Integrand size = 19, antiderivative size = 245 \[ \int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {(a+b)^8 \log (1-\sin (c+d x))}{2 d}+\frac {(a-b)^8 \log (1+\sin (c+d x))}{2 d}-\frac {b^2 \left (28 a^6+70 a^4 b^2+28 a^2 b^4+b^6\right ) \sin (c+d x)}{d}-\frac {4 a b^3 \left (7 a^4+7 a^2 b^2+b^4\right ) \sin ^2(c+d x)}{d}-\frac {b^4 \left (70 a^4+28 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{3 d}-\frac {2 a b^5 \left (7 a^2+b^2\right ) \sin ^4(c+d x)}{d}-\frac {b^6 \left (28 a^2+b^2\right ) \sin ^5(c+d x)}{5 d}-\frac {4 a b^7 \sin ^6(c+d x)}{3 d}-\frac {b^8 \sin ^7(c+d x)}{7 d} \] Output:
-1/2*(a+b)^8*ln(1-sin(d*x+c))/d+1/2*(a-b)^8*ln(1+sin(d*x+c))/d-b^2*(28*a^6 +70*a^4*b^2+28*a^2*b^4+b^6)*sin(d*x+c)/d-4*a*b^3*(7*a^4+7*a^2*b^2+b^4)*sin (d*x+c)^2/d-1/3*b^4*(70*a^4+28*a^2*b^2+b^4)*sin(d*x+c)^3/d-2*a*b^5*(7*a^2+ b^2)*sin(d*x+c)^4/d-1/5*b^6*(28*a^2+b^2)*sin(d*x+c)^5/d-4/3*a*b^7*sin(d*x+ c)^6/d-1/7*b^8*sin(d*x+c)^7/d
Time = 0.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.93 \[ \int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {b \left (-\frac {(a+b)^8 \log (1-\sin (c+d x))}{2 b}+\frac {(a-b)^8 \log (1+\sin (c+d x))}{2 b}-b \left (28 a^6+70 a^4 b^2+28 a^2 b^4+b^6\right ) \sin (c+d x)-4 a b^2 \left (7 a^4+7 a^2 b^2+b^4\right ) \sin ^2(c+d x)-\frac {1}{3} b^3 \left (70 a^4+28 a^2 b^2+b^4\right ) \sin ^3(c+d x)-2 a b^4 \left (7 a^2+b^2\right ) \sin ^4(c+d x)-\frac {1}{5} b^5 \left (28 a^2+b^2\right ) \sin ^5(c+d x)-\frac {4}{3} a b^6 \sin ^6(c+d x)-\frac {1}{7} b^7 \sin ^7(c+d x)\right )}{d} \] Input:
Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^8,x]
Output:
(b*(-1/2*((a + b)^8*Log[1 - Sin[c + d*x]])/b + ((a - b)^8*Log[1 + Sin[c + d*x]])/(2*b) - b*(28*a^6 + 70*a^4*b^2 + 28*a^2*b^4 + b^6)*Sin[c + d*x] - 4 *a*b^2*(7*a^4 + 7*a^2*b^2 + b^4)*Sin[c + d*x]^2 - (b^3*(70*a^4 + 28*a^2*b^ 2 + b^4)*Sin[c + d*x]^3)/3 - 2*a*b^4*(7*a^2 + b^2)*Sin[c + d*x]^4 - (b^5*( 28*a^2 + b^2)*Sin[c + d*x]^5)/5 - (4*a*b^6*Sin[c + d*x]^6)/3 - (b^7*Sin[c + d*x]^7)/7))/d
Time = 0.45 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, 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.
| \(\displaystyle \int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^8}{\cos (c+d x)}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {b \int \frac {(a+b \sin (c+d x))^8}{b^2-b^2 \sin ^2(c+d x)}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {\int \left (\frac {b (a-b)^8}{2 (\sin (c+d x) b+b)}-b^8 \sin ^6(c+d x)-8 a b^7 \sin ^5(c+d x)-b^6 \left (28 a^2+b^2\right ) \sin ^4(c+d x)-8 a b^5 \left (7 a^2+b^2\right ) \sin ^3(c+d x)-b^4 \left (70 a^4+28 b^2 a^2+b^4\right ) \sin ^2(c+d x)-b^2 \left (28 a^6+70 b^2 a^4+28 b^4 a^2+b^6\right )-8 a b^3 \left (7 a^4+7 b^2 a^2+b^4\right ) \sin (c+d x)+\frac {b (a+b)^8}{2 (b-b \sin (c+d x))}\right )d(b \sin (c+d x))}{b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{5} b^7 \left (28 a^2+b^2\right ) \sin ^5(c+d x)-2 a b^6 \left (7 a^2+b^2\right ) \sin ^4(c+d x)-4 a b^4 \left (7 a^4+7 a^2 b^2+b^4\right ) \sin ^2(c+d x)-\frac {1}{3} b^5 \left (70 a^4+28 a^2 b^2+b^4\right ) \sin ^3(c+d x)-b^3 \left (28 a^6+70 a^4 b^2+28 a^2 b^4+b^6\right ) \sin (c+d x)-\frac {4}{3} a b^8 \sin ^6(c+d x)-\frac {1}{2} b (a+b)^8 \log (b-b \sin (c+d x))+\frac {1}{2} b (a-b)^8 \log (b \sin (c+d x)+b)-\frac {1}{7} b^9 \sin ^7(c+d x)}{b d}\) |
Input:
Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^8,x]
Output:
(-1/2*(b*(a + b)^8*Log[b - b*Sin[c + d*x]]) + ((a - b)^8*b*Log[b + b*Sin[c + d*x]])/2 - b^3*(28*a^6 + 70*a^4*b^2 + 28*a^2*b^4 + b^6)*Sin[c + d*x] - 4*a*b^4*(7*a^4 + 7*a^2*b^2 + b^4)*Sin[c + d*x]^2 - (b^5*(70*a^4 + 28*a^2*b ^2 + b^4)*Sin[c + d*x]^3)/3 - 2*a*b^6*(7*a^2 + b^2)*Sin[c + d*x]^4 - (b^7* (28*a^2 + b^2)*Sin[c + d*x]^5)/5 - (4*a*b^8*Sin[c + d*x]^6)/3 - (b^9*Sin[c + d*x]^7)/7)/(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 = 1.36 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.16
| method | result | size |
| parallelrisch | \(\frac {53760 a b \left (a^{2}+b^{2}\right ) \left (a^{4}+6 b^{2} a^{2}+b^{4}\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-6720 \left (a +b \right )^{8} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+6720 \left (a -b \right )^{8} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-188160 \left (\left (-\frac {1}{2} a^{5} b -\frac {3}{4} a^{3} b^{3}-\frac {29}{224} a \,b^{5}\right ) \cos \left (2 d x +2 c \right )+\left (-\frac {5}{24} a^{4} b^{2}-\frac {7}{48} a^{2} b^{4}-\frac {37}{5376} b^{6}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {1}{16} a^{3} b^{3}+\frac {1}{56} a \,b^{5}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {1}{80} a^{2} b^{4}+\frac {9}{8960} b^{6}\right ) \sin \left (5 d x +5 c \right )-\frac {a \,b^{5} \cos \left (6 d x +6 c \right )}{672}-\frac {b^{6} \sin \left (7 d x +7 c \right )}{12544}+\left (a^{6}+\frac {25}{8} a^{4} b^{2}+\frac {11}{8} a^{2} b^{4}+\frac {93}{1792} b^{6}\right ) \sin \left (d x +c \right )-\frac {19 a^{5} b}{42}-\frac {31 a^{3} b^{3}}{48}-\frac {13 a \,b^{5}}{168}\right ) b^{2}}{6720 d}\) | \(284\) |
| derivativedivides | \(\frac {a^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-8 a^{7} b \ln \left (\cos \left (d x +c \right )\right )+28 a^{6} b^{2} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+56 a^{5} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+70 a^{4} b^{4} \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+56 a^{3} b^{5} \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+28 b^{6} a^{2} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+8 a \,b^{7} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+b^{8} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(329\) |
| default | \(\frac {a^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-8 a^{7} b \ln \left (\cos \left (d x +c \right )\right )+28 a^{6} b^{2} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+56 a^{5} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+70 a^{4} b^{4} \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+56 a^{3} b^{5} \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+28 b^{6} a^{2} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+8 a \,b^{7} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+b^{8} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(329\) |
| norman | \(\frac {-\frac {8 \left (14 a^{5} b^{3}+14 a^{3} b^{5}+2 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {8 \left (14 a^{5} b^{3}+14 a^{3} b^{5}+2 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}-\frac {4 \left (168 a^{5} b^{3}+224 a^{3} b^{5}+32 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {4 \left (168 a^{5} b^{3}+224 a^{3} b^{5}+32 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}-\frac {8 \left (630 a^{5} b^{3}+966 a^{3} b^{5}+170 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {8 \left (630 a^{5} b^{3}+966 a^{3} b^{5}+170 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 d}-\frac {2 \left (3360 a^{5} b^{3}+5376 a^{3} b^{5}+1024 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}-\frac {2 b^{2} \left (28 a^{6}+70 a^{4} b^{2}+28 a^{2} b^{4}+b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 b^{2} \left (28 a^{6}+70 a^{4} b^{2}+28 a^{2} b^{4}+b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{d}-\frac {2 b^{2} \left (588 a^{6}+1750 a^{4} b^{2}+700 a^{2} b^{4}+25 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {2 b^{2} \left (588 a^{6}+1750 a^{4} b^{2}+700 a^{2} b^{4}+25 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{3 d}-\frac {2 b^{2} \left (8820 a^{6}+29050 a^{4} b^{2}+12964 a^{2} b^{4}+463 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}-\frac {2 b^{2} \left (8820 a^{6}+29050 a^{4} b^{2}+12964 a^{2} b^{4}+463 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{15 d}-\frac {2 b^{2} \left (102900 a^{6}+355250 a^{4} b^{2}+170324 a^{2} b^{4}+7043 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{105 d}-\frac {2 b^{2} \left (102900 a^{6}+355250 a^{4} b^{2}+170324 a^{2} b^{4}+7043 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{105 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}+\frac {\left (a^{8}-8 a^{7} b +28 a^{6} b^{2}-56 a^{5} b^{3}+70 a^{4} b^{4}-56 a^{3} b^{5}+28 b^{6} a^{2}-8 a \,b^{7}+b^{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {\left (a^{8}+8 a^{7} b +28 a^{6} b^{2}+56 a^{5} b^{3}+70 a^{4} b^{4}+56 a^{3} b^{5}+28 b^{6} a^{2}+8 a \,b^{7}+b^{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {8 a b \left (a^{6}+7 a^{4} b^{2}+7 a^{2} b^{4}+b^{6}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(849\) |
| risch | \(\frac {b^{8} \sin \left (7 d x +7 c \right )}{448 d}+\frac {a \,b^{7} \cos \left (6 d x +6 c \right )}{24 d}+\frac {14 a^{5} b^{3} \cos \left (2 d x +2 c \right )}{d}+\frac {21 a^{3} b^{5} \cos \left (2 d x +2 c \right )}{d}+\frac {29 a \,b^{7} \cos \left (2 d x +2 c \right )}{8 d}+\frac {16 i a^{7} b c}{d}+\frac {112 i a^{5} b^{3} c}{d}+\frac {112 i a^{3} b^{5} c}{d}+\frac {16 i a \,b^{7} c}{d}-\frac {14 i {\mathrm e}^{-i \left (d x +c \right )} a^{6} b^{2}}{d}-\frac {9 \sin \left (5 d x +5 c \right ) b^{8}}{320 d}-\frac {7 \sin \left (5 d x +5 c \right ) b^{6} a^{2}}{20 d}-\frac {7 a^{3} b^{5} \cos \left (4 d x +4 c \right )}{4 d}-\frac {a \,b^{7} \cos \left (4 d x +4 c \right )}{2 d}+\frac {35 \sin \left (3 d x +3 c \right ) a^{4} b^{4}}{6 d}+\frac {49 \sin \left (3 d x +3 c \right ) b^{6} a^{2}}{12 d}+\frac {37 \sin \left (3 d x +3 c \right ) b^{8}}{192 d}+8 i a^{7} b x +56 i a^{5} b^{3} x +56 i a^{3} b^{5} x +8 i a \,b^{7} x -\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{7} b}{d}+\frac {28 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{6} b^{2}}{d}-\frac {56 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{5} b^{3}}{d}+\frac {70 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{4} b^{4}}{d}-\frac {56 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{3} b^{5}}{d}+\frac {28 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{6} a^{2}}{d}-\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a \,b^{7}}{d}-\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{7} b}{d}-\frac {28 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{6} b^{2}}{d}-\frac {56 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{5} b^{3}}{d}-\frac {70 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{4} b^{4}}{d}-\frac {56 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{3} b^{5}}{d}-\frac {28 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{6} a^{2}}{d}-\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a \,b^{7}}{d}-\frac {93 i {\mathrm e}^{-i \left (d x +c \right )} b^{8}}{128 d}+\frac {93 i {\mathrm e}^{i \left (d x +c \right )} b^{8}}{128 d}-\frac {a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{8}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{8}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{8}}{d}+\frac {77 i {\mathrm e}^{i \left (d x +c \right )} b^{6} a^{2}}{4 d}+\frac {14 i {\mathrm e}^{i \left (d x +c \right )} a^{6} b^{2}}{d}+\frac {175 i {\mathrm e}^{i \left (d x +c \right )} a^{4} b^{4}}{4 d}-\frac {175 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} b^{4}}{4 d}-\frac {77 i {\mathrm e}^{-i \left (d x +c \right )} b^{6} a^{2}}{4 d}\) | \(883\) |
Input:
int(sec(d*x+c)*(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
1/6720*(53760*a*b*(a^2+b^2)*(a^4+6*a^2*b^2+b^4)*ln(sec(1/2*d*x+1/2*c)^2)-6 720*(a+b)^8*ln(tan(1/2*d*x+1/2*c)-1)+6720*(a-b)^8*ln(tan(1/2*d*x+1/2*c)+1) -188160*((-1/2*a^5*b-3/4*a^3*b^3-29/224*a*b^5)*cos(2*d*x+2*c)+(-5/24*a^4*b ^2-7/48*a^2*b^4-37/5376*b^6)*sin(3*d*x+3*c)+(1/16*a^3*b^3+1/56*a*b^5)*cos( 4*d*x+4*c)+(1/80*a^2*b^4+9/8960*b^6)*sin(5*d*x+5*c)-1/672*a*b^5*cos(6*d*x+ 6*c)-1/12544*b^6*sin(7*d*x+7*c)+(a^6+25/8*a^4*b^2+11/8*a^2*b^4+93/1792*b^6 )*sin(d*x+c)-19/42*a^5*b-31/48*a^3*b^3-13/168*a*b^5)*b^2)/d
Time = 0.14 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.33 \[ \int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {280 \, a b^{7} \cos \left (d x + c\right )^{6} - 420 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 840 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (a^{8} - 8 \, a^{7} b + 28 \, a^{6} b^{2} - 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} - 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} - 8 \, a b^{7} + b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (a^{8} + 8 \, a^{7} b + 28 \, a^{6} b^{2} + 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} + 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} + 8 \, a b^{7} + b^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15 \, b^{8} \cos \left (d x + c\right )^{6} - 2940 \, a^{6} b^{2} - 9800 \, a^{4} b^{4} - 4508 \, a^{2} b^{6} - 176 \, b^{8} - 6 \, {\left (98 \, a^{2} b^{6} + 11 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (1225 \, a^{4} b^{4} + 1078 \, a^{2} b^{6} + 61 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{210 \, d} \] Input:
integrate(sec(d*x+c)*(a+b*sin(d*x+c))^8,x, algorithm="fricas")
Output:
1/210*(280*a*b^7*cos(d*x + c)^6 - 420*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^4 + 840*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2 + 105*(a^8 - 8*a^ 7*b + 28*a^6*b^2 - 56*a^5*b^3 + 70*a^4*b^4 - 56*a^3*b^5 + 28*a^2*b^6 - 8*a *b^7 + b^8)*log(sin(d*x + c) + 1) - 105*(a^8 + 8*a^7*b + 28*a^6*b^2 + 56*a ^5*b^3 + 70*a^4*b^4 + 56*a^3*b^5 + 28*a^2*b^6 + 8*a*b^7 + b^8)*log(-sin(d* x + c) + 1) + 2*(15*b^8*cos(d*x + c)^6 - 2940*a^6*b^2 - 9800*a^4*b^4 - 450 8*a^2*b^6 - 176*b^8 - 6*(98*a^2*b^6 + 11*b^8)*cos(d*x + c)^4 + 2*(1225*a^4 *b^4 + 1078*a^2*b^6 + 61*b^8)*cos(d*x + c)^2)*sin(d*x + c))/d
Timed out. \[ \int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)*(a+b*sin(d*x+c))**8,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.29 \[ \int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {30 \, b^{8} \sin \left (d x + c\right )^{7} + 280 \, a b^{7} \sin \left (d x + c\right )^{6} + 42 \, {\left (28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{5} + 420 \, {\left (7 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{4} + 70 \, {\left (70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{3} + 840 \, {\left (7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{2} - 105 \, {\left (a^{8} - 8 \, a^{7} b + 28 \, a^{6} b^{2} - 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} - 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} - 8 \, a b^{7} + b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (a^{8} + 8 \, a^{7} b + 28 \, a^{6} b^{2} + 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} + 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} + 8 \, a b^{7} + b^{8}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, {\left (28 \, a^{6} b^{2} + 70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )}{210 \, d} \] Input:
integrate(sec(d*x+c)*(a+b*sin(d*x+c))^8,x, algorithm="maxima")
Output:
-1/210*(30*b^8*sin(d*x + c)^7 + 280*a*b^7*sin(d*x + c)^6 + 42*(28*a^2*b^6 + b^8)*sin(d*x + c)^5 + 420*(7*a^3*b^5 + a*b^7)*sin(d*x + c)^4 + 70*(70*a^ 4*b^4 + 28*a^2*b^6 + b^8)*sin(d*x + c)^3 + 840*(7*a^5*b^3 + 7*a^3*b^5 + a* b^7)*sin(d*x + c)^2 - 105*(a^8 - 8*a^7*b + 28*a^6*b^2 - 56*a^5*b^3 + 70*a^ 4*b^4 - 56*a^3*b^5 + 28*a^2*b^6 - 8*a*b^7 + b^8)*log(sin(d*x + c) + 1) + 1 05*(a^8 + 8*a^7*b + 28*a^6*b^2 + 56*a^5*b^3 + 70*a^4*b^4 + 56*a^3*b^5 + 28 *a^2*b^6 + 8*a*b^7 + b^8)*log(sin(d*x + c) - 1) + 210*(28*a^6*b^2 + 70*a^4 *b^4 + 28*a^2*b^6 + b^8)*sin(d*x + c))/d
Time = 0.14 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.77 \[ \int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {{\left (a^{8} - 8 \, a^{7} b + 28 \, a^{6} b^{2} - 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} - 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} - 8 \, a b^{7} + b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{2 \, d} - \frac {{\left (a^{8} + 8 \, a^{7} b + 28 \, a^{6} b^{2} + 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} + 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} + 8 \, a b^{7} + b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} - \frac {15 \, b^{8} d^{6} \sin \left (d x + c\right )^{7} + 140 \, a b^{7} d^{6} \sin \left (d x + c\right )^{6} + 588 \, a^{2} b^{6} d^{6} \sin \left (d x + c\right )^{5} + 21 \, b^{8} d^{6} \sin \left (d x + c\right )^{5} + 1470 \, a^{3} b^{5} d^{6} \sin \left (d x + c\right )^{4} + 210 \, a b^{7} d^{6} \sin \left (d x + c\right )^{4} + 2450 \, a^{4} b^{4} d^{6} \sin \left (d x + c\right )^{3} + 980 \, a^{2} b^{6} d^{6} \sin \left (d x + c\right )^{3} + 35 \, b^{8} d^{6} \sin \left (d x + c\right )^{3} + 2940 \, a^{5} b^{3} d^{6} \sin \left (d x + c\right )^{2} + 2940 \, a^{3} b^{5} d^{6} \sin \left (d x + c\right )^{2} + 420 \, a b^{7} d^{6} \sin \left (d x + c\right )^{2} + 2940 \, a^{6} b^{2} d^{6} \sin \left (d x + c\right ) + 7350 \, a^{4} b^{4} d^{6} \sin \left (d x + c\right ) + 2940 \, a^{2} b^{6} d^{6} \sin \left (d x + c\right ) + 105 \, b^{8} d^{6} \sin \left (d x + c\right )}{105 \, d^{7}} \] Input:
integrate(sec(d*x+c)*(a+b*sin(d*x+c))^8,x, algorithm="giac")
Output:
1/2*(a^8 - 8*a^7*b + 28*a^6*b^2 - 56*a^5*b^3 + 70*a^4*b^4 - 56*a^3*b^5 + 2 8*a^2*b^6 - 8*a*b^7 + b^8)*log(abs(sin(d*x + c) + 1))/d - 1/2*(a^8 + 8*a^7 *b + 28*a^6*b^2 + 56*a^5*b^3 + 70*a^4*b^4 + 56*a^3*b^5 + 28*a^2*b^6 + 8*a* b^7 + b^8)*log(abs(sin(d*x + c) - 1))/d - 1/105*(15*b^8*d^6*sin(d*x + c)^7 + 140*a*b^7*d^6*sin(d*x + c)^6 + 588*a^2*b^6*d^6*sin(d*x + c)^5 + 21*b^8* d^6*sin(d*x + c)^5 + 1470*a^3*b^5*d^6*sin(d*x + c)^4 + 210*a*b^7*d^6*sin(d *x + c)^4 + 2450*a^4*b^4*d^6*sin(d*x + c)^3 + 980*a^2*b^6*d^6*sin(d*x + c) ^3 + 35*b^8*d^6*sin(d*x + c)^3 + 2940*a^5*b^3*d^6*sin(d*x + c)^2 + 2940*a^ 3*b^5*d^6*sin(d*x + c)^2 + 420*a*b^7*d^6*sin(d*x + c)^2 + 2940*a^6*b^2*d^6 *sin(d*x + c) + 7350*a^4*b^4*d^6*sin(d*x + c) + 2940*a^2*b^6*d^6*sin(d*x + c) + 105*b^8*d^6*sin(d*x + c))/d^7
Time = 15.86 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.87 \[ \int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^8}{2}+{\sin \left (c+d\,x\right )}^3\,\left (\frac {70\,a^4\,b^4}{3}+\frac {28\,a^2\,b^6}{3}+\frac {b^8}{3}\right )-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^8}{2}+{\sin \left (c+d\,x\right )}^5\,\left (\frac {28\,a^2\,b^6}{5}+\frac {b^8}{5}\right )+\sin \left (c+d\,x\right )\,\left (28\,a^6\,b^2+70\,a^4\,b^4+28\,a^2\,b^6+b^8\right )+{\sin \left (c+d\,x\right )}^2\,\left (28\,a^5\,b^3+28\,a^3\,b^5+4\,a\,b^7\right )+\frac {b^8\,{\sin \left (c+d\,x\right )}^7}{7}+{\sin \left (c+d\,x\right )}^4\,\left (14\,a^3\,b^5+2\,a\,b^7\right )+\frac {4\,a\,b^7\,{\sin \left (c+d\,x\right )}^6}{3}}{d} \] Input:
int((a + b*sin(c + d*x))^8/cos(c + d*x),x)
Output:
-((log(sin(c + d*x) - 1)*(a + b)^8)/2 + sin(c + d*x)^3*(b^8/3 + (28*a^2*b^ 6)/3 + (70*a^4*b^4)/3) - (log(sin(c + d*x) + 1)*(a - b)^8)/2 + sin(c + d*x )^5*(b^8/5 + (28*a^2*b^6)/5) + sin(c + d*x)*(b^8 + 28*a^2*b^6 + 70*a^4*b^4 + 28*a^6*b^2) + sin(c + d*x)^2*(4*a*b^7 + 28*a^3*b^5 + 28*a^5*b^3) + (b^8 *sin(c + d*x)^7)/7 + sin(c + d*x)^4*(2*a*b^7 + 14*a^3*b^5) + (4*a*b^7*sin( c + d*x)^6)/3)/d
Time = 0.17 (sec) , antiderivative size = 660, normalized size of antiderivative = 2.69 \[ \int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)*(a+b*sin(d*x+c))^8,x)
Output:
(840*log(tan((c + d*x)/2)**2 + 1)*a**7*b + 5880*log(tan((c + d*x)/2)**2 + 1)*a**5*b**3 + 5880*log(tan((c + d*x)/2)**2 + 1)*a**3*b**5 + 840*log(tan(( c + d*x)/2)**2 + 1)*a*b**7 - 105*log(tan((c + d*x)/2) - 1)*a**8 - 840*log( tan((c + d*x)/2) - 1)*a**7*b - 2940*log(tan((c + d*x)/2) - 1)*a**6*b**2 - 5880*log(tan((c + d*x)/2) - 1)*a**5*b**3 - 7350*log(tan((c + d*x)/2) - 1)* a**4*b**4 - 5880*log(tan((c + d*x)/2) - 1)*a**3*b**5 - 2940*log(tan((c + d *x)/2) - 1)*a**2*b**6 - 840*log(tan((c + d*x)/2) - 1)*a*b**7 - 105*log(tan ((c + d*x)/2) - 1)*b**8 + 105*log(tan((c + d*x)/2) + 1)*a**8 - 840*log(tan ((c + d*x)/2) + 1)*a**7*b + 2940*log(tan((c + d*x)/2) + 1)*a**6*b**2 - 588 0*log(tan((c + d*x)/2) + 1)*a**5*b**3 + 7350*log(tan((c + d*x)/2) + 1)*a** 4*b**4 - 5880*log(tan((c + d*x)/2) + 1)*a**3*b**5 + 2940*log(tan((c + d*x) /2) + 1)*a**2*b**6 - 840*log(tan((c + d*x)/2) + 1)*a*b**7 + 105*log(tan((c + d*x)/2) + 1)*b**8 - 15*sin(c + d*x)**7*b**8 - 140*sin(c + d*x)**6*a*b** 7 - 588*sin(c + d*x)**5*a**2*b**6 - 21*sin(c + d*x)**5*b**8 - 1470*sin(c + d*x)**4*a**3*b**5 - 210*sin(c + d*x)**4*a*b**7 - 2450*sin(c + d*x)**3*a** 4*b**4 - 980*sin(c + d*x)**3*a**2*b**6 - 35*sin(c + d*x)**3*b**8 - 2940*si n(c + d*x)**2*a**5*b**3 - 2940*sin(c + d*x)**2*a**3*b**5 - 420*sin(c + d*x )**2*a*b**7 - 2940*sin(c + d*x)*a**6*b**2 - 7350*sin(c + d*x)*a**4*b**4 - 2940*sin(c + d*x)*a**2*b**6 - 105*sin(c + d*x)*b**8)/(105*d)