Integrand size = 29, antiderivative size = 279 \[ \int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {843 \log (1-\sin (c+d x))}{512 a d}+\frac {6 \log (\sin (c+d x))}{a d}-\frac {2229 \log (1+\sin (c+d x))}{512 a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {a^2}{32 d (a-a \sin (c+d x))^3}+\frac {81 a}{512 d (a-a \sin (c+d x))^2}+\frac {203}{256 d (a-a \sin (c+d x))}+\frac {a^4}{160 d (a+a \sin (c+d x))^5}+\frac {11 a^3}{256 d (a+a \sin (c+d x))^4}+\frac {23 a^2}{128 d (a+a \sin (c+d x))^3}+\frac {325 a}{512 d (a+a \sin (c+d x))^2}+\frac {5}{2 d (a+a \sin (c+d x))} \]
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Time = 0.23 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 90} \[ \int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^4}{160 d (a \sin (c+d x)+a)^5}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {11 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac {a^2}{32 d (a-a \sin (c+d x))^3}+\frac {23 a^2}{128 d (a \sin (c+d x)+a)^3}+\frac {81 a}{512 d (a-a \sin (c+d x))^2}+\frac {325 a}{512 d (a \sin (c+d x)+a)^2}+\frac {203}{256 d (a-a \sin (c+d x))}+\frac {5}{2 d (a \sin (c+d x)+a)}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {843 \log (1-\sin (c+d x))}{512 a d}+\frac {6 \log (\sin (c+d x))}{a d}-\frac {2229 \log (\sin (c+d x)+1)}{512 a d} \]
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Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {a^9 \text {Subst}\left (\int \frac {a^3}{(a-x)^5 x^3 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^{12} \text {Subst}\left (\int \frac {1}{(a-x)^5 x^3 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^{12} \text {Subst}\left (\int \left (\frac {1}{64 a^9 (a-x)^5}+\frac {3}{32 a^{10} (a-x)^4}+\frac {81}{256 a^{11} (a-x)^3}+\frac {203}{256 a^{12} (a-x)^2}+\frac {843}{512 a^{13} (a-x)}+\frac {1}{a^{11} x^3}-\frac {1}{a^{12} x^2}+\frac {6}{a^{13} x}-\frac {1}{32 a^8 (a+x)^6}-\frac {11}{64 a^9 (a+x)^5}-\frac {69}{128 a^{10} (a+x)^4}-\frac {325}{256 a^{11} (a+x)^3}-\frac {5}{2 a^{12} (a+x)^2}-\frac {2229}{512 a^{13} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {843 \log (1-\sin (c+d x))}{512 a d}+\frac {6 \log (\sin (c+d x))}{a d}-\frac {2229 \log (1+\sin (c+d x))}{512 a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {a^2}{32 d (a-a \sin (c+d x))^3}+\frac {81 a}{512 d (a-a \sin (c+d x))^2}+\frac {203}{256 d (a-a \sin (c+d x))}+\frac {a^4}{160 d (a+a \sin (c+d x))^5}+\frac {11 a^3}{256 d (a+a \sin (c+d x))^4}+\frac {23 a^2}{128 d (a+a \sin (c+d x))^3}+\frac {325 a}{512 d (a+a \sin (c+d x))^2}+\frac {5}{2 d (a+a \sin (c+d x))} \\ \end{align*}
Time = 6.16 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.91 \[ \int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^9 \left (\frac {\csc (c+d x)}{a^{10}}-\frac {\csc ^2(c+d x)}{2 a^{10}}-\frac {843 \log (1-\sin (c+d x))}{512 a^{10}}+\frac {6 \log (\sin (c+d x))}{a^{10}}-\frac {2229 \log (1+\sin (c+d x))}{512 a^{10}}+\frac {1}{256 a^6 (a-a \sin (c+d x))^4}+\frac {1}{32 a^7 (a-a \sin (c+d x))^3}+\frac {81}{512 a^8 (a-a \sin (c+d x))^2}+\frac {203}{256 a^9 (a-a \sin (c+d x))}+\frac {1}{160 a^5 (a+a \sin (c+d x))^5}+\frac {11}{256 a^6 (a+a \sin (c+d x))^4}+\frac {23}{128 a^7 (a+a \sin (c+d x))^3}+\frac {325}{512 a^8 (a+a \sin (c+d x))^2}+\frac {5}{2 a^9 (a+a \sin (c+d x))}\right )}{d} \]
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Time = 3.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.59
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )}+6 \ln \left (\sin \left (d x +c \right )\right )+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{32 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {81}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {203}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {843 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {11}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {23}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {325}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{2 \left (1+\sin \left (d x +c \right )\right )}-\frac {2229 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(166\) |
default | \(\frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )}+6 \ln \left (\sin \left (d x +c \right )\right )+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{32 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {81}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {203}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {843 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {11}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {23}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {325}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{2 \left (1+\sin \left (d x +c \right )\right )}-\frac {2229 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(166\) |
risch | \(\frac {i \left (111333 \,{\mathrm e}^{5 i \left (d x +c \right )}+58336 i {\mathrm e}^{14 i \left (d x +c \right )}+3465 \,{\mathrm e}^{i \left (d x +c \right )}+85906 \,{\mathrm e}^{13 i \left (d x +c \right )}+85906 \,{\mathrm e}^{9 i \left (d x +c \right )}+182360 \,{\mathrm e}^{15 i \left (d x +c \right )}+21296 i {\mathrm e}^{16 i \left (d x +c \right )}-58336 i {\mathrm e}^{8 i \left (d x +c \right )}+31530 \,{\mathrm e}^{3 i \left (d x +c \right )}+182360 \,{\mathrm e}^{7 i \left (d x +c \right )}+111333 \,{\mathrm e}^{17 i \left (d x +c \right )}+870 i {\mathrm e}^{18 i \left (d x +c \right )}-21296 i {\mathrm e}^{6 i \left (d x +c \right )}-173828 \,{\mathrm e}^{11 i \left (d x +c \right )}+31530 \,{\mathrm e}^{19 i \left (d x +c \right )}+3465 \,{\mathrm e}^{21 i \left (d x +c \right )}+46852 i {\mathrm e}^{12 i \left (d x +c \right )}-46852 i {\mathrm e}^{10 i \left (d x +c \right )}-870 i {\mathrm e}^{4 i \left (d x +c \right )}-750 i {\mathrm e}^{20 i \left (d x +c \right )}+750 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{640 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d a}-\frac {2229 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}-\frac {843 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\) | \(356\) |
parallelrisch | \(\frac {33 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\left (\frac {281 \sin \left (11 d x +11 c \right )}{11264}-\frac {1405 \sin \left (5 d x +5 c \right )}{11264}+\frac {281 \cos \left (4 d x +4 c \right )}{704}+\frac {1967 \sin \left (7 d x +7 c \right )}{11264}+\frac {3653 \cos \left (6 d x +6 c \right )}{5632}-\frac {1967 \sin \left (d x +c \right )}{5632}-\frac {281 \sin \left (3 d x +3 c \right )}{512}+\frac {281 \cos \left (10 d x +10 c \right )}{5632}-\frac {1967 \cos \left (2 d x +2 c \right )}{2816}+\frac {843 \cos \left (8 d x +8 c \right )}{2816}+\frac {1405 \sin \left (9 d x +9 c \right )}{11264}-\frac {1967}{2816}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-\frac {5201}{2816}-\frac {5201 \cos \left (2 d x +2 c \right )}{2816}+\frac {2229 \cos \left (8 d x +8 c \right )}{2816}+\frac {3715 \sin \left (9 d x +9 c \right )}{11264}+\frac {743 \sin \left (11 d x +11 c \right )}{11264}-\frac {3715 \sin \left (5 d x +5 c \right )}{11264}+\frac {743 \cos \left (4 d x +4 c \right )}{704}+\frac {5201 \sin \left (7 d x +7 c \right )}{11264}+\frac {9659 \cos \left (6 d x +6 c \right )}{5632}-\frac {5201 \sin \left (d x +c \right )}{5632}-\frac {743 \sin \left (3 d x +3 c \right )}{512}+\frac {743 \cos \left (10 d x +10 c \right )}{5632}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {13 \cos \left (6 d x +6 c \right )}{11}+\frac {5 \sin \left (5 d x +5 c \right )}{22}-\frac {5 \sin \left (9 d x +9 c \right )}{22}-\frac {6 \cos \left (8 d x +8 c \right )}{11}-\frac {\sin \left (11 d x +11 c \right )}{22}-\frac {8 \cos \left (4 d x +4 c \right )}{11}+\frac {7 \sin \left (d x +c \right )}{11}+\sin \left (3 d x +3 c \right )+\frac {14 \cos \left (2 d x +2 c \right )}{11}-\frac {\cos \left (10 d x +10 c \right )}{11}-\frac {7 \sin \left (7 d x +7 c \right )}{22}+\frac {14}{11}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5713 \cos \left (2 d x +2 c \right )}{5280}+\frac {20353}{21120}-\frac {11201 \cos \left (4 d x +4 c \right )}{5280}+\frac {54013 \sin \left (3 d x +3 c \right )}{84480}+\frac {44101 \sin \left (d x +c \right )}{84480}-\frac {221 \cos \left (8 d x +8 c \right )}{640}+\frac {31 \sin \left (9 d x +9 c \right )}{11264}+\frac {1497 \sin \left (7 d x +7 c \right )}{56320}-\frac {2 \cos \left (10 d x +10 c \right )}{55}+\frac {131 \sin \left (11 d x +11 c \right )}{56320}+\frac {40627 \sin \left (5 d x +5 c \right )}{168960}-\frac {2213 \cos \left (6 d x +6 c \right )}{1760}\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (70+\sin \left (9 d x +9 c \right )+7 \sin \left (7 d x +7 c \right )+20 \sin \left (5 d x +5 c \right )+28 \sin \left (3 d x +3 c \right )+14 \sin \left (d x +c \right )+2 \cos \left (8 d x +8 c \right )+16 \cos \left (6 d x +6 c \right )+56 \cos \left (4 d x +4 c \right )+112 \cos \left (2 d x +2 c \right )\right )}\) | \(644\) |
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Time = 0.32 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.19 \[ \int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {6930 \, \cos \left (d x + c\right )^{10} - 1560 \, \cos \left (d x + c\right )^{8} - 2454 \, \cos \left (d x + c\right )^{6} - 884 \, \cos \left (d x + c\right )^{4} - 464 \, \cos \left (d x + c\right )^{2} + 15360 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} + {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 11145 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} + {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4215 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} + {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (375 \, \cos \left (d x + c\right )^{8} - 765 \, \cos \left (d x + c\right )^{6} - 178 \, \cos \left (d x + c\right )^{4} - 56 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) - 288}{2560 \, {\left (a d \cos \left (d x + c\right )^{10} - a d \cos \left (d x + c\right )^{8} + {\left (a d \cos \left (d x + c\right )^{10} - a d \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.92 \[ \int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (3465 \, \sin \left (d x + c\right )^{10} - 375 \, \sin \left (d x + c\right )^{9} - 16545 \, \sin \left (d x + c\right )^{8} + 735 \, \sin \left (d x + c\right )^{7} + 30303 \, \sin \left (d x + c\right )^{6} + 223 \, \sin \left (d x + c\right )^{5} - 25847 \, \sin \left (d x + c\right )^{4} - 1207 \, \sin \left (d x + c\right )^{3} + 9408 \, \sin \left (d x + c\right )^{2} + 640 \, \sin \left (d x + c\right ) - 640\right )}}{a \sin \left (d x + c\right )^{11} + a \sin \left (d x + c\right )^{10} - 4 \, a \sin \left (d x + c\right )^{9} - 4 \, a \sin \left (d x + c\right )^{8} + 6 \, a \sin \left (d x + c\right )^{7} + 6 \, a \sin \left (d x + c\right )^{6} - 4 \, a \sin \left (d x + c\right )^{5} - 4 \, a \sin \left (d x + c\right )^{4} + a \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2}} - \frac {11145 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {4215 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {15360 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{2560 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.72 \[ \int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {44580 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {16860 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {61440 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {5120 \, {\left (18 \, \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )}}{a \sin \left (d x + c\right )^{2}} - \frac {5 \, {\left (7025 \, \sin \left (d x + c\right )^{4} - 29724 \, \sin \left (d x + c\right )^{3} + 47346 \, \sin \left (d x + c\right )^{2} - 33684 \, \sin \left (d x + c\right ) + 9045\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {101791 \, \sin \left (d x + c\right )^{5} + 534555 \, \sin \left (d x + c\right )^{4} + 1126810 \, \sin \left (d x + c\right )^{3} + 1192850 \, \sin \left (d x + c\right )^{2} + 634975 \, \sin \left (d x + c\right ) + 136235}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \]
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Time = 9.53 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.94 \[ \int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {693\,{\sin \left (c+d\,x\right )}^{10}}{256}-\frac {75\,{\sin \left (c+d\,x\right )}^9}{256}-\frac {3309\,{\sin \left (c+d\,x\right )}^8}{256}+\frac {147\,{\sin \left (c+d\,x\right )}^7}{256}+\frac {30303\,{\sin \left (c+d\,x\right )}^6}{1280}+\frac {223\,{\sin \left (c+d\,x\right )}^5}{1280}-\frac {25847\,{\sin \left (c+d\,x\right )}^4}{1280}-\frac {1207\,{\sin \left (c+d\,x\right )}^3}{1280}+\frac {147\,{\sin \left (c+d\,x\right )}^2}{20}+\frac {\sin \left (c+d\,x\right )}{2}-\frac {1}{2}}{d\,\left (a\,{\sin \left (c+d\,x\right )}^{11}+a\,{\sin \left (c+d\,x\right )}^{10}-4\,a\,{\sin \left (c+d\,x\right )}^9-4\,a\,{\sin \left (c+d\,x\right )}^8+6\,a\,{\sin \left (c+d\,x\right )}^7+6\,a\,{\sin \left (c+d\,x\right )}^6-4\,a\,{\sin \left (c+d\,x\right )}^5-4\,a\,{\sin \left (c+d\,x\right )}^4+a\,{\sin \left (c+d\,x\right )}^3+a\,{\sin \left (c+d\,x\right )}^2\right )}-\frac {2229\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{512\,a\,d}-\frac {843\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{512\,a\,d}+\frac {6\,\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d} \]
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