Integrand size = 27, antiderivative size = 97 \[ \int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{10 d}+\frac {a \csc ^3(c+d x)}{3 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \]
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Time = 0.10 (sec) , antiderivative size = 97, 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} \[ \int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^{10}(c+d x)}{10 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^3(c+d x)}{3 d} \]
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Rule 14
Rule 276
Rule 2686
Rule 2687
Rule 2913
Rubi steps \begin{align*} \text {integral}& = a \int \cot ^7(c+d x) \csc ^3(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^4(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int x^2 \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 ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \text {Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac {a \text {Subst}\left (\int \left (x^7+x^9\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{10 d}+\frac {a \csc ^3(c+d x)}{3 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.33 \[ \int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc ^3(c+d x)}{3 d}+\frac {a \csc ^4(c+d x)}{4 d}-\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{2 d}+\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {3 a \csc ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d} \]
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Time = 0.41 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(88\) |
default | \(-\frac {a \left (\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(88\) |
parallelrisch | \(\frac {a \left (\sec ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-4239774-7986510 \cos \left (2 d x +2 c \right )-49770 \cos \left (8 d x +8 c \right )-860160 \sin \left (7 d x +7 c \right )-172032 \sin \left (5 d x +5 c \right )-1066275 \cos \left (6 d x +6 c \right )+2908160 \sin \left (d x +c \right )-4792320 \sin \left (3 d x +3 c \right )-3177720 \cos \left (4 d x +4 c \right )+4977 \cos \left (10 d x +10 c \right )\right )}{169114337280 d}\) | \(127\) |
risch | \(-\frac {4 i a \left (315 i {\mathrm e}^{16 i \left (d x +c \right )}+210 \,{\mathrm e}^{17 i \left (d x +c \right )}+630 i {\mathrm e}^{14 i \left (d x +c \right )}+42 \,{\mathrm e}^{15 i \left (d x +c \right )}+2205 i {\mathrm e}^{12 i \left (d x +c \right )}+1170 \,{\mathrm e}^{13 i \left (d x +c \right )}+1764 i {\mathrm e}^{10 i \left (d x +c \right )}-710 \,{\mathrm e}^{11 i \left (d x +c \right )}+2205 i {\mathrm e}^{8 i \left (d x +c \right )}+710 \,{\mathrm e}^{9 i \left (d x +c \right )}+630 i {\mathrm e}^{6 i \left (d x +c \right )}-1170 \,{\mathrm e}^{7 i \left (d x +c \right )}+315 i {\mathrm e}^{4 i \left (d x +c \right )}-42 \,{\mathrm e}^{5 i \left (d x +c \right )}-210 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}\) | \(194\) |
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Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.48 \[ \int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {630 \, a \cos \left (d x + c\right )^{6} - 630 \, a \cos \left (d x + c\right )^{4} + 315 \, a \cos \left (d x + c\right )^{2} + 8 \, {\left (105 \, a \cos \left (d x + c\right )^{6} - 126 \, a \cos \left (d x + c\right )^{4} + 72 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 63 \, a}{2520 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {840 \, a \sin \left (d x + c\right )^{7} + 630 \, a \sin \left (d x + c\right )^{6} - 1512 \, a \sin \left (d x + c\right )^{5} - 1260 \, a \sin \left (d x + c\right )^{4} + 1080 \, a \sin \left (d x + c\right )^{3} + 945 \, a \sin \left (d x + c\right )^{2} - 280 \, a \sin \left (d x + c\right ) - 252 \, a}{2520 \, d \sin \left (d x + c\right )^{10}} \]
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Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {840 \, a \sin \left (d x + c\right )^{7} + 630 \, a \sin \left (d x + c\right )^{6} - 1512 \, a \sin \left (d x + c\right )^{5} - 1260 \, a \sin \left (d x + c\right )^{4} + 1080 \, a \sin \left (d x + c\right )^{3} + 945 \, a \sin \left (d x + c\right )^{2} - 280 \, a \sin \left (d x + c\right ) - 252 \, a}{2520 \, d \sin \left (d x + c\right )^{10}} \]
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Time = 10.81 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{3}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{4}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{2}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{7}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{8}+\frac {a\,\sin \left (c+d\,x\right )}{9}+\frac {a}{10}}{d\,{\sin \left (c+d\,x\right )}^{10}} \]
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