\(\int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx\) [418]

Optimal result

Integrand size = 21, antiderivative size = 369 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {35}{8} b^4 \left (16 a^4+16 a^2 b^2+b^4\right ) x+\frac {a b \left (8 a^6-104 a^4 b^2-803 a^2 b^4-256 b^6\right ) \cos (c+d x)}{6 d}+\frac {b^2 \left (16 a^6-200 a^4 b^2-866 a^2 b^4-105 b^6\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d} \] Output:

35/8*b^4*(16*a^4+16*a^2*b^2+b^4)*x+1/6*a*b*(8*a^6-104*a^4*b^2-803*a^2*b^4- 
256*b^6)*cos(d*x+c)/d+1/24*b^2*(16*a^6-200*a^4*b^2-866*a^2*b^4-105*b^6)*co 
s(d*x+c)*sin(d*x+c)/d+1/12*a*b*(8*a^4-88*a^2*b^2-151*b^4)*cos(d*x+c)*(a+b* 
sin(d*x+c))^2/d+1/12*b*(8*a^4-72*a^2*b^2-35*b^4)*cos(d*x+c)*(a+b*sin(d*x+c 
))^3/d+1/3*a*b*(2*a^2-13*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^4/d+1/3*b*(2*a^2 
-7*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^5/d+1/3*sec(d*x+c)^3*(b+a*sin(d*x+c))* 
(a+b*sin(d*x+c))^7/d-1/3*sec(d*x+c)*(a+b*sin(d*x+c))^6*(5*a*b-(2*a^2-7*b^2 
)*sin(d*x+c))/d
 

Mathematica [A] (verified)

Time = 3.88 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.12 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\sec ^3(c+d x) \left (2048 a^7 b-7168 a^5 b^3-44800 a^3 b^5-13440 a b^7+40320 a^4 b^4 (c+d x) \cos (c+d x)+40320 a^2 b^6 (c+d x) \cos (c+d x)+2520 b^8 (c+d x) \cos (c+d x)-21504 a^5 b^3 \cos (2 (c+d x))-64512 a^3 b^5 \cos (2 (c+d x))-17472 a b^7 \cos (2 (c+d x))+13440 a^4 b^4 (c+d x) \cos (3 (c+d x))+13440 a^2 b^6 (c+d x) \cos (3 (c+d x))+840 b^8 (c+d x) \cos (3 (c+d x))-5376 a^3 b^5 \cos (4 (c+d x))-1920 a b^7 \cos (4 (c+d x))+64 a b^7 \cos (6 (c+d x))+384 a^8 \sin (c+d x)+5376 a^6 b^2 \sin (c+d x)-6720 a^2 b^6 \sin (c+d x)-525 b^8 \sin (c+d x)+128 a^8 \sin (3 (c+d x))-1792 a^6 b^2 \sin (3 (c+d x))-17920 a^4 b^4 \sin (3 (c+d x))-14560 a^2 b^6 \sin (3 (c+d x))-847 b^8 \sin (3 (c+d x))-672 a^2 b^6 \sin (5 (c+d x))-63 b^8 \sin (5 (c+d x))+3 b^8 \sin (7 (c+d x))\right )}{768 d} \] Input:

Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^8,x]
 

Output:

(Sec[c + d*x]^3*(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)] - 64 
512*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)]))/(768*d)
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 3170, 25, 3042, 3340, 27, 3042, 3232, 27, 3042, 3232, 3042, 3232, 27, 3042, 3232, 3042, 3213}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \sin (c+d x))^8}{\cos (c+d x)^4}dx\)

\(\Big \downarrow \) 3170

\(\displaystyle \frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}-\frac {1}{3} \int -\sec ^2(c+d x) (a+b \sin (c+d x))^6 \left (2 a^2-5 b \sin (c+d x) a-7 b^2\right )dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \int \sec ^2(c+d x) (a+b \sin (c+d x))^6 \left (2 a^2-5 b \sin (c+d x) a-7 b^2\right )dx+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \int \frac {(a+b \sin (c+d x))^6 \left (2 a^2-5 b \sin (c+d x) a-7 b^2\right )}{\cos (c+d x)^2}dx+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3340

\(\displaystyle \frac {1}{3} \left (-\int -6 (a+b \sin (c+d x))^5 \left (5 a b^2-b \left (2 a^2-7 b^2\right ) \sin (c+d x)\right )dx-\frac {\sec (c+d x) \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^6}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (6 \int (a+b \sin (c+d x))^5 \left (5 a b^2-b \left (2 a^2-7 b^2\right ) \sin (c+d x)\right )dx-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \left (6 \int (a+b \sin (c+d x))^5 \left (5 a b^2-b \left (2 a^2-7 b^2\right ) \sin (c+d x)\right )dx-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {1}{6} \int 5 (a+b \sin (c+d x))^4 \left (b^2 \left (4 a^2+7 b^2\right )-a b \left (2 a^2-13 b^2\right ) \sin (c+d x)\right )dx+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {5}{6} \int (a+b \sin (c+d x))^4 \left (b^2 \left (4 a^2+7 b^2\right )-a b \left (2 a^2-13 b^2\right ) \sin (c+d x)\right )dx+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {5}{6} \int (a+b \sin (c+d x))^4 \left (b^2 \left (4 a^2+7 b^2\right )-a b \left (2 a^2-13 b^2\right ) \sin (c+d x)\right )dx+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {5}{6} \left (\frac {1}{5} \int (a+b \sin (c+d x))^3 \left (3 a b^2 \left (4 a^2+29 b^2\right )-b \left (8 a^4-72 b^2 a^2-35 b^4\right ) \sin (c+d x)\right )dx+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {5}{6} \left (\frac {1}{5} \int (a+b \sin (c+d x))^3 \left (3 a b^2 \left (4 a^2+29 b^2\right )-b \left (8 a^4-72 b^2 a^2-35 b^4\right ) \sin (c+d x)\right )dx+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {5}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int 3 (a+b \sin (c+d x))^2 \left (b^2 \left (8 a^4+188 b^2 a^2+35 b^4\right )-a b \left (8 a^4-88 b^2 a^2-151 b^4\right ) \sin (c+d x)\right )dx+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {5}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int (a+b \sin (c+d x))^2 \left (b^2 \left (8 a^4+188 b^2 a^2+35 b^4\right )-a b \left (8 a^4-88 b^2 a^2-151 b^4\right ) \sin (c+d x)\right )dx+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {5}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int (a+b \sin (c+d x))^2 \left (b^2 \left (8 a^4+188 b^2 a^2+35 b^4\right )-a b \left (8 a^4-88 b^2 a^2-151 b^4\right ) \sin (c+d x)\right )dx+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {5}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int (a+b \sin (c+d x)) \left (a b^2 \left (8 a^4+740 b^2 a^2+407 b^4\right )-b \left (16 a^6-200 b^2 a^4-866 b^4 a^2-105 b^6\right ) \sin (c+d x)\right )dx+\frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{3 d}\right )+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {5}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int (a+b \sin (c+d x)) \left (a b^2 \left (8 a^4+740 b^2 a^2+407 b^4\right )-b \left (16 a^6-200 b^2 a^4-866 b^4 a^2-105 b^6\right ) \sin (c+d x)\right )dx+\frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{3 d}\right )+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {1}{3} \left (6 \left (\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {5}{6} \left (\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}+\frac {1}{5} \left (\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}+\frac {3}{4} \left (\frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {105}{2} b^4 x \left (16 a^4+16 a^2 b^2+b^4\right )+\frac {2 a b \left (8 a^6-104 a^4 b^2-803 a^2 b^4-256 b^6\right ) \cos (c+d x)}{d}+\frac {b^2 \left (16 a^6-200 a^4 b^2-866 a^2 b^4-105 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )\right )\right )\right )\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}\)

Input:

Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^8,x]
 

Output:

(Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(3*d) + (-((S 
ec[c + d*x]*(a + b*Sin[c + d*x])^6*(5*a*b - (2*a^2 - 7*b^2)*Sin[c + d*x])) 
/d) + 6*((b*(2*a^2 - 7*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^5)/(6*d) + ( 
5*((a*b*(2*a^2 - 13*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(5*d) + ((b* 
(8*a^4 - 72*a^2*b^2 - 35*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(4*d) + 
 (3*((a*b*(8*a^4 - 88*a^2*b^2 - 151*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x]) 
^2)/(3*d) + ((105*b^4*(16*a^4 + 16*a^2*b^2 + b^4)*x)/2 + (2*a*b*(8*a^6 - 1 
04*a^4*b^2 - 803*a^2*b^4 - 256*b^6)*Cos[c + d*x])/d + (b^2*(16*a^6 - 200*a 
^4*b^2 - 866*a^2*b^4 - 105*b^6)*Cos[c + d*x]*Sin[c + d*x])/(2*d))/3))/4)/5 
))/6))/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3170

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-(g*Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x 
])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) 
  Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + 
a^2*(p + 2) + a*b*(m + p + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g 
}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2* 
p] || IntegerQ[m])
 

rule 3213

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) 
*(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co 
s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free 
Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
 

rule 3232

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( 
f*(m + 1))), x] + Simp[1/(m + 1)   Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* 
d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ 
[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 
 0] && IntegerQ[2*m]
 

rule 3340

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(g* 
Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])/(f*g*(p 
 + 1))), x] + Simp[1/(g^2*(p + 1))   Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Si 
n[e + f*x])^(m - 1)*Simp[a*c*(p + 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x] 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ 
[m, 0] && LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] 
&& SimplerQ[c + d*x, a + b*x])
 

Maple [A] (verified)

Time = 5.42 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.01

method	result	size
parallelrisch	\(\frac {-3584 \left (-\frac {15}{4} a^{4} b^{3} d x -\frac {15}{4} a^{2} b^{5} d x -\frac {15}{64} b^{7} d x +a^{7}+6 a^{5} b^{2}+8 a^{3} b^{4}+\frac {16}{7} a \,b^{6}\right ) b \cos \left (3 d x +3 c \right )+\left (128 a^{8}-1792 a^{6} b^{2}-17920 a^{4} b^{4}-14560 b^{6} a^{2}-847 b^{8}\right ) \sin \left (3 d x +3 c \right )-21504 a \left (a^{4}+3 b^{2} a^{2}+\frac {13}{16} b^{4}\right ) b^{3} \cos \left (2 d x +2 c \right )+\left (-5376 a^{3} b^{5}-1920 a \,b^{7}\right ) \cos \left (4 d x +4 c \right )+\left (-672 b^{6} a^{2}-63 b^{8}\right ) \sin \left (5 d x +5 c \right )+64 \cos \left (6 d x +6 c \right ) a \,b^{7}+3 \sin \left (7 d x +7 c \right ) b^{8}-10752 \left (-\frac {15}{4} a^{4} b^{3} d x -\frac {15}{4} a^{2} b^{5} d x -\frac {15}{64} b^{7} d x +a^{7}+6 a^{5} b^{2}+8 a^{3} b^{4}+\frac {16}{7} a \,b^{6}\right ) b \cos \left (d x +c \right )+\left (384 a^{8}+5376 a^{6} b^{2}-6720 b^{6} a^{2}-525 b^{8}\right ) \sin \left (d x +c \right )+2048 a^{7} b -7168 a^{5} b^{3}-44800 a^{3} b^{5}-13440 a \,b^{7}}{192 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\)	\(371\)
derivativedivides	\(\frac {-a^{8} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{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} \sin \left (d x +c \right )^{3}}{3 \cos \left (d x +c \right )^{3}}+56 a^{5} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}\right )+70 a^{4} b^{4} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+56 a^{3} b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+28 b^{6} a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{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 \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{3}\right )+b^{8} \left (\frac {\sin \left (d x +c \right )^{9}}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}-2 \left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{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 {\sec \left (d x +c \right )^{2}}{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} \sin \left (d x +c \right )^{3}}{3 \cos \left (d x +c \right )^{3}}+56 a^{5} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}\right )+70 a^{4} b^{4} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+56 a^{3} b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+28 b^{6} a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{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 \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{3}\right )+b^{8} \left (\frac {\sin \left (d x +c \right )^{9}}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}-2 \left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{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	\(70 a^{4} b^{4} x +70 a^{2} b^{6} x +\frac {35 b^{8} x}{8}-\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{8}}{8 d}+\frac {a \,b^{7} {\mathrm e}^{3 i \left (d x +c \right )}}{3 d}+\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} b^{8}}{8 d}+\frac {i b^{8} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}-\frac {28 a^{3} b^{5} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {11 a \,b^{7} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {28 a^{3} b^{5} {\mathrm e}^{-i \left (d x +c \right )}}{d}-\frac {11 a \,b^{7} {\mathrm e}^{-i \left (d x +c \right )}}{d}-\frac {7 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{6} a^{2}}{2 d}+\frac {4 i \left (-16 i a^{7} b \,{\mathrm e}^{3 i \left (d x +c \right )}+36 i a \,b^{7} {\mathrm e}^{5 i \left (d x +c \right )}+56 i a^{5} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+36 i a \,b^{7} {\mathrm e}^{i \left (d x +c \right )}+84 i a^{5} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+56 i a \,b^{7} {\mathrm e}^{3 i \left (d x +c \right )}+168 i a^{3} b^{5} {\mathrm e}^{5 i \left (d x +c \right )}-42 a^{6} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-210 a^{4} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-126 a^{2} b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-6 b^{8} {\mathrm e}^{4 i \left (d x +c \right )}+224 i a^{3} b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+168 i a^{3} b^{5} {\mathrm e}^{i \left (d x +c \right )}+84 i a^{5} b^{3} {\mathrm e}^{i \left (d x +c \right )}+3 a^{8} {\mathrm e}^{2 i \left (d x +c \right )}-210 a^{4} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-168 a^{2} b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-9 b^{8} {\mathrm e}^{2 i \left (d x +c \right )}+a^{8}-14 a^{6} b^{2}-140 a^{4} b^{4}-98 b^{6} a^{2}-5 b^{8}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {a \,b^{7} {\mathrm e}^{-3 i \left (d x +c \right )}}{3 d}-\frac {i b^{8} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}+\frac {7 i {\mathrm e}^{2 i \left (d x +c \right )} b^{6} a^{2}}{2 d}\)	\(603\)

Input:

int(sec(d*x+c)^4*(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
 

Output:

1/192*(-3584*(-15/4*a^4*b^3*d*x-15/4*a^2*b^5*d*x-15/64*b^7*d*x+a^7+6*a^5*b 
^2+8*a^3*b^4+16/7*a*b^6)*b*cos(3*d*x+3*c)+(128*a^8-1792*a^6*b^2-17920*a^4* 
b^4-14560*a^2*b^6-847*b^8)*sin(3*d*x+3*c)-21504*a*(a^4+3*b^2*a^2+13/16*b^4 
)*b^3*cos(2*d*x+2*c)+(-5376*a^3*b^5-1920*a*b^7)*cos(4*d*x+4*c)+(-672*a^2*b 
^6-63*b^8)*sin(5*d*x+5*c)+64*cos(6*d*x+6*c)*a*b^7+3*sin(7*d*x+7*c)*b^8-107 
52*(-15/4*a^4*b^3*d*x-15/4*a^2*b^5*d*x-15/64*b^7*d*x+a^7+6*a^5*b^2+8*a^3*b 
^4+16/7*a*b^6)*b*cos(d*x+c)+(384*a^8+5376*a^6*b^2-6720*a^2*b^6-525*b^8)*si 
n(d*x+c)+2048*a^7*b-7168*a^5*b^3-44800*a^3*b^5-13440*a*b^7)/d/(cos(3*d*x+3 
*c)+3*cos(d*x+c))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.73 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {64 \, a b^{7} \cos \left (d x + c\right )^{6} + 64 \, a^{7} b + 448 \, a^{5} b^{3} + 448 \, a^{3} b^{5} + 64 \, a b^{7} + 105 \, {\left (16 \, a^{4} b^{4} + 16 \, a^{2} b^{6} + b^{8}\right )} d x \cos \left (d x + c\right )^{3} - 192 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 192 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, b^{8} \cos \left (d x + c\right )^{6} + 8 \, a^{8} + 224 \, a^{6} b^{2} + 560 \, a^{4} b^{4} + 224 \, a^{2} b^{6} + 8 \, b^{8} - 3 \, {\left (112 \, a^{2} b^{6} + 13 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (a^{8} - 14 \, a^{6} b^{2} - 140 \, a^{4} b^{4} - 98 \, a^{2} b^{6} - 5 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )^{3}} \] Input:

integrate(sec(d*x+c)^4*(a+b*sin(d*x+c))^8,x, algorithm="fricas")
 

Output:

1/24*(64*a*b^7*cos(d*x + c)^6 + 64*a^7*b + 448*a^5*b^3 + 448*a^3*b^5 + 64* 
a*b^7 + 105*(16*a^4*b^4 + 16*a^2*b^6 + b^8)*d*x*cos(d*x + c)^3 - 192*(7*a^ 
3*b^5 + 3*a*b^7)*cos(d*x + c)^4 - 192*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*c 
os(d*x + c)^2 + (6*b^8*cos(d*x + c)^6 + 8*a^8 + 224*a^6*b^2 + 560*a^4*b^4 
+ 224*a^2*b^6 + 8*b^8 - 3*(112*a^2*b^6 + 13*b^8)*cos(d*x + c)^4 + 16*(a^8 
- 14*a^6*b^2 - 140*a^4*b^4 - 98*a^2*b^6 - 5*b^8)*cos(d*x + c)^2)*sin(d*x + 
 c))/(d*cos(d*x + c)^3)
 

Sympy [F(-1)]

Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)**4*(a+b*sin(d*x+c))**8,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.89 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {224 \, a^{6} b^{2} \tan \left (d x + c\right )^{3} + 8 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{8} + 560 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4} b^{4} + 112 \, {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{2} b^{6} + 64 \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a b^{7} + {\left (8 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - \frac {3 \, {\left (13 \, \tan \left (d x + c\right )^{3} + 11 \, \tan \left (d x + c\right )\right )}}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 72 \, \tan \left (d x + c\right )\right )} b^{8} - 448 \, a^{3} b^{5} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {448 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{3}} + \frac {64 \, a^{7} b}{\cos \left (d x + c\right )^{3}}}{24 \, d} \] Input:

integrate(sec(d*x+c)^4*(a+b*sin(d*x+c))^8,x, algorithm="maxima")
 

Output:

1/24*(224*a^6*b^2*tan(d*x + c)^3 + 8*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^8 
 + 560*(tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a^4*b^4 + 112*(2*ta 
n(d*x + c)^3 + 15*d*x + 15*c - 3*tan(d*x + c)/(tan(d*x + c)^2 + 1) - 12*ta 
n(d*x + c))*a^2*b^6 + 64*(cos(d*x + c)^3 - (9*cos(d*x + c)^2 - 1)/cos(d*x 
+ c)^3 - 9*cos(d*x + c))*a*b^7 + (8*tan(d*x + c)^3 + 105*d*x + 105*c - 3*( 
13*tan(d*x + c)^3 + 11*tan(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 + 
1) - 72*tan(d*x + c))*b^8 - 448*a^3*b^5*((6*cos(d*x + c)^2 - 1)/cos(d*x + 
c)^3 + 3*cos(d*x + c)) - 448*(3*cos(d*x + c)^2 - 1)*a^5*b^3/cos(d*x + c)^3 
 + 64*a^7*b/cos(d*x + c)^3)/d
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.85 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx =\text {Too large to display} \] Input:

integrate(sec(d*x+c)^4*(a+b*sin(d*x+c))^8,x, algorithm="giac")
 

Output:

1/24*(105*(16*a^4*b^4 + 16*a^2*b^6 + b^8)*(d*x + c) - 16*(3*a^8*tan(1/2*d* 
x + 1/2*c)^5 - 210*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 168*a^2*b^6*tan(1/2*d* 
x + 1/2*c)^5 - 9*b^8*tan(1/2*d*x + 1/2*c)^5 + 24*a^7*b*tan(1/2*d*x + 1/2*c 
)^4 - 168*a^3*b^5*tan(1/2*d*x + 1/2*c)^4 - 48*a*b^7*tan(1/2*d*x + 1/2*c)^4 
 - 2*a^8*tan(1/2*d*x + 1/2*c)^3 + 112*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 + 700 
*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 + 448*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 + 22* 
b^8*tan(1/2*d*x + 1/2*c)^3 + 336*a^5*b^3*tan(1/2*d*x + 1/2*c)^2 + 672*a^3* 
b^5*tan(1/2*d*x + 1/2*c)^2 + 144*a*b^7*tan(1/2*d*x + 1/2*c)^2 + 3*a^8*tan( 
1/2*d*x + 1/2*c) - 210*a^4*b^4*tan(1/2*d*x + 1/2*c) - 168*a^2*b^6*tan(1/2* 
d*x + 1/2*c) - 9*b^8*tan(1/2*d*x + 1/2*c) + 8*a^7*b - 112*a^5*b^3 - 280*a^ 
3*b^5 - 64*a*b^7)/(tan(1/2*d*x + 1/2*c)^2 - 1)^3 + 2*(336*a^2*b^6*tan(1/2* 
d*x + 1/2*c)^7 + 33*b^8*tan(1/2*d*x + 1/2*c)^7 - 1344*a^3*b^5*tan(1/2*d*x 
+ 1/2*c)^6 - 384*a*b^7*tan(1/2*d*x + 1/2*c)^6 + 336*a^2*b^6*tan(1/2*d*x + 
1/2*c)^5 + 57*b^8*tan(1/2*d*x + 1/2*c)^5 - 4032*a^3*b^5*tan(1/2*d*x + 1/2* 
c)^4 - 1536*a*b^7*tan(1/2*d*x + 1/2*c)^4 - 336*a^2*b^6*tan(1/2*d*x + 1/2*c 
)^3 - 57*b^8*tan(1/2*d*x + 1/2*c)^3 - 4032*a^3*b^5*tan(1/2*d*x + 1/2*c)^2 
- 1664*a*b^7*tan(1/2*d*x + 1/2*c)^2 - 336*a^2*b^6*tan(1/2*d*x + 1/2*c) - 3 
3*b^8*tan(1/2*d*x + 1/2*c) - 1344*a^3*b^5 - 512*a*b^7)/(tan(1/2*d*x + 1/2* 
c)^2 + 1)^4)/d
 

Mupad [B] (verification not implemented)

Time = 17.83 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.97 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx =\text {Too large to display} \] Input:

int((a + b*sin(c + d*x))^8/cos(c + d*x)^4,x)
                                                                                    
                                                                                    
 

Output:

(tan(c/2 + (d*x)/2)^8*((304*a^7*b)/3 + (1792*a^3*b^5)/3 + (2464*a^5*b^3)/3 
) + tan(c/2 + (d*x)/2)^10*(64*a^7*b + 224*a^5*b^3) - (256*a*b^7)/3 + (16*a 
^7*b)/3 - tan(c/2 + (d*x)/2)*((35*b^8)/4 - 2*a^8 + 140*a^2*b^6 + 140*a^4*b 
^4) - tan(c/2 + (d*x)/2)^2*((256*a*b^7)/3 - (64*a^7*b)/3 + (896*a^3*b^5)/3 
 + (224*a^5*b^3)/3) + tan(c/2 + (d*x)/2)^4*(256*a*b^7 + 48*a^7*b + 896*a^3 
*b^5 + 448*a^5*b^3) + tan(c/2 + (d*x)/2)^6*(256*a*b^7 + (256*a^7*b)/3 + (4 
480*a^3*b^5)/3 + (3136*a^5*b^3)/3) - tan(c/2 + (d*x)/2)^3*((35*b^8)/6 - (2 
0*a^8)/3 + (280*a^2*b^6)/3 + (280*a^4*b^4)/3 - (224*a^6*b^2)/3) - tan(c/2 
+ (d*x)/2)^11*((35*b^8)/6 - (20*a^8)/3 + (280*a^2*b^6)/3 + (280*a^4*b^4)/3 
 - (224*a^6*b^2)/3) + tan(c/2 + (d*x)/2)^7*(8*a^8 + 17*b^8 + 784*a^2*b^6 + 
 1680*a^4*b^4 + 448*a^6*b^2) + tan(c/2 + (d*x)/2)^5*((26*a^8)/3 + (329*b^8 
)/12 + (1316*a^2*b^6)/3 + (2660*a^4*b^4)/3 + (896*a^6*b^2)/3) + tan(c/2 + 
(d*x)/2)^9*((26*a^8)/3 + (329*b^8)/12 + (1316*a^2*b^6)/3 + (2660*a^4*b^4)/ 
3 + (896*a^6*b^2)/3) - (896*a^3*b^5)/3 - (224*a^5*b^3)/3 - tan(c/2 + (d*x) 
/2)^13*((35*b^8)/4 - 2*a^8 + 140*a^2*b^6 + 140*a^4*b^4) + 16*a^7*b*tan(c/2 
 + (d*x)/2)^12)/(d*(tan(c/2 + (d*x)/2)^2 - 3*tan(c/2 + (d*x)/2)^4 - 3*tan( 
c/2 + (d*x)/2)^6 + 3*tan(c/2 + (d*x)/2)^8 + 3*tan(c/2 + (d*x)/2)^10 - tan( 
c/2 + (d*x)/2)^12 - tan(c/2 + (d*x)/2)^14 + 1)) + (35*b^4*atan((35*b^4*tan 
(c/2 + (d*x)/2)*(16*a^4 + b^4 + 16*a^2*b^2))/(35*b^8 + 560*a^2*b^6 + 560*a 
^4*b^4))*(16*a^4 + b^4 + 16*a^2*b^2))/(4*d)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.81 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx =\text {Too large to display} \] Input:

int(sec(d*x+c)^4*(a+b*sin(d*x+c))^8,x)
 

Output:

( - 64*cos(c + d*x)*sin(c + d*x)**2*a**7*b + 896*cos(c + d*x)*sin(c + d*x) 
**2*a**5*b**3 + 1680*cos(c + d*x)*sin(c + d*x)**2*a**4*b**4*c + 1680*cos(c 
 + d*x)*sin(c + d*x)**2*a**4*b**4*d*x + 3584*cos(c + d*x)*sin(c + d*x)**2* 
a**3*b**5 + 1680*cos(c + d*x)*sin(c + d*x)**2*a**2*b**6*c + 1680*cos(c + d 
*x)*sin(c + d*x)**2*a**2*b**6*d*x + 1024*cos(c + d*x)*sin(c + d*x)**2*a*b* 
*7 + 105*cos(c + d*x)*sin(c + d*x)**2*b**8*c + 105*cos(c + d*x)*sin(c + d* 
x)**2*b**8*d*x + 64*cos(c + d*x)*a**7*b - 896*cos(c + d*x)*a**5*b**3 - 168 
0*cos(c + d*x)*a**4*b**4*c - 1680*cos(c + d*x)*a**4*b**4*d*x - 3584*cos(c 
+ d*x)*a**3*b**5 - 1680*cos(c + d*x)*a**2*b**6*c - 1680*cos(c + d*x)*a**2* 
b**6*d*x - 1024*cos(c + d*x)*a*b**7 - 105*cos(c + d*x)*b**8*c - 105*cos(c 
+ d*x)*b**8*d*x + 6*sin(c + d*x)**7*b**8 + 64*sin(c + d*x)**6*a*b**7 + 336 
*sin(c + d*x)**5*a**2*b**6 + 21*sin(c + d*x)**5*b**8 + 1344*sin(c + d*x)** 
4*a**3*b**5 + 384*sin(c + d*x)**4*a*b**7 + 16*sin(c + d*x)**3*a**8 - 224*s 
in(c + d*x)**3*a**6*b**2 - 2240*sin(c + d*x)**3*a**4*b**4 - 2240*sin(c + d 
*x)**3*a**2*b**6 - 140*sin(c + d*x)**3*b**8 - 1344*sin(c + d*x)**2*a**5*b* 
*3 - 5376*sin(c + d*x)**2*a**3*b**5 - 1536*sin(c + d*x)**2*a*b**7 - 24*sin 
(c + d*x)*a**8 + 1680*sin(c + d*x)*a**4*b**4 + 1680*sin(c + d*x)*a**2*b**6 
 + 105*sin(c + d*x)*b**8 - 64*a**7*b + 896*a**5*b**3 + 3584*a**3*b**5 + 10 
24*a*b**7)/(24*cos(c + d*x)*d*(sin(c + d*x)**2 - 1))