Integrand size = 21, antiderivative size = 121 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {192 a^8 \log (1-\sin (c+d x))}{d}+\frac {129 a^8 \sin (c+d x)}{d}+\frac {36 a^8 \sin ^2(c+d x)}{d}+\frac {10 a^8 \sin ^3(c+d x)}{d}+\frac {2 a^8 \sin ^4(c+d x)}{d}+\frac {a^8 \sin ^5(c+d x)}{5 d}+\frac {64 a^9}{d (a-a \sin (c+d x))} \] Output:
192*a^8*ln(1-sin(d*x+c))/d+129*a^8*sin(d*x+c)/d+36*a^8*sin(d*x+c)^2/d+10*a ^8*sin(d*x+c)^3/d+2*a^8*sin(d*x+c)^4/d+1/5*a^8*sin(d*x+c)^5/d+64*a^9/d/(a- a*sin(d*x+c))
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {a^8 \sec ^2(c+d x) (1-\sin (c+d x)) (1+\sin (c+d x)) \left (192 \log (1-\sin (c+d x))+\frac {64}{1-\sin (c+d x)}+129 \sin (c+d x)+36 \sin ^2(c+d x)+10 \sin ^3(c+d x)+2 \sin ^4(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d} \] Input:
Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^8,x]
Output:
(a^8*Sec[c + d*x]^2*(1 - Sin[c + d*x])*(1 + Sin[c + d*x])*(192*Log[1 - Sin [c + d*x]] + 64/(1 - Sin[c + d*x]) + 129*Sin[c + d*x] + 36*Sin[c + d*x]^2 + 10*Sin[c + d*x]^3 + 2*Sin[c + d*x]^4 + Sin[c + d*x]^5/5))/d
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3146, 49, 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.
| \(\displaystyle \int \sec ^3(c+d x) (a \sin (c+d x)+a)^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^8}{\cos (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {a^3 \int \frac {(\sin (c+d x) a+a)^6}{(a-a \sin (c+d x))^2}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {a^3 \int \left (\frac {64 a^6}{(a-a \sin (c+d x))^2}-\frac {192 a^5}{a-a \sin (c+d x)}+\sin ^4(c+d x) a^4+8 \sin ^3(c+d x) a^4+30 \sin ^2(c+d x) a^4+72 \sin (c+d x) a^4+129 a^4\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \left (\frac {64 a^6}{a-a \sin (c+d x)}+\frac {1}{5} a^5 \sin ^5(c+d x)+2 a^5 \sin ^4(c+d x)+10 a^5 \sin ^3(c+d x)+36 a^5 \sin ^2(c+d x)+129 a^5 \sin (c+d x)+192 a^5 \log (a-a \sin (c+d x))\right )}{d}\) |
Input:
Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^8,x]
Output:
(a^3*(192*a^5*Log[a - a*Sin[c + d*x]] + 129*a^5*Sin[c + d*x] + 36*a^5*Sin[ c + d*x]^2 + 10*a^5*Sin[c + d*x]^3 + 2*a^5*Sin[c + d*x]^4 + (a^5*Sin[c + d *x]^5)/5 + (64*a^6)/(a - a*Sin[c + d*x])))/d
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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 + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Time = 1.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.06
| method | result | size |
| parallelrisch | \(\frac {a^{8} \left (-30720 \left (\sin \left (d x +c \right )-1\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+61440 \left (\sin \left (d x +c \right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+18 \sin \left (5 d x +5 c \right )-8095 \cos \left (2 d x +2 c \right )+166 \cos \left (4 d x +4 c \right )-\cos \left (6 d x +6 c \right )-27580 \sin \left (d x +c \right )-1130 \sin \left (3 d x +3 c \right )+7930\right )}{160 d \left (\sin \left (d x +c \right )-1\right )}\) | \(128\) |
| risch | \(-192 i a^{8} x -\frac {1093 i a^{8} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {1093 i a^{8} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {384 i a^{8} c}{d}-\frac {128 i a^{8} {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{2} d}+\frac {384 a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a^{8} \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{8} \cos \left (4 d x +4 c \right )}{4 d}-\frac {41 a^{8} \sin \left (3 d x +3 c \right )}{16 d}-\frac {19 a^{8} \cos \left (2 d x +2 c \right )}{d}\) | \(176\) |
| derivativedivides | \(\frac {a^{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 )+8 a^{8} \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 )+28 a^{8} \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 )+56 a^{8} \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 )+70 a^{8} \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^{8} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+28 a^{8} \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 )+\frac {4 a^{8}}{\cos \left (d x +c \right )^{2}}+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 )}{d}\) | \(442\) |
| default | \(\frac {a^{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 )+8 a^{8} \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 )+28 a^{8} \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 )+56 a^{8} \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 )+70 a^{8} \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^{8} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+28 a^{8} \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 )+\frac {4 a^{8}}{\cos \left (d x +c \right )^{2}}+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 )}{d}\) | \(442\) |
Input:
int(sec(d*x+c)^3*(a+a*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
1/160*a^8*(-30720*(sin(d*x+c)-1)*ln(sec(1/2*d*x+1/2*c)^2)+61440*(sin(d*x+c )-1)*ln(tan(1/2*d*x+1/2*c)-1)+18*sin(5*d*x+5*c)-8095*cos(2*d*x+2*c)+166*co s(4*d*x+4*c)-cos(6*d*x+6*c)-27580*sin(d*x+c)-1130*sin(3*d*x+3*c)+7930)/d/( sin(d*x+c)-1)
Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {4 \, a^{8} \cos \left (d x + c\right )^{6} - 172 \, a^{8} \cos \left (d x + c\right )^{4} + 2192 \, a^{8} \cos \left (d x + c\right )^{2} - 1119 \, a^{8} - 3840 \, {\left (a^{8} \sin \left (d x + c\right ) - a^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (36 \, a^{8} \cos \left (d x + c\right )^{4} - 592 \, a^{8} \cos \left (d x + c\right )^{2} - 2399 \, a^{8}\right )} \sin \left (d x + c\right )}{20 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \] Input:
integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^8,x, algorithm="fricas")
Output:
-1/20*(4*a^8*cos(d*x + c)^6 - 172*a^8*cos(d*x + c)^4 + 2192*a^8*cos(d*x + c)^2 - 1119*a^8 - 3840*(a^8*sin(d*x + c) - a^8)*log(-sin(d*x + c) + 1) - ( 36*a^8*cos(d*x + c)^4 - 592*a^8*cos(d*x + c)^2 - 2399*a^8)*sin(d*x + c))/( d*sin(d*x + c) - d)
Timed out. \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**3*(a+a*sin(d*x+c))**8,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.80 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {a^{8} \sin \left (d x + c\right )^{5} + 10 \, a^{8} \sin \left (d x + c\right )^{4} + 50 \, a^{8} \sin \left (d x + c\right )^{3} + 180 \, a^{8} \sin \left (d x + c\right )^{2} + 960 \, a^{8} \log \left (\sin \left (d x + c\right ) - 1\right ) + 645 \, a^{8} \sin \left (d x + c\right ) - \frac {320 \, a^{8}}{\sin \left (d x + c\right ) - 1}}{5 \, d} \] Input:
integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^8,x, algorithm="maxima")
Output:
1/5*(a^8*sin(d*x + c)^5 + 10*a^8*sin(d*x + c)^4 + 50*a^8*sin(d*x + c)^3 + 180*a^8*sin(d*x + c)^2 + 960*a^8*log(sin(d*x + c) - 1) + 645*a^8*sin(d*x + c) - 320*a^8/(sin(d*x + c) - 1))/d
Time = 0.14 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {192 \, a^{8} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {64 \, a^{8}}{d {\left (\sin \left (d x + c\right ) - 1\right )}} + \frac {a^{8} d^{4} \sin \left (d x + c\right )^{5} + 10 \, a^{8} d^{4} \sin \left (d x + c\right )^{4} + 50 \, a^{8} d^{4} \sin \left (d x + c\right )^{3} + 180 \, a^{8} d^{4} \sin \left (d x + c\right )^{2} + 645 \, a^{8} d^{4} \sin \left (d x + c\right )}{5 \, d^{5}} \] Input:
integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^8,x, algorithm="giac")
Output:
192*a^8*log(abs(sin(d*x + c) - 1))/d - 64*a^8/(d*(sin(d*x + c) - 1)) + 1/5 *(a^8*d^4*sin(d*x + c)^5 + 10*a^8*d^4*sin(d*x + c)^4 + 50*a^8*d^4*sin(d*x + c)^3 + 180*a^8*d^4*sin(d*x + c)^2 + 645*a^8*d^4*sin(d*x + c))/d^5
Time = 25.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.80 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {192\,a^8\,\ln \left (\sin \left (c+d\,x\right )-1\right )-\frac {64\,a^8}{\sin \left (c+d\,x\right )-1}+129\,a^8\,\sin \left (c+d\,x\right )+36\,a^8\,{\sin \left (c+d\,x\right )}^2+10\,a^8\,{\sin \left (c+d\,x\right )}^3+2\,a^8\,{\sin \left (c+d\,x\right )}^4+\frac {a^8\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \] Input:
int((a + a*sin(c + d*x))^8/cos(c + d*x)^3,x)
Output:
(192*a^8*log(sin(c + d*x) - 1) - (64*a^8)/(sin(c + d*x) - 1) + 129*a^8*sin (c + d*x) + 36*a^8*sin(c + d*x)^2 + 10*a^8*sin(c + d*x)^3 + 2*a^8*sin(c + d*x)^4 + (a^8*sin(c + d*x)^5)/5)/d
Time = 0.18 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {a^{8} \left (-960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )+960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )+1920 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )-1920 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\sin \left (d x +c \right )^{6}+9 \sin \left (d x +c \right )^{5}+40 \sin \left (d x +c \right )^{4}+130 \sin \left (d x +c \right )^{3}+465 \sin \left (d x +c \right )^{2}-965\right )}{5 d \left (\sin \left (d x +c \right )-1\right )} \] Input:
int(sec(d*x+c)^3*(a+a*sin(d*x+c))^8,x)
Output:
(a**8*( - 960*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x) + 960*log(tan((c + d*x)/2)**2 + 1) + 1920*log(tan((c + d*x)/2) - 1)*sin(c + d*x) - 1920*log( tan((c + d*x)/2) - 1) + sin(c + d*x)**6 + 9*sin(c + d*x)**5 + 40*sin(c + d *x)**4 + 130*sin(c + d*x)**3 + 465*sin(c + d*x)**2 - 965))/(5*d*(sin(c + d *x) - 1))