Integrand size = 27, antiderivative size = 113 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^{12}(c+d x)}{12 d}+\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{13}(c+d x)}{13 d} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 113, 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 = {2913, 2686, 276, 2687, 272, 45} \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^{12}(c+d x)}{12 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^{13}(c+d x)}{13 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {a \csc ^7(c+d x)}{7 d} \]
[In]
[Out]
Rule 45
Rule 272
Rule 276
Rule 2686
Rule 2687
Rule 2913
Rubi steps \begin{align*} \text {integral}& = a \int \cot ^7(c+d x) \csc ^6(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^7(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int x^6 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}-\frac {a \text {Subst}\left (\int x^7 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \text {Subst}\left (\int x^3 (1+x)^2 \, dx,x,\cot ^2(c+d x)\right )}{2 d}-\frac {a \text {Subst}\left (\int \left (-x^6+3 x^8-3 x^{10}+x^{12}\right ) \, dx,x,\csc (c+d x)\right )}{d} \\ & = \frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{13}(c+d x)}{13 d}-\frac {a \text {Subst}\left (\int \left (x^3+2 x^4+x^5\right ) \, dx,x,\cot ^2(c+d x)\right )}{2 d} \\ & = -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^{12}(c+d x)}{12 d}+\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{13}(c+d x)}{13 d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.14 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc ^6(c+d x)}{6 d}+\frac {a \csc ^7(c+d x)}{7 d}-\frac {3 a \csc ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {3 a \csc ^{10}(c+d x)}{10 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{12}(c+d x)}{12 d}-\frac {a \csc ^{13}(c+d x)}{13 d} \]
[In]
[Out]
Time = 0.60 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\csc ^{13}\left (d x +c \right )\right )}{13}+\frac {\left (\csc ^{12}\left (d x +c \right )\right )}{12}-\frac {3 \left (\csc ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\csc ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}\right )}{d}\) | \(88\) |
default | \(-\frac {a \left (\frac {\left (\csc ^{13}\left (d x +c \right )\right )}{13}+\frac {\left (\csc ^{12}\left (d x +c \right )\right )}{12}-\frac {3 \left (\csc ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\csc ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}\right )}{d}\) | \(88\) |
parallelrisch | \(-\frac {a \left (\sec ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (5269094400+281281 \sin \left (13 d x +13 c \right )+9235333120 \cos \left (2 d x +2 c \right )+21939918 \sin \left (9 d x +9 c \right )+575568994 \sin \left (7 d x +7 c \right )+1513146635 \sin \left (5 d x +5 c \right )+1124597760 \cos \left (6 d x +6 c \right )+876287412 \sin \left (d x +c \right )+2786865081 \sin \left (3 d x +3 c \right )+3748659200 \cos \left (4 d x +4 c \right )-3656653 \sin \left (11 d x +11 c \right )\right )}{2063645886382080 d}\) | \(138\) |
risch | \(-\frac {32 a \left (8580 i {\mathrm e}^{19 i \left (d x +c \right )}+5005 \,{\mathrm e}^{20 i \left (d x +c \right )}+28600 i {\mathrm e}^{17 i \left (d x +c \right )}+10010 \,{\mathrm e}^{18 i \left (d x +c \right )}+70460 i {\mathrm e}^{15 i \left (d x +c \right )}+24024 \,{\mathrm e}^{16 i \left (d x +c \right )}+80400 i {\mathrm e}^{13 i \left (d x +c \right )}+3003 \,{\mathrm e}^{14 i \left (d x +c \right )}+70460 i {\mathrm e}^{11 i \left (d x +c \right )}-3003 \,{\mathrm e}^{12 i \left (d x +c \right )}+28600 i {\mathrm e}^{9 i \left (d x +c \right )}-24024 \,{\mathrm e}^{10 i \left (d x +c \right )}+8580 i {\mathrm e}^{7 i \left (d x +c \right )}-10010 \,{\mathrm e}^{8 i \left (d x +c \right )}-5005 \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{15015 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{13}}\) | \(193\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.42 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {17160 \, a \cos \left (d x + c\right )^{6} - 11440 \, a \cos \left (d x + c\right )^{4} + 4160 \, a \cos \left (d x + c\right )^{2} + 1001 \, {\left (20 \, a \cos \left (d x + c\right )^{6} - 15 \, a \cos \left (d x + c\right )^{4} + 6 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 640 \, a}{120120 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
[In]
[Out]
Timed out. \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {20020 \, a \sin \left (d x + c\right )^{7} + 17160 \, a \sin \left (d x + c\right )^{6} - 45045 \, a \sin \left (d x + c\right )^{5} - 40040 \, a \sin \left (d x + c\right )^{4} + 36036 \, a \sin \left (d x + c\right )^{3} + 32760 \, a \sin \left (d x + c\right )^{2} - 10010 \, a \sin \left (d x + c\right ) - 9240 \, a}{120120 \, d \sin \left (d x + c\right )^{13}} \]
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {20020 \, a \sin \left (d x + c\right )^{7} + 17160 \, a \sin \left (d x + c\right )^{6} - 45045 \, a \sin \left (d x + c\right )^{5} - 40040 \, a \sin \left (d x + c\right )^{4} + 36036 \, a \sin \left (d x + c\right )^{3} + 32760 \, a \sin \left (d x + c\right )^{2} - 10010 \, a \sin \left (d x + c\right ) - 9240 \, a}{120120 \, d \sin \left (d x + c\right )^{13}} \]
[In]
[Out]
Time = 10.36 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{6}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{7}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{8}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{10}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{11}+\frac {a\,\sin \left (c+d\,x\right )}{12}+\frac {a}{13}}{d\,{\sin \left (c+d\,x\right )}^{13}} \]
[In]
[Out]