Integrand size = 27, antiderivative size = 81 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {b \csc ^3(c+d x)}{3 d}+\frac {2 b \csc ^5(c+d x)}{5 d}-\frac {b \csc ^7(c+d x)}{7 d} \]
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Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2913, 2687, 14, 2686, 276} \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \csc ^7(c+d x)}{7 d}+\frac {2 b \csc ^5(c+d x)}{5 d}-\frac {b \csc ^3(c+d x)}{3 d} \]
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Rule 14
Rule 276
Rule 2686
Rule 2687
Rule 2913
Rubi steps \begin{align*} \text {integral}& = a \int \cot ^5(c+d x) \csc ^4(c+d x) \, dx+b \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}-\frac {b \text {Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \text {Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,-\cot (c+d x)\right )}{d}-\frac {b \text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {b \csc ^3(c+d x)}{3 d}+\frac {2 b \csc ^5(c+d x)}{5 d}-\frac {b \csc ^7(c+d x)}{7 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {2 b \csc ^5(c+d x)}{5 d}+\frac {a \csc ^6(c+d x)}{3 d}-\frac {b \csc ^7(c+d x)}{7 d}-\frac {a \csc ^8(c+d x)}{8 d} \]
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Time = 0.38 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(-\frac {\frac {\left (\csc ^{8}\left (d x +c \right )\right ) a}{8}+\frac {b \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right ) a}{3}-\frac {2 b \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right ) a}{4}+\frac {b \left (\csc ^{3}\left (d x +c \right )\right )}{3}}{d}\) | \(73\) |
default | \(-\frac {\frac {\left (\csc ^{8}\left (d x +c \right )\right ) a}{8}+\frac {b \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right ) a}{3}-\frac {2 b \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right ) a}{4}+\frac {b \left (\csc ^{3}\left (d x +c \right )\right )}{3}}{d}\) | \(73\) |
parallelrisch | \(-\frac {853 \left (a \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (2 d x +2 c \right )+\frac {2561 \cos \left (4 d x +4 c \right )}{5118}+\frac {73 \cos \left (6 d x +6 c \right )}{2559}-\frac {73 \cos \left (8 d x +8 c \right )}{20472}+\frac {5975}{6824}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {16384 b \left (\cos \left (2 d x +2 c \right )+\frac {5 \cos \left (4 d x +4 c \right )}{4}+\frac {57}{28}\right )}{12795}\right ) \left (\csc ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4194304 d}\) | \(118\) |
risch | \(\frac {4 i \left (105 i a \,{\mathrm e}^{12 i \left (d x +c \right )}+70 b \,{\mathrm e}^{13 i \left (d x +c \right )}+140 i a \,{\mathrm e}^{10 i \left (d x +c \right )}-14 b \,{\mathrm e}^{11 i \left (d x +c \right )}+350 i a \,{\mathrm e}^{8 i \left (d x +c \right )}+172 b \,{\mathrm e}^{9 i \left (d x +c \right )}+140 i a \,{\mathrm e}^{6 i \left (d x +c \right )}-172 b \,{\mathrm e}^{7 i \left (d x +c \right )}+105 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+14 b \,{\mathrm e}^{5 i \left (d x +c \right )}-70 b \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}\) | \(158\) |
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Time = 0.39 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.35 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {210 \, a \cos \left (d x + c\right )^{4} - 140 \, a \cos \left (d x + c\right )^{2} + 8 \, {\left (35 \, b \cos \left (d x + c\right )^{4} - 28 \, b \cos \left (d x + c\right )^{2} + 8 \, b\right )} \sin \left (d x + c\right ) + 35 \, a}{840 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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Timed out. \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {280 \, b \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 336 \, b \sin \left (d x + c\right )^{3} - 280 \, a \sin \left (d x + c\right )^{2} + 120 \, b \sin \left (d x + c\right ) + 105 \, a}{840 \, d \sin \left (d x + c\right )^{8}} \]
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Time = 0.40 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {280 \, b \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 336 \, b \sin \left (d x + c\right )^{3} - 280 \, a \sin \left (d x + c\right )^{2} + 120 \, b \sin \left (d x + c\right ) + 105 \, a}{840 \, d \sin \left (d x + c\right )^{8}} \]
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Time = 11.53 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {280\,b\,{\sin \left (c+d\,x\right )}^5+210\,a\,{\sin \left (c+d\,x\right )}^4-336\,b\,{\sin \left (c+d\,x\right )}^3-280\,a\,{\sin \left (c+d\,x\right )}^2+120\,b\,\sin \left (c+d\,x\right )+105\,a}{840\,d\,{\sin \left (c+d\,x\right )}^8} \]
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