\(\int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx\) [48]

Optimal result

Integrand size = 21, antiderivative size = 210 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=-\frac {805 a^3 x}{128}-\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {603 a^3 \cos (c+d x) \sin (c+d x)}{128 d}-\frac {293 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \sin ^3(c+d x)}{3 d}-\frac {2 a^3 \sin ^5(c+d x)}{5 d}-\frac {3 a^3 \sin ^7(c+d x)}{7 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d} \]

[Out]

-805/128*a^3*x-1/2*a^3*arctanh(sin(d*x+c))/d+603/128*a^3*cos(d*x+c)*sin(d*x+c)/d-293/192*a^3*cos(d*x+c)^3*sin(
d*x+c)/d-1/48*a^3*cos(d*x+c)^5*sin(d*x+c)/d+1/8*a^3*cos(d*x+c)^7*sin(d*x+c)/d-1/3*a^3*sin(d*x+c)^3/d-2/5*a^3*s
in(d*x+c)^5/d-3/7*a^3*sin(d*x+c)^7/d+3*a^3*tan(d*x+c)/d+1/2*a^3*sec(d*x+c)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3957, 2951, 2717, 2715, 8, 2713, 3855, 3852, 3853} \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=-\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^3 \sin ^7(c+d x)}{7 d}-\frac {2 a^3 \sin ^5(c+d x)}{5 d}-\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac {a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}-\frac {293 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {603 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac {805 a^3 x}{128} \]

[In]

Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^8,x]

[Out]

(-805*a^3*x)/128 - (a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (603*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (293*a^3*
Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^3*Cos[c + d*x]^7*Sin[c +
d*x])/(8*d) - (a^3*Sin[c + d*x]^3)/(3*d) - (2*a^3*Sin[c + d*x]^5)/(5*d) - (3*a^3*Sin[c + d*x]^7)/(7*d) + (3*a^
3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2951

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \sin ^5(c+d x) \tan ^3(c+d x) \, dx \\ & = -\frac {\int \left (11 a^{11}+6 a^{11} \cos (c+d x)-14 a^{11} \cos ^2(c+d x)-14 a^{11} \cos ^3(c+d x)+6 a^{11} \cos ^4(c+d x)+11 a^{11} \cos ^5(c+d x)+a^{11} \cos ^6(c+d x)-3 a^{11} \cos ^7(c+d x)-a^{11} \cos ^8(c+d x)+a^{11} \sec (c+d x)-3 a^{11} \sec ^2(c+d x)-a^{11} \sec ^3(c+d x)\right ) \, dx}{a^8} \\ & = -11 a^3 x-a^3 \int \cos ^6(c+d x) \, dx+a^3 \int \cos ^8(c+d x) \, dx-a^3 \int \sec (c+d x) \, dx+a^3 \int \sec ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^7(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \, dx-\left (6 a^3\right ) \int \cos (c+d x) \, dx-\left (6 a^3\right ) \int \cos ^4(c+d x) \, dx-\left (11 a^3\right ) \int \cos ^5(c+d x) \, dx+\left (14 a^3\right ) \int \cos ^2(c+d x) \, dx+\left (14 a^3\right ) \int \cos ^3(c+d x) \, dx \\ & = -11 a^3 x-\frac {a^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {6 a^3 \sin (c+d x)}{d}+\frac {7 a^3 \cos (c+d x) \sin (c+d x)}{d}-\frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} a^3 \int \sec (c+d x) \, dx-\frac {1}{6} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{8} \left (7 a^3\right ) \int \cos ^6(c+d x) \, dx-\frac {1}{2} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx+\left (7 a^3\right ) \int 1 \, dx-\frac {\left (3 a^3\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}+\frac {\left (11 a^3\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (14 a^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = -4 a^3 x-\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{4 d}-\frac {41 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \sin ^3(c+d x)}{3 d}-\frac {2 a^3 \sin ^5(c+d x)}{5 d}-\frac {3 a^3 \sin ^7(c+d x)}{7 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{8} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{48} \left (35 a^3\right ) \int \cos ^4(c+d x) \, dx-\frac {1}{4} \left (9 a^3\right ) \int 1 \, dx \\ & = -\frac {25 a^3 x}{4}-\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {71 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {293 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \sin ^3(c+d x)}{3 d}-\frac {2 a^3 \sin ^5(c+d x)}{5 d}-\frac {3 a^3 \sin ^7(c+d x)}{7 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{16} \left (5 a^3\right ) \int 1 \, dx+\frac {1}{64} \left (35 a^3\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {105 a^3 x}{16}-\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {603 a^3 \cos (c+d x) \sin (c+d x)}{128 d}-\frac {293 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \sin ^3(c+d x)}{3 d}-\frac {2 a^3 \sin ^5(c+d x)}{5 d}-\frac {3 a^3 \sin ^7(c+d x)}{7 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{128} \left (35 a^3\right ) \int 1 \, dx \\ & = -\frac {805 a^3 x}{128}-\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {603 a^3 \cos (c+d x) \sin (c+d x)}{128 d}-\frac {293 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \sin ^3(c+d x)}{3 d}-\frac {2 a^3 \sin ^5(c+d x)}{5 d}-\frac {3 a^3 \sin ^7(c+d x)}{7 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.74 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=\frac {a^3 \sec ^2(c+d x) \left (-1352400 c-1352400 d x-215040 \text {arctanh}(\sin (c+d x)) \cos ^2(c+d x)-1352400 (c+d x) \cos (2 (c+d x))+173600 \sin (c+d x)+1052520 \sin (2 (c+d x))-11648 \sin (3 (c+d x))+175280 \sin (4 (c+d x))+22784 \sin (5 (c+d x))-18095 \sin (6 (c+d x))-6288 \sin (7 (c+d x))+770 \sin (8 (c+d x))+720 \sin (9 (c+d x))+105 \sin (10 (c+d x))\right )}{430080 d} \]

[In]

Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^8,x]

[Out]

(a^3*Sec[c + d*x]^2*(-1352400*c - 1352400*d*x - 215040*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 - 1352400*(c + d*x
)*Cos[2*(c + d*x)] + 173600*Sin[c + d*x] + 1052520*Sin[2*(c + d*x)] - 11648*Sin[3*(c + d*x)] + 175280*Sin[4*(c
+ d*x)] + 22784*Sin[5*(c + d*x)] - 18095*Sin[6*(c + d*x)] - 6288*Sin[7*(c + d*x)] + 770*Sin[8*(c + d*x)] + 72
0*Sin[9*(c + d*x)] + 105*Sin[10*(c + d*x)]))/(430080*d)

Maple [A] (verified)

Time = 2.48 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.94

method	result	size
parallelrisch	\(\frac {a^{3} \left (107520 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-107520 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-1352400 d x \cos \left (2 d x +2 c \right )-1352400 d x +22784 \sin \left (5 d x +5 c \right )-18095 \sin \left (6 d x +6 c \right )-6288 \sin \left (7 d x +7 c \right )+770 \sin \left (8 d x +8 c \right )+720 \sin \left (9 d x +9 c \right )+105 \sin \left (10 d x +10 c \right )+173600 \sin \left (d x +c \right )+1052520 \sin \left (2 d x +2 c \right )-11648 \sin \left (3 d x +3 c \right )+175280 \sin \left (4 d x +4 c \right )\right )}{215040 d \left (1+\cos \left (2 d x +2 c \right )\right )}\)	\(197\)
derivativedivides	\(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{7}}{2}+\frac {7 \sin \left (d x +c \right )^{5}}{10}+\frac {7 \sin \left (d x +c \right )^{3}}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\)	\(272\)
default	\(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{7}}{2}+\frac {7 \sin \left (d x +c \right )^{5}}{10}+\frac {7 \sin \left (d x +c \right )^{3}}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\)	\(272\)
parts	\(\frac {a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{7}}{2}+\frac {7 \sin \left (d x +c \right )^{5}}{10}+\frac {7 \sin \left (d x +c \right )^{3}}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )}{d}\)	\(280\)
risch	\(-\frac {805 a^{3} x}{128}+\frac {127 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {127 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}-\frac {i a^{3} \left ({\mathrm e}^{3 i \left (d x +c \right )}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}-6\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {67 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{384 d}+\frac {47 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {67 i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{384 d}-\frac {47 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {a^{3} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {3 a^{3} \sin \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}-\frac {23 a^{3} \sin \left (5 d x +5 c \right )}{320 d}-\frac {23 a^{3} \sin \left (4 d x +4 c \right )}{128 d}\)	\(298\)

[In]

int((a+a*sec(d*x+c))^3*sin(d*x+c)^8,x,method=_RETURNVERBOSE)

[Out]

1/215040*a^3*(107520*(1+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)-1)-107520*(1+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*
c)[phone]*d*x*cos(2*d*x+2*c)-1352400*d*x+22784*sin(5*d*x+5*c)-18095*sin(6*d*x+6*c)-6288*sin(7*d*x+7*c)+770
*sin(8*d*x+8*c)+720*sin(9*d*x+9*c)+105*sin(10*d*x+10*c)+173600*sin(d*x+c)+1052520*sin(2*d*x+2*c)-11648*sin(3*d
*x+3*c)+175280*sin(4*d*x+4*c))/d/(1+cos(2*d*x+2*c))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.97 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=-\frac {84525 \, a^{3} d x \cos \left (d x + c\right )^{2} + 3360 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3360 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (1680 \, a^{3} \cos \left (d x + c\right )^{9} + 5760 \, a^{3} \cos \left (d x + c\right )^{8} - 280 \, a^{3} \cos \left (d x + c\right )^{7} - 22656 \, a^{3} \cos \left (d x + c\right )^{6} - 20510 \, a^{3} \cos \left (d x + c\right )^{5} + 32512 \, a^{3} \cos \left (d x + c\right )^{4} + 63315 \, a^{3} \cos \left (d x + c\right )^{3} - 15616 \, a^{3} \cos \left (d x + c\right )^{2} + 40320 \, a^{3} \cos \left (d x + c\right ) + 6720 \, a^{3}\right )} \sin \left (d x + c\right )}{13440 \, d \cos \left (d x + c\right )^{2}} \]

[In]

integrate((a+a*sec(d*x+c))^3*sin(d*x+c)^8,x, algorithm="fricas")

[Out]

-1/13440*(84525*a^3*d*x*cos(d*x + c)^2 + 3360*a^3*cos(d*x + c)^2*log(sin(d*x + c) + 1) - 3360*a^3*cos(d*x + c)
^2*log(-sin(d*x + c) + 1) - (1680*a^3*cos(d*x + c)^9 + 5760*a^3*cos(d*x + c)^8 - 280*a^3*cos(d*x + c)^7 - 2265
6*a^3*cos(d*x + c)^6 - 20510*a^3*cos(d*x + c)^5 + 32512*a^3*cos(d*x + c)^4 + 63315*a^3*cos(d*x + c)^3 - 15616*
a^3*cos(d*x + c)^2 + 40320*a^3*cos(d*x + c) + 6720*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2)

Sympy [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+a*sec(d*x+c))**3*sin(d*x+c)**8,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.39 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=-\frac {1536 \, {\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a^{3} - 1792 \, {\left (12 \, \sin \left (d x + c\right )^{5} + 40 \, \sin \left (d x + c\right )^{3} - \frac {30 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 180 \, \sin \left (d x + c\right )\right )} a^{3} - 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 6720 \, {\left (105 \, d x + 105 \, c - \frac {87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} a^{3}}{107520 \, d} \]

[In]

integrate((a+a*sec(d*x+c))^3*sin(d*x+c)^8,x, algorithm="maxima")

[Out]

-1/107520*(1536*(30*sin(d*x + c)^7 + 42*sin(d*x + c)^5 + 70*sin(d*x + c)^3 - 105*log(sin(d*x + c) + 1) + 105*l
og(sin(d*x + c) - 1) + 210*sin(d*x + c))*a^3 - 1792*(12*sin(d*x + c)^5 + 40*sin(d*x + c)^3 - 30*sin(d*x + c)/(
sin(d*x + c)^2 - 1) - 105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1) + 180*sin(d*x + c))*a^3 - 35*(128*
sin(2*d*x + 2*c)^3 + 840*d*x + 840*c + 3*sin(8*d*x + 8*c) + 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*a^3 +
6720*(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))*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.16 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=-\frac {84525 \, {\left (d x + c\right )} a^{3} + 6720 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6720 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {13440 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {2 \, {\left (44205 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 303065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 841981 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1123793 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 487983 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 490749 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 267225 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 44205 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8}}}{13440 \, d} \]

[In]

integrate((a+a*sec(d*x+c))^3*sin(d*x+c)^8,x, algorithm="giac")

[Out]

-1/13440*(84525*(d*x + c)*a^3 + 6720*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 6720*a^3*log(abs(tan(1/2*d*x + 1
/2*c) - 1)) + 13440*(5*a^3*tan(1/2*d*x + 1/2*c)^3 - 7*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2
+ 2*(44205*a^3*tan(1/2*d*x + 1/2*c)^15 + 303065*a^3*tan(1/2*d*x + 1/2*c)^13 + 841981*a^3*tan(1/2*d*x + 1/2*c)
^11 + 1123793*a^3*tan(1/2*d*x + 1/2*c)^9 + 487983*a^3*tan(1/2*d*x + 1/2*c)^7 - 490749*a^3*tan(1/2*d*x + 1/2*c)
^5 - 267225*a^3*tan(1/2*d*x + 1/2*c)^3 - 44205*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^8)/d

Mupad [B] (verification not implemented)

Time = 14.84 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.52 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=\frac {-\frac {741\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{64}-\frac {12469\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{192}-\frac {5027\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{40}-\frac {19211\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{420}+\frac {199977\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{1120}+\frac {877061\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3360}+\frac {10233\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{140}+\frac {6243\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}+\frac {4967\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}+\frac {869\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {805\,a^3\,x}{128} \]

[In]

int(sin(c + d*x)^8*(a + a/cos(c + d*x))^3,x)

[Out]

((4967*a^3*tan(c/2 + (d*x)/2)^3)/64 + (6243*a^3*tan(c/2 + (d*x)/2)^5)/40 + (10233*a^3*tan(c/2 + (d*x)/2)^7)/14
0 + (877061*a^3*tan(c/2 + (d*x)/2)^9)/3360 + (199977*a^3*tan(c/2 + (d*x)/2)^11)/1120 - (19211*a^3*tan(c/2 + (d
*x)/2)^13)/420 - (5027*a^3*tan(c/2 + (d*x)/2)^15)/40 - (12469*a^3*tan(c/2 + (d*x)/2)^17)/192 - (741*a^3*tan(c/
2 + (d*x)/2)^19)/64 + (869*a^3*tan(c/2 + (d*x)/2))/64)/(d*(6*tan(c/2 + (d*x)/2)^2 + 13*tan(c/2 + (d*x)/2)^4 +
8*tan(c/2 + (d*x)/2)^6 - 14*tan(c/2 + (d*x)/2)^8 - 28*tan(c/2 + (d*x)/2)^10 - 14*tan(c/2 + (d*x)/2)^12 + 8*tan
(c/2 + (d*x)/2)^14 + 13*tan(c/2 + (d*x)/2)^16 + 6*tan(c/2 + (d*x)/2)^18 + tan(c/2 + (d*x)/2)^20 + 1)) - (a^3*a
tanh(tan(c/2 + (d*x)/2)))/d - (805*a^3*x)/128