Integrand size = 29, antiderivative size = 303 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 b \left (a^2+2 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \left (2 a^2+21 b^2\right ) \cot (c+d x)}{35 d}-\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac {\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac {\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d} \]
[Out]
Time = 0.58 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2972, 3126, 3110, 3100, 2827, 3852, 8, 3855} \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 b \left (a^2+2 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \left (2 a^2+21 b^2\right ) \cot (c+d x)}{35 d}+\frac {\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}-\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d} \]
[In]
[Out]
Rule 8
Rule 2827
Rule 2972
Rule 3100
Rule 3110
Rule 3126
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}-\frac {\int \csc ^6(c+d x) (a+b \sin (c+d x))^3 \left (6 \left (8 a^2-b^2\right )+3 a b \sin (c+d x)-3 \left (14 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{42 a^2} \\ & = \frac {\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}-\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x))^2 \left (3 b \left (53 a^2-6 b^2\right )-6 a \left (3 a^2-b^2\right ) \sin (c+d x)-9 b \left (18 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{210 a^2} \\ & = \frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac {\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}-\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x)) \left (-18 \left (4 a^4-19 a^2 b^2+2 b^4\right )-3 a b \left (81 a^2-2 b^2\right ) \sin (c+d x)-3 b^2 \left (163 a^2-6 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{840 a^2} \\ & = -\frac {\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac {\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}+\frac {\int \csc ^3(c+d x) \left (9 b \left (105 a^4-116 a^2 b^2+12 b^4\right )+72 a^3 \left (2 a^2+21 b^2\right ) \sin (c+d x)+9 b^3 \left (163 a^2-6 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2520 a^2} \\ & = -\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac {\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac {\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}+\frac {\int \csc ^2(c+d x) \left (144 a^3 \left (2 a^2+21 b^2\right )+945 a^2 b \left (a^2+2 b^2\right ) \sin (c+d x)\right ) \, dx}{5040 a^2} \\ & = -\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac {\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac {\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}+\frac {1}{16} \left (3 b \left (a^2+2 b^2\right )\right ) \int \csc (c+d x) \, dx+\frac {1}{35} \left (a \left (2 a^2+21 b^2\right )\right ) \int \csc ^2(c+d x) \, dx \\ & = -\frac {3 b \left (a^2+2 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}-\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac {\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac {\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}-\frac {\left (a \left (2 a^2+21 b^2\right )\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{35 d} \\ & = -\frac {3 b \left (a^2+2 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \left (2 a^2+21 b^2\right ) \cot (c+d x)}{35 d}-\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac {\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac {\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d} \\ \end{align*}
Time = 1.38 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.07 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {56 a \left (14 a^2-3 b^2\right ) \cos (3 (c+d x)) \csc ^7(c+d x)+112 a^3 \cos (5 (c+d x)) \csc ^7(c+d x)-504 a b^2 \cos (5 (c+d x)) \csc ^7(c+d x)-16 a^3 \cos (7 (c+d x)) \csc ^7(c+d x)-168 a b^2 \cos (7 (c+d x)) \csc ^7(c+d x)+3360 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+6720 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3360 a^2 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-6720 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+70 \cot (c+d x) \csc ^6(c+d x) \left (12 a \left (2 a^2+b^2\right )+b \left (31 a^2-18 b^2\right ) \sin (c+d x)\right )+1540 a^2 b \csc ^7(c+d x) \sin (4 (c+d x))+840 b^3 \csc ^7(c+d x) \sin (4 (c+d x))+105 a^2 b \csc ^7(c+d x) \sin (6 (c+d x))-350 b^3 \csc ^7(c+d x) \sin (6 (c+d x))}{17920 d} \]
[In]
[Out]
Time = 0.68 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{2} b \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+b^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(243\) |
default | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{2} b \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+b^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(243\) |
parallelrisch | \(\frac {\frac {3 b \left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {3 \left (a^{3} \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )+\frac {7 \cos \left (3 d x +3 c \right )}{15}+\frac {\cos \left (5 d x +5 c \right )}{15}-\frac {\cos \left (7 d x +7 c \right )}{105}\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {13 b \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )+\frac {47 \cos \left (3 d x +3 c \right )}{78}+\frac {\cos \left (5 d x +5 c \right )}{26}\right ) a^{2} \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+16 b^{2} \left (\cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{2}+\frac {\cos \left (5 d x +5 c \right )}{10}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+8 b^{3} \left (\cos \left (d x +c \right )+\frac {5 \cos \left (3 d x +3 c \right )}{3}\right )\right ) \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}}{d}\) | \(253\) |
risch | \(-\frac {-2240 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-448 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-224 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+1680 i a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-3360 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+3696 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+5040 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-6720 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-672 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+350 b^{3} {\mathrm e}^{13 i \left (d x +c \right )}-840 b^{3} {\mathrm e}^{11 i \left (d x +c \right )}-630 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+336 i a \,b^{2}+630 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+32 i a^{3}+840 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-350 b^{3} {\mathrm e}^{i \left (d x +c \right )}-105 a^{2} b \,{\mathrm e}^{13 i \left (d x +c \right )}-1540 a^{2} b \,{\mathrm e}^{11 i \left (d x +c \right )}-1085 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}-1120 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}+1085 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-1120 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+1540 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+105 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}}{280 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(461\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.15 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {32 \, {\left (2 \, a^{3} + 21 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} - 224 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left ({\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - a^{2} b - 2 \, b^{3} + 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 105 \, {\left ({\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - a^{2} b - 2 \, b^{3} + 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 70 \, {\left ({\left (3 \, a^{2} b - 10 \, b^{3}\right )} \cos \left (d x + c\right )^{5} + 8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1120 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
[In]
[Out]
Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.69 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {35 \, a^{2} b {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 70 \, b^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {672 \, a b^{2}}{\tan \left (d x + c\right )^{5}} - \frac {32 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}}}{1120 \, d} \]
[In]
[Out]
none
Time = 0.46 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.50 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 35 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 84 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 70 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 420 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 560 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 840 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 840 \, {\left (a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2178 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4356 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 840 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 560 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 420 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 70 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 84 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{4480 \, d} \]
[In]
[Out]
Time = 10.59 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.18 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^3}{128}+\frac {3\,a\,b^2}{32}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,a\,b^2}{160}-\frac {a^3}{640}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,a^2\,b}{128}+\frac {b^3}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {3\,a^2\,b}{128}-\frac {b^3}{64}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2\,b}{16}+\frac {3\,b^3}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {12\,a\,b^2}{5}-\frac {a^3}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (3\,a^3+24\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^2\,b-2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,a^2\,b+16\,b^3\right )+\frac {a^3}{7}+a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^3}{128}+\frac {3\,a\,b^2}{16}\right )}{d}+\frac {a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d} \]
[In]
[Out]