Integrand size = 21, antiderivative size = 369 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {35}{8} b^4 \left (16 a^4+16 a^2 b^2+b^4\right ) x+\frac {a b \left (8 a^6-104 a^4 b^2-803 a^2 b^4-256 b^6\right ) \cos (c+d x)}{6 d}+\frac {b^2 \left (16 a^6-200 a^4 b^2-866 a^2 b^4-105 b^6\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d} \]
[Out]
Time = 0.44 (sec) , antiderivative size = 369, 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} \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}-\frac {\sec (c+d x) \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^6}{3 d}+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac {35}{8} b^4 x \left (16 a^4+16 a^2 b^2+b^4\right )+\frac {a b \left (8 a^6-104 a^4 b^2-803 a^2 b^4-256 b^6\right ) \cos (c+d x)}{6 d}+\frac {b^2 \left (16 a^6-200 a^4 b^2-866 a^2 b^4-105 b^6\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d} \]
[In]
[Out]
Rule 2770
Rule 2813
Rule 2832
Rule 2940
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {1}{3} \int \sec ^2(c+d x) (a+b \sin (c+d x))^6 \left (-2 a^2+7 b^2+5 a b \sin (c+d x)\right ) \, dx \\ & = \frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}+\frac {1}{3} \int (a+b \sin (c+d x))^5 \left (30 a b^2-6 b \left (2 a^2-7 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}+\frac {1}{18} \int (a+b \sin (c+d x))^4 \left (30 b^2 \left (4 a^2+7 b^2\right )-30 a b \left (2 a^2-13 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}+\frac {1}{90} \int (a+b \sin (c+d x))^3 \left (90 a b^2 \left (4 a^2+29 b^2\right )-30 b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}+\frac {1}{360} \int (a+b \sin (c+d x))^2 \left (90 b^2 \left (8 a^4+188 a^2 b^2+35 b^4\right )-90 a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}+\frac {\int (a+b \sin (c+d x)) \left (90 a b^2 \left (8 a^4+740 a^2 b^2+407 b^4\right )-90 b \left (16 a^6-200 a^4 b^2-866 a^2 b^4-105 b^6\right ) \sin (c+d x)\right ) \, dx}{1080} \\ & = \frac {35}{8} b^4 \left (16 a^4+16 a^2 b^2+b^4\right ) x+\frac {a b \left (8 a^6-104 a^4 b^2-803 a^2 b^4-256 b^6\right ) \cos (c+d x)}{6 d}+\frac {b^2 \left (16 a^6-200 a^4 b^2-866 a^2 b^4-105 b^6\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d} \\ \end{align*}
Time = 2.82 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.12 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\sec ^3(c+d x) \left (2048 a^7 b-7168 a^5 b^3-44800 a^3 b^5-13440 a b^7+40320 a^4 b^4 (c+d x) \cos (c+d x)+40320 a^2 b^6 (c+d x) \cos (c+d x)+2520 b^8 (c+d x) \cos (c+d x)-21504 a^5 b^3 \cos (2 (c+d x))-64512 a^3 b^5 \cos (2 (c+d x))-17472 a b^7 \cos (2 (c+d x))+13440 a^4 b^4 (c+d x) \cos (3 (c+d x))+13440 a^2 b^6 (c+d x) \cos (3 (c+d x))+840 b^8 (c+d x) \cos (3 (c+d x))-5376 a^3 b^5 \cos (4 (c+d x))-1920 a b^7 \cos (4 (c+d x))+64 a b^7 \cos (6 (c+d x))+384 a^8 \sin (c+d x)+5376 a^6 b^2 \sin (c+d x)-6720 a^2 b^6 \sin (c+d x)-525 b^8 \sin (c+d x)+128 a^8 \sin (3 (c+d x))-1792 a^6 b^2 \sin (3 (c+d x))-17920 a^4 b^4 \sin (3 (c+d x))-14560 a^2 b^6 \sin (3 (c+d x))-847 b^8 \sin (3 (c+d x))-672 a^2 b^6 \sin (5 (c+d x))-63 b^8 \sin (5 (c+d x))+3 b^8 \sin (7 (c+d x))\right )}{768 d} \]
[In]
[Out]
Time = 4.27 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.01
method | result | size |
parallelrisch | \(\frac {-3584 b \left (-\frac {15}{4} a^{4} b^{3} d x -\frac {15}{4} a^{2} b^{5} d x -\frac {15}{64} b^{7} d x +a^{7}+6 b^{2} a^{5}+8 b^{4} a^{3}+\frac {16}{7} b^{6} a \right ) \cos \left (3 d x +3 c \right )+\left (128 a^{8}-1792 a^{6} b^{2}-17920 a^{4} b^{4}-14560 a^{2} b^{6}-847 b^{8}\right ) \sin \left (3 d x +3 c \right )-21504 \left (a^{4}+3 a^{2} b^{2}+\frac {13}{16} b^{4}\right ) a \,b^{3} \cos \left (2 d x +2 c \right )+\left (-5376 a^{3} b^{5}-1920 a \,b^{7}\right ) \cos \left (4 d x +4 c \right )+\left (-672 a^{2} b^{6}-63 b^{8}\right ) \sin \left (5 d x +5 c \right )+64 \cos \left (6 d x +6 c \right ) a \,b^{7}+3 \sin \left (7 d x +7 c \right ) b^{8}-10752 b \left (-\frac {15}{4} a^{4} b^{3} d x -\frac {15}{4} a^{2} b^{5} d x -\frac {15}{64} b^{7} d x +a^{7}+6 b^{2} a^{5}+8 b^{4} a^{3}+\frac {16}{7} b^{6} a \right ) \cos \left (d x +c \right )+\left (384 a^{8}+5376 a^{6} b^{2}-6720 a^{2} b^{6}-525 b^{8}\right ) \sin \left (d x +c \right )+2048 a^{7} b -7168 a^{5} b^{3}-44800 a^{3} b^{5}-13440 a \,b^{7}}{192 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(371\) |
derivativedivides | \(\frac {-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+\frac {8 a^{7} b}{3 \cos \left (d x +c \right )^{3}}+\frac {28 a^{6} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+70 a^{4} b^{4} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\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 )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \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 )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \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 )}{3}\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{\cos \left (d x +c \right )}-2 \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}{8}+\frac {35 c}{8}\right )}{d}\) | \(495\) |
default | \(\frac {-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+\frac {8 a^{7} b}{3 \cos \left (d x +c \right )^{3}}+\frac {28 a^{6} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+70 a^{4} b^{4} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\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 )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \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 )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \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 )}{3}\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{\cos \left (d x +c \right )}-2 \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}{8}+\frac {35 c}{8}\right )}{d}\) | \(495\) |
risch | \(70 x \,a^{4} b^{4}+70 x \,a^{2} b^{6}+\frac {35 b^{8} x}{8}+\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} b^{8}}{8 d}+\frac {a \,b^{7} {\mathrm e}^{3 i \left (d x +c \right )}}{3 d}-\frac {7 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{2} b^{6}}{2 d}+\frac {7 i {\mathrm e}^{2 i \left (d x +c \right )} a^{2} b^{6}}{2 d}-\frac {28 a^{3} b^{5} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {11 a \,b^{7} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {28 a^{3} b^{5} {\mathrm e}^{-i \left (d x +c \right )}}{d}-\frac {11 a \,b^{7} {\mathrm e}^{-i \left (d x +c \right )}}{d}-\frac {i b^{8} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}-\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{8}}{8 d}+\frac {a \,b^{7} {\mathrm e}^{-3 i \left (d x +c \right )}}{3 d}+\frac {i b^{8} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}+\frac {4 i \left (168 i a^{3} b^{5} {\mathrm e}^{i \left (d x +c \right )}+84 i a^{5} b^{3} {\mathrm e}^{i \left (d x +c \right )}-16 i a^{7} b \,{\mathrm e}^{3 i \left (d x +c \right )}+224 i a^{3} b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+36 i a \,b^{7} {\mathrm e}^{5 i \left (d x +c \right )}+56 i a \,b^{7} {\mathrm e}^{3 i \left (d x +c \right )}+56 i a^{5} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-42 a^{6} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-210 a^{4} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-126 a^{2} b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-6 b^{8} {\mathrm e}^{4 i \left (d x +c \right )}+36 i a \,b^{7} {\mathrm e}^{i \left (d x +c \right )}+168 i a^{3} b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+84 i a^{5} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+3 a^{8} {\mathrm e}^{2 i \left (d x +c \right )}-210 a^{4} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-168 a^{2} b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-9 b^{8} {\mathrm e}^{2 i \left (d x +c \right )}+a^{8}-14 a^{6} b^{2}-140 a^{4} b^{4}-98 a^{2} b^{6}-5 b^{8}\right )}{3 d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )^{3}}\) | \(603\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.73 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {64 \, a b^{7} \cos \left (d x + c\right )^{6} + 64 \, a^{7} b + 448 \, a^{5} b^{3} + 448 \, a^{3} b^{5} + 64 \, a b^{7} + 105 \, {\left (16 \, a^{4} b^{4} + 16 \, a^{2} b^{6} + b^{8}\right )} d x \cos \left (d x + c\right )^{3} - 192 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 192 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, b^{8} \cos \left (d x + c\right )^{6} + 8 \, a^{8} + 224 \, a^{6} b^{2} + 560 \, a^{4} b^{4} + 224 \, a^{2} b^{6} + 8 \, b^{8} - 3 \, {\left (112 \, a^{2} b^{6} + 13 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (a^{8} - 14 \, a^{6} b^{2} - 140 \, a^{4} b^{4} - 98 \, a^{2} b^{6} - 5 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )^{3}} \]
[In]
[Out]
Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.89 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {224 \, a^{6} b^{2} \tan \left (d x + c\right )^{3} + 8 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{8} + 560 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4} b^{4} + 112 \, {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{2} b^{6} + 64 \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a b^{7} + {\left (8 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - \frac {3 \, {\left (13 \, \tan \left (d x + c\right )^{3} + 11 \, \tan \left (d x + c\right )\right )}}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 72 \, \tan \left (d x + c\right )\right )} b^{8} - 448 \, a^{3} b^{5} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {448 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{3}} + \frac {64 \, a^{7} b}{\cos \left (d x + c\right )^{3}}}{24 \, d} \]
[In]
[Out]
none
Time = 0.44 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.85 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {105 \, {\left (16 \, a^{4} b^{4} + 16 \, a^{2} b^{6} + b^{8}\right )} {\left (d x + c\right )} - \frac {16 \, {\left (3 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 210 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 168 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 48 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 700 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 448 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 22 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 336 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 672 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 210 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 168 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a^{7} b - 112 \, a^{5} b^{3} - 280 \, a^{3} b^{5} - 64 \, a b^{7}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} + \frac {2 \, {\left (336 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1344 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 384 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 336 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 57 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4032 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1536 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 57 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4032 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1664 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 336 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 33 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1344 \, a^{3} b^{5} - 512 \, a b^{7}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
[In]
[Out]
Time = 6.97 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.97 \[ \int \sec ^4(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 (\frac {304\,a^7\,b}{3}+\frac {2464\,a^5\,b^3}{3}+\frac {1792\,a^3\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (64\,a^7\,b+224\,a^5\,b^3\right )-\frac {256\,a\,b^7}{3}+\frac {16\,a^7\,b}{3}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-2\,a^8+140\,a^4\,b^4+140\,a^2\,b^6+\frac {35\,b^8}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-\frac {64\,a^7\,b}{3}+\frac {224\,a^5\,b^3}{3}+\frac {896\,a^3\,b^5}{3}+\frac {256\,a\,b^7}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (48\,a^7\,b+448\,a^5\,b^3+896\,a^3\,b^5+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {256\,a^7\,b}{3}+\frac {3136\,a^5\,b^3}{3}+\frac {4480\,a^3\,b^5}{3}+256\,a\,b^7\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-\frac {20\,a^8}{3}-\frac {224\,a^6\,b^2}{3}+\frac {280\,a^4\,b^4}{3}+\frac {280\,a^2\,b^6}{3}+\frac {35\,b^8}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-\frac {20\,a^8}{3}-\frac {224\,a^6\,b^2}{3}+\frac {280\,a^4\,b^4}{3}+\frac {280\,a^2\,b^6}{3}+\frac {35\,b^8}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (8\,a^8+448\,a^6\,b^2+1680\,a^4\,b^4+784\,a^2\,b^6+17\,b^8\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {26\,a^8}{3}+\frac {896\,a^6\,b^2}{3}+\frac {2660\,a^4\,b^4}{3}+\frac {1316\,a^2\,b^6}{3}+\frac {329\,b^8}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {26\,a^8}{3}+\frac {896\,a^6\,b^2}{3}+\frac {2660\,a^4\,b^4}{3}+\frac {1316\,a^2\,b^6}{3}+\frac {329\,b^8}{12}\right )-\frac {896\,a^3\,b^5}{3}-\frac {224\,a^5\,b^3}{3}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (-2\,a^8+140\,a^4\,b^4+140\,a^2\,b^6+\frac {35\,b^8}{4}\right )+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}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {35\,b^4\,\mathrm {atan}\left (\frac {35\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (16\,a^4+16\,a^2\,b^2+b^4\right )}{560\,a^4\,b^4+560\,a^2\,b^6+35\,b^8}\right )\,\left (16\,a^4+16\,a^2\,b^2+b^4\right )}{4\,d} \]
[In]
[Out]