Integrand size = 29, antiderivative size = 162 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {\sec ^3(c+d x)}{a^3 d}-\frac {3 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {7 \tan (c+d x)}{a^3 d}+\frac {5 \tan ^3(c+d x)}{a^3 d}+\frac {13 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d} \]
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Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2954, 2952, 3852, 2702, 308, 213, 2700, 276, 2686, 30} \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}+\frac {13 \tan ^5(c+d x)}{5 a^3 d}+\frac {5 \tan ^3(c+d x)}{a^3 d}+\frac {7 \tan (c+d x)}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}-\frac {3 \sec ^5(c+d x)}{5 a^3 d}-\frac {\sec ^3(c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d} \]
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Rule 30
Rule 213
Rule 276
Rule 308
Rule 2686
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))^3 \, dx}{a^6} \\ & = \frac {\int \left (3 a^3 \sec ^8(c+d x)-3 a^3 \csc (c+d x) \sec ^8(c+d x)+a^3 \csc ^2(c+d x) \sec ^8(c+d x)-a^3 \sec ^7(c+d x) \tan (c+d x)\right ) \, dx}{a^6} \\ & = \frac {\int \csc ^2(c+d x) \sec ^8(c+d x) \, dx}{a^3}-\frac {\int \sec ^7(c+d x) \tan (c+d x) \, dx}{a^3}+\frac {3 \int \sec ^8(c+d x) \, dx}{a^3}-\frac {3 \int \csc (c+d x) \sec ^8(c+d x) \, dx}{a^3} \\ & = -\frac {\text {Subst}\left (\int x^6 \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^4}{x^2} \, dx,x,\tan (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \frac {x^8}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{a^3 d} \\ & = -\frac {\sec ^7(c+d x)}{7 a^3 d}+\frac {3 \tan (c+d x)}{a^3 d}+\frac {3 \tan ^3(c+d x)}{a^3 d}+\frac {9 \tan ^5(c+d x)}{5 a^3 d}+\frac {3 \tan ^7(c+d x)}{7 a^3 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^3 d}-\frac {3 \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^3 d} \\ & = -\frac {\cot (c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {\sec ^3(c+d x)}{a^3 d}-\frac {3 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {7 \tan (c+d x)}{a^3 d}+\frac {5 \tan ^3(c+d x)}{a^3 d}+\frac {13 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}-\frac {3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {\sec ^3(c+d x)}{a^3 d}-\frac {3 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {7 \tan (c+d x)}{a^3 d}+\frac {5 \tan ^3(c+d x)}{a^3 d}+\frac {13 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(351\) vs. \(2(162)=324\).
Time = 1.17 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.17 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\csc ^3(c+d x) \left (-966-440 \cos (2 (c+d x))-2640 \cos (3 (c+d x))+846 \cos (4 (c+d x))+176 \cos (5 (c+d x))-1575 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+14 \cos (c+d x) \left (176+105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+1575 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 \cos (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1316 \sin (c+d x)+3520 \sin (2 (c+d x))+2100 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (2 (c+d x))-2100 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (2 (c+d x))-1380 \sin (3 (c+d x))-1056 \sin (4 (c+d x))-630 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))+630 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))+176 \sin (5 (c+d x))\right )}{140 a^3 d \left (\csc ^2\left (\frac {1}{2} (c+d x)\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\sin (c+d x))^3} \]
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Time = 0.95 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {92}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {26}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {31}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {49}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {111}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{2 d \,a^{3}}\) | \(164\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {92}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {26}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {31}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {49}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {111}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{2 d \,a^{3}}\) | \(164\) |
parallelrisch | \(\frac {-210 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1785 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6720 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10010 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4200 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6006 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+9296 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+5079 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1144}{70 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}\) | \(186\) |
risch | \(-\frac {2 \left (-1540 \,{\mathrm e}^{7 i \left (d x +c \right )}+630 i {\mathrm e}^{8 i \left (d x +c \right )}+966 \,{\mathrm e}^{5 i \left (d x +c \right )}-1890 i {\mathrm e}^{6 i \left (d x +c \right )}+1980 \,{\mathrm e}^{3 i \left (d x +c \right )}-951 \,{\mathrm e}^{i \left (d x +c \right )}+2010 i {\mathrm e}^{2 i \left (d x +c \right )}-176 i-574 i {\mathrm e}^{4 i \left (d x +c \right )}+105 \,{\mathrm e}^{9 i \left (d x +c \right )}\right )}{35 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) d \,a^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}\) | \(197\) |
norman | \(\frac {-\frac {143 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {1}{2 a d}-\frac {60 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {51 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {96 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {572 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{35 d a}+\frac {664 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {429 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {5079 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{70 d a}}{a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(239\) |
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Time = 0.27 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.64 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {846 \, \cos \left (d x + c\right )^{4} - 956 \, \cos \left (d x + c\right )^{2} + 105 \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} - {\left (3 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 105 \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} - {\left (3 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (176 \, \cos \left (d x + c\right )^{4} - 477 \, \cos \left (d x + c\right )^{2} + 15\right )} \sin \left (d x + c\right ) + 40}{70 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + 4 \, a^{3} d \cos \left (d x + c\right ) - {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\csc ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (154) = 308\).
Time = 0.22 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.44 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {934 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3854 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6566 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3556 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3710 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {7070 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {4270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {1015 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 35}{\frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}} + \frac {210 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {35 \, \sin \left (d x + c\right )}{a^{3} {\left (\cos \left (d x + c\right ) + 1\right )}}}{70 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {35 \, {\left (12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 17 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} a^{3}} + \frac {3885 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 19880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 57120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 41671 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16632 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2931}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{280 \, d} \]
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Time = 11.88 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.69 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}-\frac {-29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-122\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-202\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-106\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {508\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {938\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+\frac {3854\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}+\frac {934\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}+1}{d\,\left (-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d} \]
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