Integrand size = 29, antiderivative size = 164 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}+\frac {5 \tan (c+d x)}{a^2 d}+\frac {3 \tan ^3(c+d x)}{a^2 d}+\frac {7 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^7(c+d x)}{7 a^2 d} \]
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Time = 0.24 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2954, 2952, 3852, 2702, 308, 213, 2700, 276} \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {7 \tan ^5(c+d x)}{5 a^2 d}+\frac {3 \tan ^3(c+d x)}{a^2 d}+\frac {5 \tan (c+d x)}{a^2 d}-\frac {\cot (c+d x)}{a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d} \]
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Rule 213
Rule 276
Rule 308
Rule 2700
Rule 2702
Rule 2952
Rule 2954
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc ^2(c+d x) \sec ^8(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \sec ^8(c+d x)-2 a^2 \csc (c+d x) \sec ^8(c+d x)+a^2 \csc ^2(c+d x) \sec ^8(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \sec ^8(c+d x) \, dx}{a^2}+\frac {\int \csc ^2(c+d x) \sec ^8(c+d x) \, dx}{a^2}-\frac {2 \int \csc (c+d x) \sec ^8(c+d x) \, dx}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^4}{x^2} \, dx,x,\tan (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{a^2 d}-\frac {2 \text {Subst}\left (\int \frac {x^8}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{a^2 d}+\frac {3 \tan ^5(c+d x)}{5 a^2 d}+\frac {\tan ^7(c+d x)}{7 a^2 d}+\frac {\text {Subst}\left (\int \left (4+\frac {1}{x^2}+6 x^2+4 x^4+x^6\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}-\frac {2 \text {Subst}\left (\int \left (1+x^2+x^4+x^6+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = -\frac {\cot (c+d x)}{a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}+\frac {5 \tan (c+d x)}{a^2 d}+\frac {3 \tan ^3(c+d x)}{a^2 d}+\frac {7 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {2 \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}+\frac {5 \tan (c+d x)}{a^2 d}+\frac {3 \tan ^3(c+d x)}{a^2 d}+\frac {7 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^7(c+d x)}{7 a^2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(442\) vs. \(2(164)=328\).
Time = 6.26 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.70 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {16 \left (-\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {1}{768 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{384 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {13 \sin \left (\frac {1}{2} (c+d x)\right )}{384 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{224 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^7}-\frac {1}{448 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {3 \sin \left (\frac {1}{2} (c+d x)\right )}{140 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}-\frac {3}{280 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {997 \sin \left (\frac {1}{2} (c+d x)\right )}{13440 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {997}{26880 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4777 \sin \left (\frac {1}{2} (c+d x)\right )}{13440 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{32 d}\right )}{a^2} \]
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Time = 1.15 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {5}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {48}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {107}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {59}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {75}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{2 d \,a^{2}}\) | \(194\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {5}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {48}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {107}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {59}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {75}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{2 d \,a^{2}}\) | \(194\) |
parallelrisch | \(\frac {-420 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3570 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7560 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1575 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18480 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11172 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12992 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16409 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-816 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+5422 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2248}{210 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}\) | \(214\) |
risch | \(-\frac {4 \left (-385 \,{\mathrm e}^{9 i \left (d x +c \right )}+420 i {\mathrm e}^{10 i \left (d x +c \right )}-1274 \,{\mathrm e}^{7 i \left (d x +c \right )}+560 i {\mathrm e}^{8 i \left (d x +c \right )}-616 i {\mathrm e}^{6 i \left (d x +c \right )}+1249 \,{\mathrm e}^{3 i \left (d x +c \right )}-1816 i {\mathrm e}^{4 i \left (d x +c \right )}+759 \,{\mathrm e}^{i \left (d x +c \right )}-444 i {\mathrm e}^{2 i \left (d x +c \right )}+216 i-454 \,{\mathrm e}^{5 i \left (d x +c \right )}+105 \,{\mathrm e}^{11 i \left (d x +c \right )}\right )}{105 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d \,a^{2}}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{2}}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{2}}\) | \(220\) |
norman | \(\frac {\frac {1}{2 a d}+\frac {266 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {93 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {17 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {53 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {57 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {2711 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{420 d a}+\frac {926 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d a}-\frac {71 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d a}-\frac {2329 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140 d a}-\frac {9269 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{210 d a}}{a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\) | \(277\) |
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Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.52 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {432 \, \cos \left (d x + c\right )^{6} - 660 \, \cos \left (d x + c\right )^{4} + 98 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (2 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} + {\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left (2 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} + {\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (327 \, \cos \left (d x + c\right )^{4} - 41 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) + 25}{105 \, {\left (2 \, a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} + {\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (154) = 308\).
Time = 0.23 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.93 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {1828 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3847 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1656 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {12734 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {7952 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {9702 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {12600 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {315 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {5460 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {2205 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 105}{\frac {a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {14 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {a^{2} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}} + \frac {420 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {105 \, \sin \left (d x + c\right )}{a^{2} {\left (\cos \left (d x + c\right ) + 1\right )}}}{210 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.24 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {1680 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {420 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}} - \frac {420 \, {\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {35 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 14\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {7875 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 41055 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 94640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 119630 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 87507 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 34979 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6122}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \]
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Time = 12.03 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.02 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-120\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {462\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {1136\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}+\frac {12734\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{105}+\frac {552\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{35}-\frac {3847\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}-\frac {1828\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}-1}{d\,\left (-2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+16\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+28\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-28\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-16\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2\,d} \]
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