Integrand size = 21, antiderivative size = 320 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 d}+\frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}-\frac {a \left (13-\frac {3 a^2}{b^2}\right ) b^7 \sin ^4(c+d x)}{8 d}+\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d} \]
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Time = 0.21 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2747, 753, 833, 815, 647, 31} \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}-\frac {a b^7 \left (13-\frac {3 a^2}{b^2}\right ) \sin ^4(c+d x)}{8 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{4 d} \]
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Rule 31
Rule 647
Rule 753
Rule 815
Rule 833
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {(a+x)^8}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {b^3 \text {Subst}\left (\int \frac {(a+x)^6 \left (-3 a^2+7 b^2+4 a x\right )}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d} \\ & = \frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {b \text {Subst}\left (\int \frac {(a+x)^4 \left (3 a^4+2 a^2 b^2+35 b^4-4 a \left (3 a^2-13 b^2\right ) x\right )}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d} \\ & = \frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {b \text {Subst}\left (\int \left (5 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right )+4 a \left (15 a^4-77 a^2 b^2-48 b^4\right ) x+5 \left (9 a^4-42 a^2 b^2-7 b^4\right ) x^2+4 a \left (3 a^2-13 b^2\right ) x^3+\frac {3 a^8-28 a^6 b^2+210 a^4 b^4+420 a^2 b^6+35 b^8+64 a b^4 \left (7 a^2+3 b^2\right ) x}{b^2-x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d} \\ & = \frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac {a b^5 \left (3 a^2-13 b^2\right ) \sin ^4(c+d x)}{8 d}+\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {b \text {Subst}\left (\int \frac {3 a^8-28 a^6 b^2+210 a^4 b^4+420 a^2 b^6+35 b^8+64 a b^4 \left (7 a^2+3 b^2\right ) x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d} \\ & = \frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac {a b^5 \left (3 a^2-13 b^2\right ) \sin ^4(c+d x)}{8 d}+\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {\left ((a+b)^6 \left (3 a^2-18 a b+35 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}-\frac {\left ((a-b)^6 \left (3 a^2+18 a b+35 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{16 d} \\ & = -\frac {(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 d}+\frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac {a b^5 \left (3 a^2-13 b^2\right ) \sin ^4(c+d x)}{8 d}+\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d} \\ \end{align*}
Time = 3.31 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.61 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {3 \left (a^2-b^2\right )^2 \left ((a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))-(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (1+\sin (c+d x))\right )+6 b^2 \left (-108 a^{10}+234 a^8 b^2-28 a^6 b^4-595 a^4 b^6+350 a^2 b^8+35 b^{10}\right ) \sin (c+d x)-24 a b^3 \left (63 a^8-21 a^6 b^2+88 a^4 b^4-8 a^2 b^6-24 b^8\right ) \sin ^2(c+d x)+14 b^4 \left (-162 a^8-144 a^6 b^2-85 a^4 b^4+50 a^2 b^6+5 b^8\right ) \sin ^3(c+d x)-12 a b^5 \left (189 a^6+333 a^4 b^2-8 a^2 b^4-24 b^6\right ) \sin ^4(c+d x)+42 b^6 \left (-36 a^6-87 a^4 b^2+10 a^2 b^4+b^6\right ) \sin ^5(c+d x)-24 a b^7 \left (27 a^4+79 a^2 b^2-8 b^4\right ) \sin ^6(c+d x)+6 b^8 \left (-27 a^4-90 a^2 b^2+5 b^4\right ) \sin ^7(c+d x)-6 a b^9 \left (3 a^2+11 b^2\right ) \sin ^8(c+d x)+12 \left (a^2-b^2\right ) \sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^9+6 \sec ^2(c+d x) (a+b \sin (c+d x))^9 \left (9 a^2 b+5 b^3-a \left (3 a^2+11 b^2\right ) \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^2 d} \]
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Time = 2.57 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.36
method | result | size |
parallelrisch | \(\frac {5376 \left (a^{2}+\frac {3 b^{2}}{7}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a \,b^{5} \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36 \left (a^{2}-6 a b +\frac {35}{3} b^{2}\right ) \left (a +b \right )^{6} \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+36 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a -b \right )^{6} \left (a^{2}+6 a b +\frac {35}{3} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-192 \left (a^{6}+7 a^{4} b^{2}+7 a^{2} b^{4}+\frac {9}{8} b^{6}\right ) a b \cos \left (2 d x +2 c \right )+\left (-48 a^{7} b +336 a^{5} b^{3}+1008 a^{3} b^{5}+288 a \,b^{7}\right ) \cos \left (4 d x +4 c \right )+\left (18 a^{8}-168 a^{6} b^{2}-2100 a^{4} b^{4}-2520 a^{2} b^{6}-189 b^{8}\right ) \sin \left (3 d x +3 c \right )+\left (-336 a^{2} b^{6}-35 b^{8}\right ) \sin \left (5 d x +5 c \right )+24 \cos \left (6 d x +6 c \right ) a \,b^{7}+\sin \left (7 d x +7 c \right ) b^{8}+\left (66 a^{8}+1176 a^{6} b^{2}+1260 a^{4} b^{4}-840 a^{2} b^{6}-105 b^{8}\right ) \sin \left (d x +c \right )+240 a^{7} b +1008 a^{5} b^{3}+336 a^{3} b^{5}-96 a \,b^{7}}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(434\) |
derivativedivides | \(\frac {a^{8} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {2 a^{7} b}{\cos \left (d x +c \right )^{4}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {14 a^{5} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{4}}+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{5}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+56 a^{3} b^{5} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}-\frac {35 \sin \left (d x +c \right )}{8}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(544\) |
default | \(\frac {a^{8} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {2 a^{7} b}{\cos \left (d x +c \right )^{4}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {14 a^{5} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{4}}+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{5}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+56 a^{3} b^{5} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}-\frac {35 \sin \left (d x +c \right )}{8}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(544\) |
risch | \(\text {Expression too large to display}\) | \(1022\) |
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Time = 0.34 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.14 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {192 \, a b^{7} \cos \left (d x + c\right )^{6} - 96 \, a b^{7} \cos \left (d x + c\right )^{4} + 96 \, a^{7} b + 672 \, a^{5} b^{3} + 672 \, a^{3} b^{5} + 96 \, a b^{7} + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 192 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, b^{8} \cos \left (d x + c\right )^{6} + 6 \, a^{8} + 168 \, a^{6} b^{2} + 420 \, a^{4} b^{4} + 168 \, a^{2} b^{6} + 6 \, b^{8} - 16 \, {\left (42 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} - 350 \, a^{4} b^{4} - 252 \, a^{2} b^{6} - 13 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.09 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {16 \, b^{8} \sin \left (d x + c\right )^{3} + 192 \, a b^{7} \sin \left (d x + c\right )^{2} - 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 48 \, {\left (28 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (d x + c\right ) - \frac {6 \, {\left (16 \, a^{7} b - 112 \, a^{5} b^{3} - 336 \, a^{3} b^{5} - 80 \, a b^{7} - {\left (3 \, a^{8} - 28 \, a^{6} b^{2} - 350 \, a^{4} b^{4} - 252 \, a^{2} b^{6} - 13 \, b^{8}\right )} \sin \left (d x + c\right )^{3} + 32 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \sin \left (d x + c\right )^{2} + {\left (5 \, a^{8} + 28 \, a^{6} b^{2} - 210 \, a^{4} b^{4} - 196 \, a^{2} b^{6} - 11 \, b^{8}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{48 \, d} \]
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Time = 0.43 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.34 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {16 \, b^{8} \sin \left (d x + c\right )^{3} + 192 \, a b^{7} \sin \left (d x + c\right )^{2} + 1344 \, a^{2} b^{6} \sin \left (d x + c\right ) + 144 \, b^{8} \sin \left (d x + c\right ) - 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (336 \, a^{3} b^{5} \sin \left (d x + c\right )^{4} + 144 \, a b^{7} \sin \left (d x + c\right )^{4} - 3 \, a^{8} \sin \left (d x + c\right )^{3} + 28 \, a^{6} b^{2} \sin \left (d x + c\right )^{3} + 350 \, a^{4} b^{4} \sin \left (d x + c\right )^{3} + 252 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} + 13 \, b^{8} \sin \left (d x + c\right )^{3} + 224 \, a^{5} b^{3} \sin \left (d x + c\right )^{2} - 224 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} - 192 \, a b^{7} \sin \left (d x + c\right )^{2} + 5 \, a^{8} \sin \left (d x + c\right ) + 28 \, a^{6} b^{2} \sin \left (d x + c\right ) - 210 \, a^{4} b^{4} \sin \left (d x + c\right ) - 196 \, a^{2} b^{6} \sin \left (d x + c\right ) - 11 \, b^{8} \sin \left (d x + c\right ) + 16 \, a^{7} b - 112 \, a^{5} b^{3} + 64 \, a b^{7}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{48 \, d} \]
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Time = 4.95 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.95 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^6\,\left (3\,a^2+18\,a\,b+35\,b^2\right )}{16\,d}-\frac {b^8\,{\sin \left (c+d\,x\right )}^3}{3\,d}-\frac {\sin \left (c+d\,x\right )\,\left (28\,a^2\,b^6+3\,b^8\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (-\frac {5\,a^8}{8}-\frac {7\,a^6\,b^2}{2}+\frac {105\,a^4\,b^4}{4}+\frac {49\,a^2\,b^6}{2}+\frac {11\,b^8}{8}\right )-{\sin \left (c+d\,x\right )}^3\,\left (-\frac {3\,a^8}{8}+\frac {7\,a^6\,b^2}{2}+\frac {175\,a^4\,b^4}{4}+\frac {63\,a^2\,b^6}{2}+\frac {13\,b^8}{8}\right )+10\,a\,b^7-2\,a^7\,b-{\sin \left (c+d\,x\right )}^2\,\left (28\,a^5\,b^3+56\,a^3\,b^5+12\,a\,b^7\right )+42\,a^3\,b^5+14\,a^5\,b^3}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^2+1\right )}-\frac {4\,a\,b^7\,{\sin \left (c+d\,x\right )}^2}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^6\,\left (3\,a^2-18\,a\,b+35\,b^2\right )}{16\,d} \]
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