Integrand size = 21, antiderivative size = 227 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {(a-7 b) (a+b)^7 \log (1-\sin (c+d x))}{4 d}+\frac {(a-b)^7 (a+7 b) \log (1+\sin (c+d x))}{4 d}+\frac {(a+b)^8}{4 d (1-\sin (c+d x))}+\frac {b^4 \left (70 a^4+56 a^2 b^2+3 b^4\right ) \sin (c+d x)}{d}+\frac {4 a b^5 \left (7 a^2+2 b^2\right ) \sin ^2(c+d x)}{d}+\frac {2 b^6 \left (14 a^2+b^2\right ) \sin ^3(c+d x)}{3 d}+\frac {2 a b^7 \sin ^4(c+d x)}{d}+\frac {b^8 \sin ^5(c+d x)}{5 d}-\frac {(a-b)^8}{4 d (1+\sin (c+d x))} \] Output:
-1/4*(a-7*b)*(a+b)^7*ln(1-sin(d*x+c))/d+1/4*(a-b)^7*(a+7*b)*ln(1+sin(d*x+c ))/d+1/4*(a+b)^8/d/(1-sin(d*x+c))+b^4*(70*a^4+56*a^2*b^2+3*b^4)*sin(d*x+c) /d+4*a*b^5*(7*a^2+2*b^2)*sin(d*x+c)^2/d+2/3*b^6*(14*a^2+b^2)*sin(d*x+c)^3/ d+2*a*b^7*sin(d*x+c)^4/d+1/5*b^8*sin(d*x+c)^5/d-1/4*(a-b)^8/d/(1+sin(d*x+c ))
Time = 2.41 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.61 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\frac {1}{2} b \left (a^2-b^2\right ) \left ((a-7 b) (a+b)^7 \log (1-\sin (c+d x))-(a-b)^7 (a+7 b) \log (1+\sin (c+d x))\right )+b^3 \left (-36 a^8-182 a^6 b^2+70 a^4 b^4+133 a^2 b^6+7 b^8\right ) \sin (c+d x)-4 a b^4 \left (21 a^6+14 a^4 b^2-22 a^2 b^4-6 b^6\right ) \sin ^2(c+d x)+\frac {7}{3} b^5 \left (-54 a^6+10 a^4 b^2+19 a^2 b^4+b^6\right ) \sin ^3(c+d x)-2 a b^6 \left (63 a^4-22 a^2 b^2-6 b^4\right ) \sin ^4(c+d x)+\frac {7}{5} b^7 \left (-60 a^4+19 a^2 b^2+b^4\right ) \sin ^5(c+d x)-4 a b^8 \left (9 a^2-2 b^2\right ) \sin ^6(c+d x)+b^9 \left (-9 a^2+b^2\right ) \sin ^7(c+d x)-a b^{10} \sin ^8(c+d x)+b \sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^9}{2 b \left (-a^2+b^2\right ) d} \] Input:
Integrate[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^8,x]
Output:
((b*(a^2 - b^2)*((a - 7*b)*(a + b)^7*Log[1 - Sin[c + d*x]] - (a - b)^7*(a + 7*b)*Log[1 + Sin[c + d*x]]))/2 + b^3*(-36*a^8 - 182*a^6*b^2 + 70*a^4*b^4 + 133*a^2*b^6 + 7*b^8)*Sin[c + d*x] - 4*a*b^4*(21*a^6 + 14*a^4*b^2 - 22*a ^2*b^4 - 6*b^6)*Sin[c + d*x]^2 + (7*b^5*(-54*a^6 + 10*a^4*b^2 + 19*a^2*b^4 + b^6)*Sin[c + d*x]^3)/3 - 2*a*b^6*(63*a^4 - 22*a^2*b^2 - 6*b^4)*Sin[c + d*x]^4 + (7*b^7*(-60*a^4 + 19*a^2*b^2 + b^4)*Sin[c + d*x]^5)/5 - 4*a*b^8*( 9*a^2 - 2*b^2)*Sin[c + d*x]^6 + b^9*(-9*a^2 + b^2)*Sin[c + d*x]^7 - a*b^10 *Sin[c + d*x]^8 + b*Sec[c + d*x]^2*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x ])^9)/(2*b*(-a^2 + b^2)*d)
Time = 0.48 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3147, 477, 2009}
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.
| \(\displaystyle \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^8}{\cos (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {b^3 \int \frac {(a+b \sin (c+d x))^8}{\left (b^2-b^2 \sin ^2(c+d x)\right )^2}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {\int \left (\frac {b^2 (a-b)^8}{4 (\sin (c+d x) b+b)^2}+\frac {b (a+7 b) (a-b)^7}{4 (\sin (c+d x) b+b)}+b^8 \sin ^4(c+d x)+8 a b^7 \sin ^3(c+d x)+2 b^6 \left (14 a^2+b^2\right ) \sin ^2(c+d x)+b^4 \left (70 a^4+56 b^2 a^2+3 b^4\right )+8 a b^5 \left (7 a^2+2 b^2\right ) \sin (c+d x)+\frac {(a-7 b) b (a+b)^7}{4 (b-b \sin (c+d x))}+\frac {b^2 (a+b)^8}{4 (b-b \sin (c+d x))^2}\right )d(b \sin (c+d x))}{b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2}{3} b^7 \left (14 a^2+b^2\right ) \sin ^3(c+d x)+4 a b^6 \left (7 a^2+2 b^2\right ) \sin ^2(c+d x)+b^5 \left (70 a^4+56 a^2 b^2+3 b^4\right ) \sin (c+d x)+2 a b^8 \sin ^4(c+d x)+\frac {b^2 (a+b)^8}{4 (b-b \sin (c+d x))}-\frac {b^2 (a-b)^8}{4 (b \sin (c+d x)+b)}-\frac {1}{4} b (a-7 b) (a+b)^7 \log (b-b \sin (c+d x))+\frac {1}{4} b (a-b)^7 (a+7 b) \log (b \sin (c+d x)+b)+\frac {1}{5} b^9 \sin ^5(c+d x)}{b d}\) |
Input:
Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^8,x]
Output:
(-1/4*((a - 7*b)*b*(a + b)^7*Log[b - b*Sin[c + d*x]]) + ((a - b)^7*b*(a + 7*b)*Log[b + b*Sin[c + d*x]])/4 + b^5*(70*a^4 + 56*a^2*b^2 + 3*b^4)*Sin[c + d*x] + 4*a*b^6*(7*a^2 + 2*b^2)*Sin[c + d*x]^2 + (2*b^7*(14*a^2 + b^2)*Si n[c + d*x]^3)/3 + 2*a*b^8*Sin[c + d*x]^4 + (b^9*Sin[c + d*x]^5)/5 + (b^2*( a + b)^8)/(4*(b - b*Sin[c + d*x])) - ((a - b)^8*b^2)/(4*(b + b*Sin[c + d*x ])))/(b*d)
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ a^p Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 ]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & & NiceSqrtQ[-b/a] && !FractionalPowerFactorQ[Rt[-b/a, 2]]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 2.20 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.57
| method | result | size |
| parallelrisch | \(\frac {-26880 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (a^{4}+2 b^{2} a^{2}+\frac {3}{7} b^{4}\right ) a \,b^{3} \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-240 \left (a -7 b \right ) \left (a +b \right )^{7} \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+240 \left (a +7 b \right ) \left (a -b \right )^{7} \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-1920 a \left (a^{6}+7 a^{4} b^{2}+7 a^{2} b^{4}+\frac {33}{32} b^{6}\right ) b \cos \left (2 d x +2 c \right )+\left (16800 a^{4} b^{4}+14000 b^{6} a^{2}+763 b^{8}\right ) \sin \left (3 d x +3 c \right )+\left (-3360 a^{3} b^{5}-1080 a \,b^{7}\right ) \cos \left (4 d x +4 c \right )+\left (-560 b^{6} a^{2}-49 b^{8}\right ) \sin \left (5 d x +5 c \right )+60 \cos \left (6 d x +6 c \right ) a \,b^{7}+3 \sin \left (7 d x +7 c \right ) b^{8}+\left (480 a^{8}+13440 a^{6} b^{2}+50400 a^{4} b^{4}+28000 b^{6} a^{2}+1295 b^{8}\right ) \sin \left (d x +c \right )+1920 a^{7} b +13440 a^{5} b^{3}+16800 a^{3} b^{5}+3000 a \,b^{7}}{480 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(356\) |
| derivativedivides | \(\frac {a^{8} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {4 a^{7} b}{\cos \left (d x +c \right )^{2}}+28 a^{6} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+56 a^{5} b^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+70 a^{4} b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{3}}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+56 a^{3} b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )+28 b^{6} a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+8 a \,b^{7} \left (\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{6}}{2}+\frac {3 \sin \left (d x +c \right )^{4}}{4}+\frac {3 \sin \left (d x +c \right )^{2}}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+b^{8} \left (\frac {\sin \left (d x +c \right )^{9}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{7}}{2}+\frac {7 \sin \left (d x +c \right )^{5}}{10}+\frac {7 \sin \left (d x +c \right )^{3}}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(459\) |
| default | \(\frac {a^{8} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {4 a^{7} b}{\cos \left (d x +c \right )^{2}}+28 a^{6} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+56 a^{5} b^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+70 a^{4} b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{3}}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+56 a^{3} b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )+28 b^{6} a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+8 a \,b^{7} \left (\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{6}}{2}+\frac {3 \sin \left (d x +c \right )^{4}}{4}+\frac {3 \sin \left (d x +c \right )^{2}}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+b^{8} \left (\frac {\sin \left (d x +c \right )^{9}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{7}}{2}+\frac {7 \sin \left (d x +c \right )^{5}}{10}+\frac {7 \sin \left (d x +c \right )^{3}}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(459\) |
| risch | \(\text {Expression too large to display}\) | \(1031\) |
Input:
int(sec(d*x+c)^3*(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
1/480*(-26880*(1+cos(2*d*x+2*c))*(a^4+2*b^2*a^2+3/7*b^4)*a*b^3*ln(sec(1/2* d*x+1/2*c)^2)-240*(a-7*b)*(a+b)^7*(1+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c) -1)+240*(a+7*b)*(a-b)^7*(1+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)+1)-1920*a *(a^6+7*a^4*b^2+7*a^2*b^4+33/32*b^6)*b*cos(2*d*x+2*c)+(16800*a^4*b^4+14000 *a^2*b^6+763*b^8)*sin(3*d*x+3*c)+(-3360*a^3*b^5-1080*a*b^7)*cos(4*d*x+4*c) +(-560*a^2*b^6-49*b^8)*sin(5*d*x+5*c)+60*cos(6*d*x+6*c)*a*b^7+3*sin(7*d*x+ 7*c)*b^8+(480*a^8+13440*a^6*b^2+50400*a^4*b^4+28000*a^2*b^6+1295*b^8)*sin( d*x+c)+1920*a^7*b+13440*a^5*b^3+16800*a^3*b^5+3000*a*b^7)/d/(1+cos(2*d*x+2 *c))
Time = 0.14 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.62 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {120 \, a b^{7} \cos \left (d x + c\right )^{6} + 240 \, a^{7} b + 1680 \, a^{5} b^{3} + 1680 \, a^{3} b^{5} + 240 \, a b^{7} - 240 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (a^{8} - 28 \, a^{6} b^{2} + 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} + 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} + 48 \, a b^{7} - 7 \, b^{8}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (a^{8} - 28 \, a^{6} b^{2} - 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} - 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} - 48 \, a b^{7} - 7 \, b^{8}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (8 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, b^{8} \cos \left (d x + c\right )^{6} + 15 \, a^{8} + 420 \, a^{6} b^{2} + 1050 \, a^{4} b^{4} + 420 \, a^{2} b^{6} + 15 \, b^{8} - 8 \, {\left (35 \, a^{2} b^{6} + 4 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (525 \, a^{4} b^{4} + 490 \, a^{2} b^{6} + 29 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{2}} \] Input:
integrate(sec(d*x+c)^3*(a+b*sin(d*x+c))^8,x, algorithm="fricas")
Output:
1/60*(120*a*b^7*cos(d*x + c)^6 + 240*a^7*b + 1680*a^5*b^3 + 1680*a^3*b^5 + 240*a*b^7 - 240*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^4 + 15*(a^8 - 28*a^6*b ^2 + 112*a^5*b^3 - 210*a^4*b^4 + 224*a^3*b^5 - 140*a^2*b^6 + 48*a*b^7 - 7* b^8)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - 15*(a^8 - 28*a^6*b^2 - 112*a^5 *b^3 - 210*a^4*b^4 - 224*a^3*b^5 - 140*a^2*b^6 - 48*a*b^7 - 7*b^8)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 105*(8*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2 + 2*(6*b^8*cos(d*x + c)^6 + 15*a^8 + 420*a^6*b^2 + 1050*a^4*b^4 + 420*a^2* b^6 + 15*b^8 - 8*(35*a^2*b^6 + 4*b^8)*cos(d*x + c)^4 + 4*(525*a^4*b^4 + 49 0*a^2*b^6 + 29*b^8)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^2)
Timed out. \[ \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**3*(a+b*sin(d*x+c))**8,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.42 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {12 \, b^{8} \sin \left (d x + c\right )^{5} + 120 \, a b^{7} \sin \left (d x + c\right )^{4} + 40 \, {\left (14 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{3} + 240 \, {\left (7 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \sin \left (d x + c\right )^{2} + 15 \, {\left (a^{8} - 28 \, a^{6} b^{2} + 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} + 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} + 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (a^{8} - 28 \, a^{6} b^{2} - 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} - 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} - 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 60 \, {\left (70 \, a^{4} b^{4} + 56 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (d x + c\right ) - \frac {30 \, {\left (8 \, a^{7} b + 56 \, a^{5} b^{3} + 56 \, a^{3} b^{5} + 8 \, a b^{7} + {\left (a^{8} + 28 \, a^{6} b^{2} + 70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{2} - 1}}{60 \, d} \] Input:
integrate(sec(d*x+c)^3*(a+b*sin(d*x+c))^8,x, algorithm="maxima")
Output:
1/60*(12*b^8*sin(d*x + c)^5 + 120*a*b^7*sin(d*x + c)^4 + 40*(14*a^2*b^6 + b^8)*sin(d*x + c)^3 + 240*(7*a^3*b^5 + 2*a*b^7)*sin(d*x + c)^2 + 15*(a^8 - 28*a^6*b^2 + 112*a^5*b^3 - 210*a^4*b^4 + 224*a^3*b^5 - 140*a^2*b^6 + 48*a *b^7 - 7*b^8)*log(sin(d*x + c) + 1) - 15*(a^8 - 28*a^6*b^2 - 112*a^5*b^3 - 210*a^4*b^4 - 224*a^3*b^5 - 140*a^2*b^6 - 48*a*b^7 - 7*b^8)*log(sin(d*x + c) - 1) + 60*(70*a^4*b^4 + 56*a^2*b^6 + 3*b^8)*sin(d*x + c) - 30*(8*a^7*b + 56*a^5*b^3 + 56*a^3*b^5 + 8*a*b^7 + (a^8 + 28*a^6*b^2 + 70*a^4*b^4 + 28 *a^2*b^6 + b^8)*sin(d*x + c))/(sin(d*x + c)^2 - 1))/d
Time = 0.15 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.72 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {{\left (a^{8} - 28 \, a^{6} b^{2} + 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} + 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} + 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{4 \, d} - \frac {{\left (a^{8} - 28 \, a^{6} b^{2} - 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} - 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} - 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{4 \, d} - \frac {8 \, a^{7} b + 56 \, a^{5} b^{3} + 56 \, a^{3} b^{5} + 8 \, a b^{7} + {\left (a^{8} + 28 \, a^{6} b^{2} + 70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )}{2 \, d {\left (\sin \left (d x + c\right ) + 1\right )} {\left (\sin \left (d x + c\right ) - 1\right )}} + \frac {3 \, b^{8} d^{4} \sin \left (d x + c\right )^{5} + 30 \, a b^{7} d^{4} \sin \left (d x + c\right )^{4} + 140 \, a^{2} b^{6} d^{4} \sin \left (d x + c\right )^{3} + 10 \, b^{8} d^{4} \sin \left (d x + c\right )^{3} + 420 \, a^{3} b^{5} d^{4} \sin \left (d x + c\right )^{2} + 120 \, a b^{7} d^{4} \sin \left (d x + c\right )^{2} + 1050 \, a^{4} b^{4} d^{4} \sin \left (d x + c\right ) + 840 \, a^{2} b^{6} d^{4} \sin \left (d x + c\right ) + 45 \, b^{8} d^{4} \sin \left (d x + c\right )}{15 \, d^{5}} \] Input:
integrate(sec(d*x+c)^3*(a+b*sin(d*x+c))^8,x, algorithm="giac")
Output:
1/4*(a^8 - 28*a^6*b^2 + 112*a^5*b^3 - 210*a^4*b^4 + 224*a^3*b^5 - 140*a^2* b^6 + 48*a*b^7 - 7*b^8)*log(abs(sin(d*x + c) + 1))/d - 1/4*(a^8 - 28*a^6*b ^2 - 112*a^5*b^3 - 210*a^4*b^4 - 224*a^3*b^5 - 140*a^2*b^6 - 48*a*b^7 - 7* b^8)*log(abs(sin(d*x + c) - 1))/d - 1/2*(8*a^7*b + 56*a^5*b^3 + 56*a^3*b^5 + 8*a*b^7 + (a^8 + 28*a^6*b^2 + 70*a^4*b^4 + 28*a^2*b^6 + b^8)*sin(d*x + c))/(d*(sin(d*x + c) + 1)*(sin(d*x + c) - 1)) + 1/15*(3*b^8*d^4*sin(d*x + c)^5 + 30*a*b^7*d^4*sin(d*x + c)^4 + 140*a^2*b^6*d^4*sin(d*x + c)^3 + 10*b ^8*d^4*sin(d*x + c)^3 + 420*a^3*b^5*d^4*sin(d*x + c)^2 + 120*a*b^7*d^4*sin (d*x + c)^2 + 1050*a^4*b^4*d^4*sin(d*x + c) + 840*a^2*b^6*d^4*sin(d*x + c) + 45*b^8*d^4*sin(d*x + c))/d^5
Time = 15.42 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.13 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {28\,a^2\,b^6}{3}+\frac {2\,b^8}{3}\right )}{d}+\frac {b^8\,{\sin \left (c+d\,x\right )}^5}{5\,d}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (28\,a^3\,b^5+8\,a\,b^7\right )}{d}+\frac {\sin \left (c+d\,x\right )\,\left (70\,a^4\,b^4+56\,a^2\,b^6+3\,b^8\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {a^8}{2}+14\,a^6\,b^2+35\,a^4\,b^4+14\,a^2\,b^6+\frac {b^8}{2}\right )+4\,a\,b^7+4\,a^7\,b+28\,a^3\,b^5+28\,a^5\,b^3}{d\,\left ({\sin \left (c+d\,x\right )}^2-1\right )}+\frac {2\,a\,b^7\,{\sin \left (c+d\,x\right )}^4}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^7\,\left (a-7\,b\right )}{4\,d}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^7\,\left (a+7\,b\right )}{4\,d} \] Input:
int((a + b*sin(c + d*x))^8/cos(c + d*x)^3,x)
Output:
(sin(c + d*x)^3*((2*b^8)/3 + (28*a^2*b^6)/3))/d + (b^8*sin(c + d*x)^5)/(5* d) + (sin(c + d*x)^2*(8*a*b^7 + 28*a^3*b^5))/d + (sin(c + d*x)*(3*b^8 + 56 *a^2*b^6 + 70*a^4*b^4))/d - (sin(c + d*x)*(a^8/2 + b^8/2 + 14*a^2*b^6 + 35 *a^4*b^4 + 14*a^6*b^2) + 4*a*b^7 + 4*a^7*b + 28*a^3*b^5 + 28*a^5*b^3)/(d*( sin(c + d*x)^2 - 1)) + (2*a*b^7*sin(c + d*x)^4)/d - (log(sin(c + d*x) - 1) *(a + b)^7*(a - 7*b))/(4*d) + (log(sin(c + d*x) + 1)*(a - b)^7*(a + 7*b))/ (4*d)
Time = 0.17 (sec) , antiderivative size = 1161, normalized size of antiderivative = 5.11 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)^3*(a+b*sin(d*x+c))^8,x)
Output:
( - 1680*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**5*b**3 - 3360*log (tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**3*b**5 - 720*log(tan((c + d*x )/2)**2 + 1)*sin(c + d*x)**2*a*b**7 + 1680*log(tan((c + d*x)/2)**2 + 1)*a* *5*b**3 + 3360*log(tan((c + d*x)/2)**2 + 1)*a**3*b**5 + 720*log(tan((c + d *x)/2)**2 + 1)*a*b**7 - 15*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**8 + 420*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**6*b**2 + 1680*log(tan(( c + d*x)/2) - 1)*sin(c + d*x)**2*a**5*b**3 + 3150*log(tan((c + d*x)/2) - 1 )*sin(c + d*x)**2*a**4*b**4 + 3360*log(tan((c + d*x)/2) - 1)*sin(c + d*x)* *2*a**3*b**5 + 2100*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**2*b**6 + 720*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a*b**7 + 105*log(tan((c + d* x)/2) - 1)*sin(c + d*x)**2*b**8 + 15*log(tan((c + d*x)/2) - 1)*a**8 - 420* log(tan((c + d*x)/2) - 1)*a**6*b**2 - 1680*log(tan((c + d*x)/2) - 1)*a**5* b**3 - 3150*log(tan((c + d*x)/2) - 1)*a**4*b**4 - 3360*log(tan((c + d*x)/2 ) - 1)*a**3*b**5 - 2100*log(tan((c + d*x)/2) - 1)*a**2*b**6 - 720*log(tan( (c + d*x)/2) - 1)*a*b**7 - 105*log(tan((c + d*x)/2) - 1)*b**8 + 15*log(tan ((c + d*x)/2) + 1)*sin(c + d*x)**2*a**8 - 420*log(tan((c + d*x)/2) + 1)*si n(c + d*x)**2*a**6*b**2 + 1680*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a **5*b**3 - 3150*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**4*b**4 + 3360 *log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**3*b**5 - 2100*log(tan((c + d *x)/2) + 1)*sin(c + d*x)**2*a**2*b**6 + 720*log(tan((c + d*x)/2) + 1)*s...