\(\int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx\) [417]

Optimal result

Integrand size = 21, antiderivative size = 349 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {7}{16} b^2 \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right ) x+\frac {a b \left (40 a^6+1664 a^4 b^2+2789 a^2 b^4+512 b^6\right ) \cos (c+d x)}{20 d}+\frac {b^2 \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d} \] Output:

-7/16*b^2*(64*a^6+240*a^4*b^2+120*a^2*b^4+5*b^6)*x+1/20*a*b*(40*a^6+1664*a 
^4*b^2+2789*a^2*b^4+512*b^6)*cos(d*x+c)/d+1/80*b^2*(80*a^6+2248*a^4*b^2+25 
02*a^2*b^4+175*b^6)*cos(d*x+c)*sin(d*x+c)/d+1/40*a*b*(40*a^4+624*a^2*b^2+3 
37*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^2/d+1/120*b*(120*a^4+992*a^2*b^2+175*b 
^4)*cos(d*x+c)*(a+b*sin(d*x+c))^3/d+1/30*a*b*(30*a^2+113*b^2)*cos(d*x+c)*( 
a+b*sin(d*x+c))^4/d+1/6*b*(6*a^2+7*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^5/d+a* 
b*cos(d*x+c)*(a+b*sin(d*x+c))^6/d+sec(d*x+c)*(b+a*sin(d*x+c))*(a+b*sin(d*x 
+c))^7/d
 

Mathematica [A] (verified)

Time = 3.83 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.90 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\sec (c+d x) \left (15360 a^7 b+161280 a^5 b^3+201600 a^3 b^5+33600 a b^7-840 b^2 \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right ) (c+d x) \cos (c+d x)+1120 \left (48 a^5 b^3+80 a^3 b^5+15 a b^7\right ) \cos (2 (c+d x))-4480 a^3 b^5 \cos (4 (c+d x))-1344 a b^7 \cos (4 (c+d x))+96 a b^7 \cos (6 (c+d x))+1920 a^8 \sin (c+d x)+53760 a^6 b^2 \sin (c+d x)+151200 a^4 b^4 \sin (c+d x)+67200 a^2 b^6 \sin (c+d x)+2625 b^8 \sin (c+d x)+16800 a^4 b^4 \sin (3 (c+d x))+12600 a^2 b^6 \sin (3 (c+d x))+630 b^8 \sin (3 (c+d x))-840 a^2 b^6 \sin (5 (c+d x))-70 b^8 \sin (5 (c+d x))+5 b^8 \sin (7 (c+d x))\right )}{1920 d} \] Input:

Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^8,x]
 

Output:

(Sec[c + d*x]*(15360*a^7*b + 161280*a^5*b^3 + 201600*a^3*b^5 + 33600*a*b^7 
 - 840*b^2*(64*a^6 + 240*a^4*b^2 + 120*a^2*b^4 + 5*b^6)*(c + d*x)*Cos[c + 
d*x] + 1120*(48*a^5*b^3 + 80*a^3*b^5 + 15*a*b^7)*Cos[2*(c + d*x)] - 4480*a 
^3*b^5*Cos[4*(c + d*x)] - 1344*a*b^7*Cos[4*(c + d*x)] + 96*a*b^7*Cos[6*(c 
+ d*x)] + 1920*a^8*Sin[c + d*x] + 53760*a^6*b^2*Sin[c + d*x] + 151200*a^4* 
b^4*Sin[c + d*x] + 67200*a^2*b^6*Sin[c + d*x] + 2625*b^8*Sin[c + d*x] + 16 
800*a^4*b^4*Sin[3*(c + d*x)] + 12600*a^2*b^6*Sin[3*(c + d*x)] + 630*b^8*Si 
n[3*(c + d*x)] - 840*a^2*b^6*Sin[5*(c + d*x)] - 70*b^8*Sin[5*(c + d*x)] + 
5*b^8*Sin[7*(c + d*x)]))/(1920*d)
 

Rubi [A] (verified)

Time = 1.59 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.06, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 3170, 27, 3042, 3232, 3042, 3232, 3042, 3232, 3042, 3232, 27, 3042, 3232, 27, 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.

\(\displaystyle \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \sin (c+d x))^8}{\cos (c+d x)^2}dx\)

\(\Big \downarrow \) 3170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3232

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \int (a+b \sin (c+d x))^4 \left (\left (108 a^2+35 b^2\right ) b^2+a \left (30 a^2+113 b^2\right ) \sin (c+d x) b\right )dx-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \int (a+b \sin (c+d x))^4 \left (\left (108 a^2+35 b^2\right ) b^2+a \left (30 a^2+113 b^2\right ) \sin (c+d x) b\right )dx-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int (a+b \sin (c+d x))^3 \left (33 a \left (20 a^2+19 b^2\right ) b^2+\left (120 a^4+992 b^2 a^2+175 b^4\right ) \sin (c+d x) b\right )dx-\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int (a+b \sin (c+d x))^3 \left (33 a \left (20 a^2+19 b^2\right ) b^2+\left (120 a^4+992 b^2 a^2+175 b^4\right ) \sin (c+d x) b\right )dx-\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int 3 (a+b \sin (c+d x))^2 \left (\left (1000 a^4+1828 b^2 a^2+175 b^4\right ) b^2+3 a \left (40 a^4+624 b^2 a^2+337 b^4\right ) \sin (c+d x) b\right )dx-\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int (a+b \sin (c+d x))^2 \left (\left (1000 a^4+1828 b^2 a^2+175 b^4\right ) b^2+3 a \left (40 a^4+624 b^2 a^2+337 b^4\right ) \sin (c+d x) b\right )dx-\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int (a+b \sin (c+d x))^2 \left (\left (1000 a^4+1828 b^2 a^2+175 b^4\right ) b^2+3 a \left (40 a^4+624 b^2 a^2+337 b^4\right ) \sin (c+d x) b\right )dx-\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int 3 (a+b \sin (c+d x)) \left (a \left (1080 a^4+3076 b^2 a^2+849 b^4\right ) b^2+\left (80 a^6+2248 b^2 a^4+2502 b^4 a^2+175 b^6\right ) \sin (c+d x) b\right )dx-\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\int (a+b \sin (c+d x)) \left (a \left (1080 a^4+3076 b^2 a^2+849 b^4\right ) b^2+\left (80 a^6+2248 b^2 a^4+2502 b^4 a^2+175 b^6\right ) \sin (c+d x) b\right )dx-\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\int (a+b \sin (c+d x)) \left (a \left (1080 a^4+3076 b^2 a^2+849 b^4\right ) b^2+\left (80 a^6+2248 b^2 a^4+2502 b^4 a^2+175 b^6\right ) \sin (c+d x) b\right )dx-\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}-7 \left (\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (-\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{d}-\frac {2 a b \left (40 a^6+1664 a^4 b^2+2789 a^2 b^4+512 b^6\right ) \cos (c+d x)}{d}-\frac {b^2 \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {35}{2} b^2 x \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right )\right )-\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{5 d}\right )-\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}\right )-\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{7 d}\right )\)

Input:

Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^8,x]
 

Output:

(Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/d - 7*(-1/7*(a* 
b*Cos[c + d*x]*(a + b*Sin[c + d*x])^6)/d + (-1/6*(b*(6*a^2 + 7*b^2)*Cos[c 
+ d*x]*(a + b*Sin[c + d*x])^5)/d + (-1/5*(a*b*(30*a^2 + 113*b^2)*Cos[c + d 
*x]*(a + b*Sin[c + d*x])^4)/d + (-1/4*(b*(120*a^4 + 992*a^2*b^2 + 175*b^4) 
*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/d + (3*((35*b^2*(64*a^6 + 240*a^4*b^ 
2 + 120*a^2*b^4 + 5*b^6)*x)/2 - (2*a*b*(40*a^6 + 1664*a^4*b^2 + 2789*a^2*b 
^4 + 512*b^6)*Cos[c + d*x])/d - (b^2*(80*a^6 + 2248*a^4*b^2 + 2502*a^2*b^4 
 + 175*b^6)*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a*b*(40*a^4 + 624*a^2*b^2 
+ 337*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/d))/4)/5)/6)/7)
 

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 5.98 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.85

method	result	size
parallelrisch	\(\frac {\left (161280 a^{5} b^{3}+268800 a^{3} b^{5}+50400 a \,b^{7}\right ) \cos \left (2 d x +2 c \right )+\left (50400 a^{4} b^{4}+37800 b^{6} a^{2}+1890 b^{8}\right ) \sin \left (3 d x +3 c \right )+\left (-13440 a^{3} b^{5}-4032 a \,b^{7}\right ) \cos \left (4 d x +4 c \right )+\left (-2520 b^{6} a^{2}-210 b^{8}\right ) \sin \left (5 d x +5 c \right )+288 \cos \left (6 d x +6 c \right ) a \,b^{7}+15 \sin \left (7 d x +7 c \right ) b^{8}+107520 \left (-\frac {3}{2} a^{6} b d x -\frac {45}{8} a^{4} b^{3} d x -\frac {45}{16} a^{2} b^{5} d x -\frac {15}{128} b^{7} d x +a^{7}+\frac {22}{3} a^{5} b^{2}+8 a^{3} b^{4}+\frac {48}{35} a \,b^{6}\right ) b \cos \left (d x +c \right )+\left (5760 a^{8}+161280 a^{6} b^{2}+453600 a^{4} b^{4}+201600 b^{6} a^{2}+7875 b^{8}\right ) \sin \left (d x +c \right )+46080 a^{7} b +483840 a^{5} b^{3}+604800 a^{3} b^{5}+100800 a \,b^{7}}{5760 d \cos \left (d x +c \right )}\)	\(298\)
derivativedivides	\(\frac {a^{8} \tan \left (d x +c \right )+\frac {8 a^{7} b}{\cos \left (d x +c \right )}+28 a^{6} b^{2} \left (\tan \left (d x +c \right )-d x -c \right )+56 a^{5} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+70 a^{4} b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+56 a^{3} b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+28 b^{6} a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+8 a \,b^{7} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )+b^{8} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )}{d}\)	\(406\)
default	\(\frac {a^{8} \tan \left (d x +c \right )+\frac {8 a^{7} b}{\cos \left (d x +c \right )}+28 a^{6} b^{2} \left (\tan \left (d x +c \right )-d x -c \right )+56 a^{5} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+70 a^{4} b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+56 a^{3} b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+28 b^{6} a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+8 a \,b^{7} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )+b^{8} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )}{d}\)	\(406\)
risch	\(-28 a^{6} b^{2} x -105 a^{4} b^{4} x -\frac {105 a^{2} b^{6} x}{2}-\frac {35 b^{8} x}{16}+\frac {35 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} b^{4}}{4 d}-\frac {47 i {\mathrm e}^{2 i \left (d x +c \right )} b^{8}}{128 d}-\frac {7 i {\mathrm e}^{2 i \left (d x +c \right )} b^{6} a^{2}}{d}+\frac {28 a^{5} b^{3} {\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {49 a^{3} b^{5} {\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {19 a \,b^{7} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {28 a^{5} b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {49 a^{3} b^{5} {\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {19 a \,b^{7} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {35 i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} b^{4}}{4 d}+\frac {2 i \left (a^{8}+28 a^{6} b^{2}+70 a^{4} b^{4}+28 b^{6} a^{2}+b^{8}-8 i a^{7} b \,{\mathrm e}^{i \left (d x +c \right )}-56 i a^{5} b^{3} {\mathrm e}^{i \left (d x +c \right )}-56 i a^{3} b^{5} {\mathrm e}^{i \left (d x +c \right )}-8 i a \,b^{7} {\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {47 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{8}}{128 d}+\frac {7 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{6} a^{2}}{d}+\frac {b^{8} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a \,b^{7} \cos \left (5 d x +5 c \right )}{10 d}-\frac {7 \sin \left (4 d x +4 c \right ) b^{6} a^{2}}{8 d}-\frac {5 \sin \left (4 d x +4 c \right ) b^{8}}{64 d}-\frac {14 a^{3} b^{5} \cos \left (3 d x +3 c \right )}{3 d}-\frac {3 a \,b^{7} \cos \left (3 d x +3 c \right )}{2 d}\)	\(499\)

Input:

int(sec(d*x+c)^2*(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
 

Output:

1/5760*((161280*a^5*b^3+268800*a^3*b^5+50400*a*b^7)*cos(2*d*x+2*c)+(50400* 
a^4*b^4+37800*a^2*b^6+1890*b^8)*sin(3*d*x+3*c)+(-13440*a^3*b^5-4032*a*b^7) 
*cos(4*d*x+4*c)+(-2520*a^2*b^6-210*b^8)*sin(5*d*x+5*c)+288*cos(6*d*x+6*c)* 
a*b^7+15*sin(7*d*x+7*c)*b^8+107520*(-3/2*a^6*b*d*x-45/8*a^4*b^3*d*x-45/16* 
a^2*b^5*d*x-15/128*b^7*d*x+a^7+22/3*a^5*b^2+8*a^3*b^4+48/35*a*b^6)*b*cos(d 
*x+c)+(5760*a^8+161280*a^6*b^2+453600*a^4*b^4+201600*a^2*b^6+7875*b^8)*sin 
(d*x+c)+46080*a^7*b+483840*a^5*b^3+604800*a^3*b^5+100800*a*b^7)/d/cos(d*x+ 
c)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.76 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {384 \, a b^{7} \cos \left (d x + c\right )^{6} + 1920 \, a^{7} b + 13440 \, a^{5} b^{3} + 13440 \, a^{3} b^{5} + 1920 \, a b^{7} - 640 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 105 \, {\left (64 \, a^{6} b^{2} + 240 \, a^{4} b^{4} + 120 \, a^{2} b^{6} + 5 \, b^{8}\right )} d x \cos \left (d x + c\right ) + 1920 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (8 \, b^{8} \cos \left (d x + c\right )^{6} + 48 \, a^{8} + 1344 \, a^{6} b^{2} + 3360 \, a^{4} b^{4} + 1344 \, a^{2} b^{6} + 48 \, b^{8} - 2 \, {\left (168 \, a^{2} b^{6} + 19 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (560 \, a^{4} b^{4} + 504 \, a^{2} b^{6} + 29 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )} \] Input:

integrate(sec(d*x+c)^2*(a+b*sin(d*x+c))^8,x, algorithm="fricas")
 

Output:

1/240*(384*a*b^7*cos(d*x + c)^6 + 1920*a^7*b + 13440*a^5*b^3 + 13440*a^3*b 
^5 + 1920*a*b^7 - 640*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^4 - 105*(64*a^6*b 
^2 + 240*a^4*b^4 + 120*a^2*b^6 + 5*b^8)*d*x*cos(d*x + c) + 1920*(7*a^5*b^3 
 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2 + 5*(8*b^8*cos(d*x + c)^6 + 48*a^8 
 + 1344*a^6*b^2 + 3360*a^4*b^4 + 1344*a^2*b^6 + 48*b^8 - 2*(168*a^2*b^6 + 
19*b^8)*cos(d*x + c)^4 + 3*(560*a^4*b^4 + 504*a^2*b^6 + 29*b^8)*cos(d*x + 
c)^2)*sin(d*x + c))/(d*cos(d*x + c))
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)**2*(a+b*sin(d*x+c))**8,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.00 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {6720 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{6} b^{2} + 8400 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{4} b^{4} + 4480 \, {\left (\cos \left (d x + c\right )^{3} - \frac {3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a^{3} b^{5} + 840 \, {\left (15 \, d x + 15 \, c - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a^{2} b^{6} - 384 \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} + \frac {5}{\cos \left (d x + c\right )} + 15 \, \cos \left (d x + c\right )\right )} a b^{7} + 5 \, {\left (105 \, d x + 105 \, c - \frac {87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} b^{8} - 13440 \, a^{5} b^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - 240 \, a^{8} \tan \left (d x + c\right ) - \frac {1920 \, a^{7} b}{\cos \left (d x + c\right )}}{240 \, d} \] Input:

integrate(sec(d*x+c)^2*(a+b*sin(d*x+c))^8,x, algorithm="maxima")
 

Output:

-1/240*(6720*(d*x + c - tan(d*x + c))*a^6*b^2 + 8400*(3*d*x + 3*c - tan(d* 
x + c)/(tan(d*x + c)^2 + 1) - 2*tan(d*x + c))*a^4*b^4 + 4480*(cos(d*x + c) 
^3 - 3/cos(d*x + c) - 6*cos(d*x + c))*a^3*b^5 + 840*(15*d*x + 15*c - (9*ta 
n(d*x + c)^3 + 7*tan(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 + 1) - 8 
*tan(d*x + c))*a^2*b^6 - 384*(cos(d*x + c)^5 - 5*cos(d*x + c)^3 + 5/cos(d* 
x + c) + 15*cos(d*x + c))*a*b^7 + 5*(105*d*x + 105*c - (87*tan(d*x + c)^5 
+ 136*tan(d*x + c)^3 + 57*tan(d*x + c))/(tan(d*x + c)^6 + 3*tan(d*x + c)^4 
 + 3*tan(d*x + c)^2 + 1) - 48*tan(d*x + c))*b^8 - 13440*a^5*b^3*(1/cos(d*x 
 + c) + cos(d*x + c)) - 240*a^8*tan(d*x + c) - 1920*a^7*b/cos(d*x + c))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (335) = 670\).

Time = 0.19 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.29 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^2*(a+b*sin(d*x+c))^8,x, algorithm="giac")
 

Output:

-1/240*(105*(64*a^6*b^2 + 240*a^4*b^4 + 120*a^2*b^6 + 5*b^8)*(d*x + c) + 4 
80*(a^8*tan(1/2*d*x + 1/2*c) + 28*a^6*b^2*tan(1/2*d*x + 1/2*c) + 70*a^4*b^ 
4*tan(1/2*d*x + 1/2*c) + 28*a^2*b^6*tan(1/2*d*x + 1/2*c) + b^8*tan(1/2*d*x 
 + 1/2*c) + 8*a^7*b + 56*a^5*b^3 + 56*a^3*b^5 + 8*a*b^7)/(tan(1/2*d*x + 1/ 
2*c)^2 - 1) + 2*(8400*a^4*b^4*tan(1/2*d*x + 1/2*c)^11 + 5880*a^2*b^6*tan(1 
/2*d*x + 1/2*c)^11 + 285*b^8*tan(1/2*d*x + 1/2*c)^11 - 13440*a^5*b^3*tan(1 
/2*d*x + 1/2*c)^10 - 13440*a^3*b^5*tan(1/2*d*x + 1/2*c)^10 - 1920*a*b^7*ta 
n(1/2*d*x + 1/2*c)^10 + 25200*a^4*b^4*tan(1/2*d*x + 1/2*c)^9 + 24360*a^2*b 
^6*tan(1/2*d*x + 1/2*c)^9 + 1295*b^8*tan(1/2*d*x + 1/2*c)^9 - 67200*a^5*b^ 
3*tan(1/2*d*x + 1/2*c)^8 - 94080*a^3*b^5*tan(1/2*d*x + 1/2*c)^8 - 13440*a* 
b^7*tan(1/2*d*x + 1/2*c)^8 + 16800*a^4*b^4*tan(1/2*d*x + 1/2*c)^7 + 18480* 
a^2*b^6*tan(1/2*d*x + 1/2*c)^7 + 1650*b^8*tan(1/2*d*x + 1/2*c)^7 - 134400* 
a^5*b^3*tan(1/2*d*x + 1/2*c)^6 - 224000*a^3*b^5*tan(1/2*d*x + 1/2*c)^6 - 4 
2240*a*b^7*tan(1/2*d*x + 1/2*c)^6 - 16800*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 
 18480*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 1650*b^8*tan(1/2*d*x + 1/2*c)^5 - 
134400*a^5*b^3*tan(1/2*d*x + 1/2*c)^4 - 241920*a^3*b^5*tan(1/2*d*x + 1/2*c 
)^4 - 49920*a*b^7*tan(1/2*d*x + 1/2*c)^4 - 25200*a^4*b^4*tan(1/2*d*x + 1/2 
*c)^3 - 24360*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 - 1295*b^8*tan(1/2*d*x + 1/2* 
c)^3 - 67200*a^5*b^3*tan(1/2*d*x + 1/2*c)^2 - 120960*a^3*b^5*tan(1/2*d*x + 
 1/2*c)^2 - 23424*a*b^7*tan(1/2*d*x + 1/2*c)^2 - 8400*a^4*b^4*tan(1/2*d...
 

Mupad [B] (verification not implemented)

Time = 17.68 (sec) , antiderivative size = 767, normalized size of antiderivative = 2.20 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx =\text {Too large to display} \] Input:

int((a + b*sin(c + d*x))^8/cos(c + d*x)^2,x)
 

Output:

(tan(c/2 + (d*x)/2)^8*(240*a^7*b + (1792*a^3*b^5)/3 + 1120*a^5*b^3) + tan( 
c/2 + (d*x)/2)^10*(96*a^7*b + 224*a^5*b^3) + tan(c/2 + (d*x)/2)*(2*a^8 + ( 
35*b^8)/8 + 105*a^2*b^6 + 210*a^4*b^4 + 56*a^6*b^2) + (256*a*b^7)/5 + 16*a 
^7*b + tan(c/2 + (d*x)/2)^2*(256*a*b^7 + 96*a^7*b + (4480*a^3*b^5)/3 + 112 
0*a^5*b^3) + tan(c/2 + (d*x)/2)^4*((2304*a*b^7)/5 + 240*a^7*b + 2688*a^3*b 
^5 + 2240*a^5*b^3) + tan(c/2 + (d*x)/2)^6*(256*a*b^7 + 320*a^7*b + (6272*a 
^3*b^5)/3 + 2240*a^5*b^3) + tan(c/2 + (d*x)/2)^13*(2*a^8 + (35*b^8)/8 + 10 
5*a^2*b^6 + 210*a^4*b^4 + 56*a^6*b^2) + tan(c/2 + (d*x)/2)^3*(12*a^8 + (24 
5*b^8)/12 + 490*a^2*b^6 + 980*a^4*b^4 + 336*a^6*b^2) + tan(c/2 + (d*x)/2)^ 
11*(12*a^8 + (245*b^8)/12 + 490*a^2*b^6 + 980*a^4*b^4 + 336*a^6*b^2) + tan 
(c/2 + (d*x)/2)^5*(30*a^8 + (791*b^8)/24 + 791*a^2*b^6 + 2030*a^4*b^4 + 84 
0*a^6*b^2) + tan(c/2 + (d*x)/2)^9*(30*a^8 + (791*b^8)/24 + 791*a^2*b^6 + 2 
030*a^4*b^4 + 840*a^6*b^2) + tan(c/2 + (d*x)/2)^7*(40*a^8 + (25*b^8)/2 + 8 
12*a^2*b^6 + 2520*a^4*b^4 + 1120*a^6*b^2) + (896*a^3*b^5)/3 + 224*a^5*b^3 
+ 16*a^7*b*tan(c/2 + (d*x)/2)^12)/(d*(5*tan(c/2 + (d*x)/2)^2 + 9*tan(c/2 + 
 (d*x)/2)^4 + 5*tan(c/2 + (d*x)/2)^6 - 5*tan(c/2 + (d*x)/2)^8 - 9*tan(c/2 
+ (d*x)/2)^10 - 5*tan(c/2 + (d*x)/2)^12 - tan(c/2 + (d*x)/2)^14 + 1)) - (7 
*b^2*atan((7*b^2*tan(c/2 + (d*x)/2)*(64*a^6 + 5*b^6 + 120*a^2*b^4 + 240*a^ 
4*b^2))/(35*b^8 + 840*a^2*b^6 + 1680*a^4*b^4 + 448*a^6*b^2))*(64*a^6 + 5*b 
^6 + 120*a^2*b^4 + 240*a^4*b^2))/(8*d)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.30 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {-1920 \cos \left (d x +c \right ) a^{7} b -26880 \cos \left (d x +c \right ) a^{5} b^{3}-35840 \cos \left (d x +c \right ) a^{3} b^{5}-6144 \cos \left (d x +c \right ) a \,b^{7}-40 \sin \left (d x +c \right )^{7} b^{8}-70 \sin \left (d x +c \right )^{5} b^{8}-175 \sin \left (d x +c \right )^{3} b^{8}+525 \sin \left (d x +c \right ) b^{8}-525 \cos \left (d x +c \right ) b^{8} c -6720 \cos \left (d x +c \right ) a^{6} b^{2} c -25200 \cos \left (d x +c \right ) a^{4} b^{4} c -12600 \cos \left (d x +c \right ) a^{2} b^{6} c -525 \cos \left (d x +c \right ) b^{8} d x -384 \sin \left (d x +c \right )^{6} a \,b^{7}-1680 \sin \left (d x +c \right )^{5} a^{2} b^{6}-4480 \sin \left (d x +c \right )^{4} a^{3} b^{5}-768 \sin \left (d x +c \right )^{4} a \,b^{7}-8400 \sin \left (d x +c \right )^{3} a^{4} b^{4}-4200 \sin \left (d x +c \right )^{3} a^{2} b^{6}-13440 \sin \left (d x +c \right )^{2} a^{5} b^{3}-17920 \sin \left (d x +c \right )^{2} a^{3} b^{5}-3072 \sin \left (d x +c \right )^{2} a \,b^{7}+6720 \sin \left (d x +c \right ) a^{6} b^{2}+25200 \sin \left (d x +c \right ) a^{4} b^{4}+12600 \sin \left (d x +c \right ) a^{2} b^{6}-6720 \cos \left (d x +c \right ) a^{6} b^{2} d x -25200 \cos \left (d x +c \right ) a^{4} b^{4} d x -12600 \cos \left (d x +c \right ) a^{2} b^{6} d x +1920 a^{7} b +26880 a^{5} b^{3}+35840 a^{3} b^{5}+6144 a \,b^{7}+240 \sin \left (d x +c \right ) a^{8}}{240 \cos \left (d x +c \right ) d} \] Input:

int(sec(d*x+c)^2*(a+b*sin(d*x+c))^8,x)
 

Output:

( - 1920*cos(c + d*x)*a**7*b - 6720*cos(c + d*x)*a**6*b**2*c - 6720*cos(c 
+ d*x)*a**6*b**2*d*x - 26880*cos(c + d*x)*a**5*b**3 - 25200*cos(c + d*x)*a 
**4*b**4*c - 25200*cos(c + d*x)*a**4*b**4*d*x - 35840*cos(c + d*x)*a**3*b* 
*5 - 12600*cos(c + d*x)*a**2*b**6*c - 12600*cos(c + d*x)*a**2*b**6*d*x - 6 
144*cos(c + d*x)*a*b**7 - 525*cos(c + d*x)*b**8*c - 525*cos(c + d*x)*b**8* 
d*x - 40*sin(c + d*x)**7*b**8 - 384*sin(c + d*x)**6*a*b**7 - 1680*sin(c + 
d*x)**5*a**2*b**6 - 70*sin(c + d*x)**5*b**8 - 4480*sin(c + d*x)**4*a**3*b* 
*5 - 768*sin(c + d*x)**4*a*b**7 - 8400*sin(c + d*x)**3*a**4*b**4 - 4200*si 
n(c + d*x)**3*a**2*b**6 - 175*sin(c + d*x)**3*b**8 - 13440*sin(c + d*x)**2 
*a**5*b**3 - 17920*sin(c + d*x)**2*a**3*b**5 - 3072*sin(c + d*x)**2*a*b**7 
 + 240*sin(c + d*x)*a**8 + 6720*sin(c + d*x)*a**6*b**2 + 25200*sin(c + d*x 
)*a**4*b**4 + 12600*sin(c + d*x)*a**2*b**6 + 525*sin(c + d*x)*b**8 + 1920* 
a**7*b + 26880*a**5*b**3 + 35840*a**3*b**5 + 6144*a*b**7)/(240*cos(c + d*x 
)*d)