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {\int \csc ^4(c+d x) \left (72 b \left (24 a^4-25 a^2 b^2+4 b^4\right )+315 a^3 \left (a^2+8 b^2\right ) \sin (c+d x)+48 b^3 \left (53 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6720 a^2} \\ & = -\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {\int \csc ^3(c+d x) \left (945 a^3 \left (a^2+8 b^2\right )+576 a^2 b \left (6 a^2+7 b^2\right ) \sin (c+d x)\right ) \, dx}{20160 a^2} \\ & = -\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {1}{35} \left (b \left (6 a^2+7 b^2\right )\right ) \int \csc ^2(c+d x) \, dx+\frac {1}{64} \left (3 a \left (a^2+8 b^2\right )\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {1}{128} \left (3 a \left (a^2+8 b^2\right )\right ) \int \csc (c+d x) \, dx-\frac {\left (b \left (6 a^2+7 b^2\right )\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{35 d} \\ & = -\frac {3 a \left (a^2+8 b^2\right ) \text {arctanh}(\cos (c+d x))}{128 d}-\frac {b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d} \\ \end{align*}
Time = 2.07 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.80 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {6720 a \left (a^2+8 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-6720 a \left (a^2+8 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^8(c+d x) \left (35 a \left (671 a^2+248 b^2\right ) \cos (c+d x)+35 \left (333 a^3+104 a b^2\right ) \cos (3 (c+d x))+805 a^3 \cos (5 (c+d x))-11480 a b^2 \cos (5 (c+d x))-105 a^3 \cos (7 (c+d x))-840 a b^2 \cos (7 (c+d x))+21504 a^2 b \sin (2 (c+d x))+2688 b^3 \sin (2 (c+d x))+16128 a^2 b \sin (4 (c+d x))+896 b^3 \sin (4 (c+d x))+3072 a^2 b \sin (6 (c+d x))-896 b^3 \sin (6 (c+d x))-384 a^2 b \sin (8 (c+d x))-448 b^3 \sin (8 (c+d x))\right )}{286720 d} \]
[In]
[Out]
Time = 0.67 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{2} b \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+3 a \,b^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {b^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}}{d}\) | \(280\) |
default | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{2} b \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+3 a \,b^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {b^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}}{d}\) | \(280\) |
parallelrisch | \(\frac {240 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +5040 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -1680 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+1680 a \,b^{2} \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+13440 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-5040 a^{2} b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+1680 a \,b^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4480 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}-1680 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +1680 a^{2} b \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1680 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+1680 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-35 a^{3} \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2240 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+560 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+35 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-560 a \,b^{2} \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-240 a^{2} b \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-448 b^{3} \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+448 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+2240 b^{3} \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-280 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+280 a^{3} \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-336 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +336 a^{2} b \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4480 b^{3} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{71680 d}\) | \(428\) |
risch | \(\frac {105 a^{3} {\mathrm e}^{i \left (d x +c \right )}-805 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-23485 a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-11655 a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-5376 i a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+26880 i a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-43008 i a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-6144 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+26880 i a^{2} b \,{\mathrm e}^{12 i \left (d x +c \right )}+31360 i b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-3640 a \,b^{2} {\mathrm e}^{11 i \left (d x +c \right )}-2688 i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-8680 a \,b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-27776 i b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+768 i a^{2} b -8680 a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-22400 i b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-3640 a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+11480 a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+11648 i b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+840 a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+840 a \,b^{2} {\mathrm e}^{15 i \left (d x +c \right )}+11480 a \,b^{2} {\mathrm e}^{13 i \left (d x +c \right )}-11655 a^{3} {\mathrm e}^{11 i \left (d x +c \right )}-23485 a^{3} {\mathrm e}^{9 i \left (d x +c \right )}+896 i b^{3}+105 a^{3} {\mathrm e}^{15 i \left (d x +c \right )}-805 a^{3} {\mathrm e}^{13 i \left (d x +c \right )}+13440 i b^{3} {\mathrm e}^{12 i \left (d x +c \right )}-4480 i b^{3} {\mathrm e}^{14 i \left (d x +c \right )}}{2240 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{16 d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{16 d}\) | \(533\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.15 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {210 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} - 70 \, {\left (11 \, a^{3} - 40 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 770 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 210 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right ) - 105 \, {\left ({\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{8} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 8 \, a b^{2} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left ({\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{8} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 8 \, a b^{2} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 256 \, {\left ({\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{7} - 7 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{8960 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
[In]
[Out]
Timed out. \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.74 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {35 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 280 \, a b^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1792 \, b^{3}}{\tan \left (d x + c\right )^{5}} - \frac {768 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2} b}{\tan \left (d x + c\right )^{7}}}{8960 \, d} \]
[In]
[Out]
none
Time = 0.45 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.37 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 240 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 560 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 336 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 448 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1680 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5040 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4480 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1680 \, {\left (a^{3} + 8 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {4566 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 36528 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 5040 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4480 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1680 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 448 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 35 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{71680 \, d} \]
[In]
[Out]
Time = 10.56 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.14 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3}{256}+\frac {3\,a\,b^2}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {3\,a^2\,b}{128}+\frac {b^3}{32}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,a^2\,b}{640}-\frac {b^3}{160}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^3}{128}+\frac {3\,a\,b^2}{16}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^3+6\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (6\,a^2\,b+8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {6\,a^2\,b}{5}-\frac {8\,b^3}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (18\,a^2\,b+16\,b^3\right )-\frac {a^3}{8}-\frac {6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{7}-2\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{256\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {9\,a^2\,b}{128}+\frac {b^3}{16}\right )}{d}-\frac {3\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d} \]
[In]
[Out]