Integrand size = 21, antiderivative size = 404 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=b^8 x+\frac {4 a b \left (24 a^6-88 a^4 b^2+125 a^2 b^4-96 b^6\right ) \cos (c+d x)}{105 d}+\frac {b^2 \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \cos (c+d x) \sin (c+d x)}{105 d}+\frac {2 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}+\frac {2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d} \]
[Out]
Time = 0.54 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2770, 2940, 2832, 2813} \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}+\frac {2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {2 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac {4 a b \left (24 a^6-88 a^4 b^2+125 a^2 b^4-96 b^6\right ) \cos (c+d x)}{105 d}+\frac {b^2 \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \sin (c+d x) \cos (c+d x)}{105 d}+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}+b^8 x \]
[In]
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Rule 2770
Rule 2813
Rule 2832
Rule 2940
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {1}{7} \int \sec ^6(c+d x) (a+b \sin (c+d x))^6 \left (-6 a^2+7 b^2+a b \sin (c+d x)\right ) \, dx \\ & = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}+\frac {1}{35} \int \sec ^4(c+d x) (a+b \sin (c+d x))^5 \left (2 a \left (12 a^2-11 b^2\right )-2 b \left (6 a^2-7 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {1}{105} \int \sec ^2(c+d x) (a+b \sin (c+d x))^4 \left (-2 \left (24 a^4+8 a^2 b^2-35 b^4\right )+6 a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac {1}{105} \int (a+b \sin (c+d x))^3 \left (24 a b^2 \left (12 a^2-11 b^2\right )-8 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac {1}{420} \int (a+b \sin (c+d x))^2 \left (24 b^2 \left (24 a^4-52 a^2 b^2+35 b^4\right )-24 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {2 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}+\frac {2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac {\int (a+b \sin (c+d x)) \left (24 a b^2 \left (24 a^4-76 a^2 b^2+87 b^4\right )-24 b \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \sin (c+d x)\right ) \, dx}{1260} \\ & = b^8 x+\frac {4 a b \left (24 a^6-88 a^4 b^2+125 a^2 b^4-96 b^6\right ) \cos (c+d x)}{105 d}+\frac {b^2 \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \cos (c+d x) \sin (c+d x)}{105 d}+\frac {2 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}+\frac {2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d} \\ \end{align*}
Time = 4.26 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.19 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\sec ^7(c+d x) \left (7680 a^7 b+16128 a^5 b^3+25536 a^3 b^5-5088 a b^7+3675 b^8 (c+d x) \cos (c+d x)-37632 a^5 b^3 \cos (2 (c+d x))-12544 a^3 b^5 \cos (2 (c+d x))-14448 a b^7 \cos (2 (c+d x))+2205 b^8 (c+d x) \cos (3 (c+d x))+15680 a^3 b^5 \cos (4 (c+d x))-3360 a b^7 \cos (4 (c+d x))+735 b^8 (c+d x) \cos (5 (c+d x))-1680 a b^7 \cos (6 (c+d x))+105 b^8 (c+d x) \cos (7 (c+d x))+1680 a^8 \sin (c+d x)+23520 a^6 b^2 \sin (c+d x)+44100 a^4 b^4 \sin (c+d x)+14700 a^2 b^6 \sin (c+d x)+1008 a^8 \sin (3 (c+d x))-4704 a^6 b^2 \sin (3 (c+d x))-20580 a^4 b^4 \sin (3 (c+d x))-8820 a^2 b^6 \sin (3 (c+d x))-1176 b^8 \sin (3 (c+d x))+336 a^8 \sin (5 (c+d x))-1568 a^6 b^2 \sin (5 (c+d x))+2940 a^4 b^4 \sin (5 (c+d x))+2940 a^2 b^6 \sin (5 (c+d x))-392 b^8 \sin (5 (c+d x))+48 a^8 \sin (7 (c+d x))-224 a^6 b^2 \sin (7 (c+d x))+420 a^4 b^4 \sin (7 (c+d x))-420 a^2 b^6 \sin (7 (c+d x))-176 b^8 \sin (7 (c+d x))\right )}{6720 d} \]
[In]
[Out]
Time = 2.33 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.34
method | result | size |
parallelrisch | \(\frac {\left (105 b^{8} d x +1120 a^{5} b^{3}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-210 a^{8}+210 b^{8}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-735 b^{8} d x -1680 a^{7} b -7840 a^{5} b^{3}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (420 a^{8}-7840 a^{6} b^{2}-1540 b^{8}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2205 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{8} d +\left (-1806 a^{8}-6272 a^{6} b^{2}-47040 a^{4} b^{4}+4942 b^{8}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3675 b^{8} d x -8400 a^{7} b -62720 a^{5} b^{3}-62720 a^{3} b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1272 a^{8}-25536 a^{6} b^{2}-40320 a^{4} b^{4}-53760 a^{2} b^{6}-9144 b^{8}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7840 \left (-\frac {15}{32} b^{5} d x +a^{5}+4 a^{3} b^{2}+\frac {24}{7} a \,b^{4}\right ) b^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1806 a^{8}-6272 a^{6} b^{2}-47040 a^{4} b^{4}+4942 b^{8}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2205 b^{8} d x -5040 a^{7} b -32928 a^{5} b^{3}-18816 a^{3} b^{5}+16128 a \,b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (420 a^{8}-7840 a^{6} b^{2}-1540 b^{8}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (735 b^{8} d x +3136 a^{5} b^{3}+6272 a^{3} b^{5}-5376 a \,b^{7}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-210 a^{8}+210 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-105 b^{8} d x -240 a^{7} b -448 a^{5} b^{3}-896 a^{3} b^{5}+768 a \,b^{7}}{105 d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{7}}\) | \(541\) |
derivativedivides | \(\frac {-a^{8} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {8 a^{7} b}{7 \cos \left (d x +c \right )^{7}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )}+\frac {\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 )}{35}\right )+\frac {4 a^{2} b^{6} \left (\sin ^{7}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{7}}+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}-\frac {\sin ^{8}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{8}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{8}\left (d x +c \right )}{7 \cos \left (d x +c \right )}-\frac {\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 )}{7}\right )+b^{8} \left (\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(567\) |
default | \(\frac {-a^{8} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {8 a^{7} b}{7 \cos \left (d x +c \right )^{7}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )}+\frac {\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 )}{35}\right )+\frac {4 a^{2} b^{6} \left (\sin ^{7}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{7}}+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}-\frac {\sin ^{8}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{8}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{8}\left (d x +c \right )}{7 \cos \left (d x +c \right )}-\frac {\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 )}{7}\right )+b^{8} \left (\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(567\) |
risch | \(b^{8} x -\frac {8 \left (-4032 a^{5} b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+770 i b^{8} {\mathrm e}^{8 i \left (d x +c \right )}+770 i b^{8} {\mathrm e}^{6 i \left (d x +c \right )}-252 i a^{8} {\mathrm e}^{4 i \left (d x +c \right )}+1568 a^{3} b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+1806 a \,b^{7} {\mathrm e}^{5 i \left (d x +c \right )}+210 a \,b^{7} {\mathrm e}^{i \left (d x +c \right )}-6384 a^{3} b^{5} {\mathrm e}^{7 i \left (d x +c \right )}+1272 a \,b^{7} {\mathrm e}^{7 i \left (d x +c \right )}+4704 a^{5} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-105 i a^{4} b^{4}-735 i a^{4} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+735 i a^{2} b^{6} {\mathrm e}^{12 i \left (d x +c \right )}-3675 i a^{4} b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+3920 i a^{6} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-7350 i a^{4} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+1470 i a^{4} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+2205 i a^{2} b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+392 i a^{6} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 i a^{8}+44 i b^{8}-1960 a^{3} b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+420 a \,b^{7} {\mathrm e}^{3 i \left (d x +c \right )}+609 i b^{8} {\mathrm e}^{4 i \left (d x +c \right )}+1568 a^{3} b^{5} {\mathrm e}^{9 i \left (d x +c \right )}+56 i a^{6} b^{2}+105 i a^{2} b^{6}-84 i a^{8} {\mathrm e}^{2 i \left (d x +c \right )}+203 i b^{8} {\mathrm e}^{2 i \left (d x +c \right )}+1806 a \,b^{7} {\mathrm e}^{9 i \left (d x +c \right )}-1960 a^{3} b^{5} {\mathrm e}^{11 i \left (d x +c \right )}+315 i b^{8} {\mathrm e}^{10 i \left (d x +c \right )}-420 i a^{8} {\mathrm e}^{6 i \left (d x +c \right )}+4704 a^{5} b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-1920 a^{7} b \,{\mathrm e}^{7 i \left (d x +c \right )}+210 a \,b^{7} {\mathrm e}^{13 i \left (d x +c \right )}+420 a \,b^{7} {\mathrm e}^{11 i \left (d x +c \right )}+105 i b^{8} {\mathrm e}^{12 i \left (d x +c \right )}-1960 i a^{6} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+1176 i a^{6} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3675 i a^{4} b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+3675 i a^{2} b^{6} {\mathrm e}^{8 i \left (d x +c \right )}\right )}{105 d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )^{7}}\) | \(672\) |
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Time = 0.29 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.76 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {105 \, b^{8} d x \cos \left (d x + c\right )^{7} - 840 \, a b^{7} \cos \left (d x + c\right )^{6} + 120 \, a^{7} b + 840 \, a^{5} b^{3} + 840 \, a^{3} b^{5} + 120 \, a b^{7} + 280 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 168 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (15 \, a^{8} + 420 \, a^{6} b^{2} + 1050 \, a^{4} b^{4} + 420 \, a^{2} b^{6} + 15 \, b^{8} + 4 \, {\left (12 \, a^{8} - 56 \, a^{6} b^{2} + 105 \, a^{4} b^{4} - 105 \, a^{2} b^{6} - 44 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (12 \, a^{8} - 56 \, a^{6} b^{2} + 105 \, a^{4} b^{4} + 630 \, a^{2} b^{6} + 61 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (3 \, a^{8} - 14 \, a^{6} b^{2} - 280 \, a^{4} b^{4} - 210 \, a^{2} b^{6} - 11 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{7}} \]
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Timed out. \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.77 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {420 \, a^{2} b^{6} \tan \left (d x + c\right )^{7} + 3 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{8} + 28 \, {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a^{6} b^{2} + 210 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 7 \, \tan \left (d x + c\right )^{5}\right )} a^{4} b^{4} + {\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} b^{8} - \frac {168 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{7}} + \frac {56 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 42 \, \cos \left (d x + c\right )^{2} + 15\right )} a^{3} b^{5}}{\cos \left (d x + c\right )^{7}} - \frac {24 \, {\left (35 \, \cos \left (d x + c\right )^{6} - 35 \, \cos \left (d x + c\right )^{4} + 21 \, \cos \left (d x + c\right )^{2} - 5\right )} a b^{7}}{\cos \left (d x + c\right )^{7}} + \frac {120 \, a^{7} b}{\cos \left (d x + c\right )^{7}}}{105 \, d} \]
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Time = 0.45 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.80 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Too large to display} \]
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Time = 8.66 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.35 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=b^8\,x-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-4\,a^8+\frac {224\,a^6\,b^2}{3}+\frac {44\,b^8}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-4\,a^8+\frac {224\,a^6\,b^2}{3}+\frac {44\,b^8}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (2\,a^8-2\,b^8\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {224\,a^5\,b^3}{5}-\frac {896\,a^3\,b^5}{15}+\frac {256\,a\,b^7}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (448\,a^5\,b^3+\frac {896\,a^3\,b^5}{3}+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (80\,a^7\,b+224\,a^5\,b^3+\frac {1792\,a^3\,b^5}{3}\right )-\frac {256\,a\,b^7}{35}+\frac {16\,a^7\,b}{7}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (48\,a^7\,b+\frac {448\,a^5\,b^3}{5}+\frac {896\,a^3\,b^5}{5}-\frac {768\,a\,b^7}{5}\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^8-2\,b^8\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (-\frac {424\,a^8}{35}+\frac {1216\,a^6\,b^2}{5}+384\,a^4\,b^4+512\,a^2\,b^6+\frac {3048\,b^8}{35}\right )+\frac {128\,a^3\,b^5}{15}-\frac {32\,a^5\,b^3}{5}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {86\,a^8}{5}+\frac {896\,a^6\,b^2}{15}+448\,a^4\,b^4-\frac {706\,b^8}{15}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {86\,a^8}{5}+\frac {896\,a^6\,b^2}{15}+448\,a^4\,b^4-\frac {706\,b^8}{15}\right )+224\,a^5\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+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}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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