Integrand size = 21, antiderivative size = 349 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {7}{16} b^2 \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right ) x+\frac {a b \left (40 a^6+1664 a^4 b^2+2789 a^2 b^4+512 b^6\right ) \cos (c+d x)}{20 d}+\frac {b^2 \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d} \]
[Out]
Time = 0.39 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2770, 2832, 2813} \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {a b \left (40 a^6+1664 a^4 b^2+2789 a^2 b^4+512 b^6\right ) \cos (c+d x)}{20 d}+\frac {b^2 \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \sin (c+d x) \cos (c+d x)}{80 d}-\frac {7}{16} b^2 x \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right )+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d} \]
[In]
[Out]
Rule 2770
Rule 2813
Rule 2832
Rubi steps \begin{align*} \text {integral}& = \frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\int (a+b \sin (c+d x))^6 \left (7 b^2+7 a b \sin (c+d x)\right ) \, dx \\ & = \frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac {1}{7} \int (a+b \sin (c+d x))^5 \left (91 a b^2+7 b \left (6 a^2+7 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac {1}{42} \int (a+b \sin (c+d x))^4 \left (7 b^2 \left (108 a^2+35 b^2\right )+7 a b \left (30 a^2+113 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac {1}{210} \int (a+b \sin (c+d x))^3 \left (231 a b^2 \left (20 a^2+19 b^2\right )+7 b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac {1}{840} \int (a+b \sin (c+d x))^2 \left (21 b^2 \left (1000 a^4+1828 a^2 b^2+175 b^4\right )+63 a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac {\int (a+b \sin (c+d x)) \left (63 a b^2 \left (1080 a^4+3076 a^2 b^2+849 b^4\right )+63 b \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \sin (c+d x)\right ) \, dx}{2520} \\ & = -\frac {7}{16} b^2 \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right ) x+\frac {a b \left (40 a^6+1664 a^4 b^2+2789 a^2 b^4+512 b^6\right ) \cos (c+d x)}{20 d}+\frac {b^2 \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d} \\ \end{align*}
Time = 2.71 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.90 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\sec (c+d x) \left (15360 a^7 b+161280 a^5 b^3+201600 a^3 b^5+33600 a b^7-840 b^2 \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right ) (c+d x) \cos (c+d x)+1120 \left (48 a^5 b^3+80 a^3 b^5+15 a b^7\right ) \cos (2 (c+d x))-4480 a^3 b^5 \cos (4 (c+d x))-1344 a b^7 \cos (4 (c+d x))+96 a b^7 \cos (6 (c+d x))+1920 a^8 \sin (c+d x)+53760 a^6 b^2 \sin (c+d x)+151200 a^4 b^4 \sin (c+d x)+67200 a^2 b^6 \sin (c+d x)+2625 b^8 \sin (c+d x)+16800 a^4 b^4 \sin (3 (c+d x))+12600 a^2 b^6 \sin (3 (c+d x))+630 b^8 \sin (3 (c+d x))-840 a^2 b^6 \sin (5 (c+d x))-70 b^8 \sin (5 (c+d x))+5 b^8 \sin (7 (c+d x))\right )}{1920 d} \]
[In]
[Out]
Time = 4.16 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.85
method | result | size |
parallelrisch | \(\frac {\left (161280 a^{5} b^{3}+268800 a^{3} b^{5}+50400 a \,b^{7}\right ) \cos \left (2 d x +2 c \right )+\left (50400 a^{4} b^{4}+37800 a^{2} b^{6}+1890 b^{8}\right ) \sin \left (3 d x +3 c \right )+\left (-13440 a^{3} b^{5}-4032 a \,b^{7}\right ) \cos \left (4 d x +4 c \right )+\left (-2520 a^{2} b^{6}-210 b^{8}\right ) \sin \left (5 d x +5 c \right )+288 \cos \left (6 d x +6 c \right ) a \,b^{7}+15 \sin \left (7 d x +7 c \right ) b^{8}+107520 \left (-\frac {3}{2} a^{6} b d x -\frac {45}{8} a^{4} b^{3} d x -\frac {45}{16} a^{2} b^{5} d x -\frac {15}{128} b^{7} d x +a^{7}+\frac {22}{3} b^{2} a^{5}+8 b^{4} a^{3}+\frac {48}{35} b^{6} a \right ) b \cos \left (d x +c \right )+\left (5760 a^{8}+161280 a^{6} b^{2}+453600 a^{4} b^{4}+201600 a^{2} b^{6}+7875 b^{8}\right ) \sin \left (d x +c \right )+46080 a^{7} b +483840 a^{5} b^{3}+604800 a^{3} b^{5}+100800 a \,b^{7}}{5760 d \cos \left (d x +c \right )}\) | \(298\) |
derivativedivides | \(\frac {a^{8} \tan \left (d x +c \right )+\frac {8 a^{7} b}{\cos \left (d x +c \right )}+28 a^{6} b^{2} \left (\tan \left (d x +c \right )-d x -c \right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )}{d}\) | \(406\) |
default | \(\frac {a^{8} \tan \left (d x +c \right )+\frac {8 a^{7} b}{\cos \left (d x +c \right )}+28 a^{6} b^{2} \left (\tan \left (d x +c \right )-d x -c \right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )}{d}\) | \(406\) |
risch | \(-28 x \,a^{6} b^{2}-105 x \,a^{4} b^{4}-\frac {105 x \,a^{2} b^{6}}{2}-\frac {35 b^{8} x}{16}-\frac {47 i {\mathrm e}^{2 i \left (d x +c \right )} b^{8}}{128 d}-\frac {35 i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} b^{4}}{4 d}-\frac {7 i {\mathrm e}^{2 i \left (d x +c \right )} a^{2} b^{6}}{d}+\frac {28 a^{5} b^{3} {\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {49 a^{3} b^{5} {\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {19 a \,b^{7} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {28 a^{5} b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {49 a^{3} b^{5} {\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {19 a \,b^{7} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {2 i \left (a^{8}+28 a^{6} b^{2}+70 a^{4} b^{4}+28 a^{2} b^{6}+b^{8}-8 i a^{7} b \,{\mathrm e}^{i \left (d x +c \right )}-56 i a^{5} b^{3} {\mathrm e}^{i \left (d x +c \right )}-56 i a^{3} b^{5} {\mathrm e}^{i \left (d x +c \right )}-8 i a \,b^{7} {\mathrm e}^{i \left (d x +c \right )}\right )}{d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}+\frac {35 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} b^{4}}{4 d}+\frac {47 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{8}}{128 d}+\frac {7 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{2} b^{6}}{d}+\frac {\sin \left (6 d x +6 c \right ) b^{8}}{192 d}+\frac {a \,b^{7} \cos \left (5 d x +5 c \right )}{10 d}-\frac {7 \sin \left (4 d x +4 c \right ) a^{2} b^{6}}{8 d}-\frac {5 \sin \left (4 d x +4 c \right ) b^{8}}{64 d}-\frac {14 a^{3} b^{5} \cos \left (3 d x +3 c \right )}{3 d}-\frac {3 a \,b^{7} \cos \left (3 d x +3 c \right )}{2 d}\) | \(499\) |
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.76 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {384 \, a b^{7} \cos \left (d x + c\right )^{6} + 1920 \, a^{7} b + 13440 \, a^{5} b^{3} + 13440 \, a^{3} b^{5} + 1920 \, a b^{7} - 640 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 105 \, {\left (64 \, a^{6} b^{2} + 240 \, a^{4} b^{4} + 120 \, a^{2} b^{6} + 5 \, b^{8}\right )} d x \cos \left (d x + c\right ) + 1920 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (8 \, b^{8} \cos \left (d x + c\right )^{6} + 48 \, a^{8} + 1344 \, a^{6} b^{2} + 3360 \, a^{4} b^{4} + 1344 \, a^{2} b^{6} + 48 \, b^{8} - 2 \, {\left (168 \, a^{2} b^{6} + 19 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (560 \, a^{4} b^{4} + 504 \, a^{2} b^{6} + 29 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )} \]
[In]
[Out]
Timed out. \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.00 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {6720 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{6} b^{2} + 8400 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{4} b^{4} + 4480 \, {\left (\cos \left (d x + c\right )^{3} - \frac {3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a^{3} b^{5} + 840 \, {\left (15 \, d x + 15 \, c - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a^{2} b^{6} - 384 \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} + \frac {5}{\cos \left (d x + c\right )} + 15 \, \cos \left (d x + c\right )\right )} a b^{7} + 5 \, {\left (105 \, d x + 105 \, c - \frac {87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} b^{8} - 13440 \, a^{5} b^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - 240 \, a^{8} \tan \left (d x + c\right ) - \frac {1920 \, a^{7} b}{\cos \left (d x + c\right )}}{240 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (335) = 670\).
Time = 0.41 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.29 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 7.23 (sec) , antiderivative size = 767, normalized size of antiderivative = 2.20 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (240\,a^7\,b+1120\,a^5\,b^3+\frac {1792\,a^3\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (96\,a^7\,b+224\,a^5\,b^3\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^8+56\,a^6\,b^2+210\,a^4\,b^4+105\,a^2\,b^6+\frac {35\,b^8}{8}\right )+\frac {256\,a\,b^7}{5}+16\,a^7\,b+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (96\,a^7\,b+1120\,a^5\,b^3+\frac {4480\,a^3\,b^5}{3}+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (240\,a^7\,b+2240\,a^5\,b^3+2688\,a^3\,b^5+\frac {2304\,a\,b^7}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (320\,a^7\,b+2240\,a^5\,b^3+\frac {6272\,a^3\,b^5}{3}+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (2\,a^8+56\,a^6\,b^2+210\,a^4\,b^4+105\,a^2\,b^6+\frac {35\,b^8}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (12\,a^8+336\,a^6\,b^2+980\,a^4\,b^4+490\,a^2\,b^6+\frac {245\,b^8}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (12\,a^8+336\,a^6\,b^2+980\,a^4\,b^4+490\,a^2\,b^6+\frac {245\,b^8}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (30\,a^8+840\,a^6\,b^2+2030\,a^4\,b^4+791\,a^2\,b^6+\frac {791\,b^8}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (30\,a^8+840\,a^6\,b^2+2030\,a^4\,b^4+791\,a^2\,b^6+\frac {791\,b^8}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (40\,a^8+1120\,a^6\,b^2+2520\,a^4\,b^4+812\,a^2\,b^6+\frac {25\,b^8}{2}\right )+\frac {896\,a^3\,b^5}{3}+224\,a^5\,b^3+16\,a^7\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,b^2\,\mathrm {atan}\left (\frac {7\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (64\,a^6+240\,a^4\,b^2+120\,a^2\,b^4+5\,b^6\right )}{448\,a^6\,b^2+1680\,a^4\,b^4+840\,a^2\,b^6+35\,b^8}\right )\,\left (64\,a^6+240\,a^4\,b^2+120\,a^2\,b^4+5\,b^6\right )}{8\,d} \]
[In]
[Out]