\(\int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx\) [419]

Optimal result

Integrand size = 21, antiderivative size = 423 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {1}{256} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) x-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}+\frac {\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \cos (c+d x) \sin (c+d x)}{256 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d} \]

[Out]

1/256*(128*a^8+896*a^6*b^2+1120*a^4*b^4+280*a^2*b^6+7*b^8)*x-11/40320*a*b*(1792*a^6+10536*a^4*b^2+9588*a^2*b^4
+1289*b^6)*cos(d*x+c)^3/d+1/256*(128*a^8+896*a^6*b^2+1120*a^4*b^4+280*a^2*b^6+7*b^8)*cos(d*x+c)*sin(d*x+c)/d-1
/13440*b*(6272*a^6+28088*a^4*b^2+15956*a^2*b^4+735*b^6)*cos(d*x+c)^3*(a+b*sin(d*x+c))/d-13/3360*a*b*(112*a^4+3
48*a^2*b^2+101*b^4)*cos(d*x+c)^3*(a+b*sin(d*x+c))^2/d-1/2016*b*(784*a^4+1500*a^2*b^2+147*b^4)*cos(d*x+c)^3*(a+
b*sin(d*x+c))^3/d-1/336*a*b*(112*a^2+109*b^2)*cos(d*x+c)^3*(a+b*sin(d*x+c))^4/d-1/240*b*(64*a^2+21*b^2)*cos(d*
x+c)^3*(a+b*sin(d*x+c))^5/d-17/90*a*b*cos(d*x+c)^3*(a+b*sin(d*x+c))^6/d-1/10*b*cos(d*x+c)^3*(a+b*sin(d*x+c))^7
/d

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2771, 2941, 2748, 2715, 8} \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}+\frac {\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \sin (c+d x) \cos (c+d x)}{256 d}+\frac {1}{256} x \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d} \]

[In]

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

[Out]

((128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*x)/256 - (11*a*b*(1792*a^6 + 10536*a^4*b^2 + 958
8*a^2*b^4 + 1289*b^6)*Cos[c + d*x]^3)/(40320*d) + ((128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8
)*Cos[c + d*x]*Sin[c + d*x])/(256*d) - (b*(6272*a^6 + 28088*a^4*b^2 + 15956*a^2*b^4 + 735*b^6)*Cos[c + d*x]^3*
(a + b*Sin[c + d*x]))/(13440*d) - (13*a*b*(112*a^4 + 348*a^2*b^2 + 101*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x]
)^2)/(3360*d) - (b*(784*a^4 + 1500*a^2*b^2 + 147*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(2016*d) - (a*b*(
112*a^2 + 109*b^2)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(336*d) - (b*(64*a^2 + 21*b^2)*Cos[c + d*x]^3*(a + b
*Sin[c + d*x])^5)/(240*d) - (17*a*b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^6)/(90*d) - (b*Cos[c + d*x]^3*(a + b*S
in[c + d*x])^7)/(10*d)

Rule 8

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

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 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2771

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-b)*(
g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Dist[1/(m + p), Int[(g*Cos[e + f*x]
)^p*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1)*Sin[e + f*x]), x], x] /; FreeQ
[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ
[m])

Rule 2941

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x
] + Dist[1/(m + p + 1), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(m + p + 1) + b*d*m + (a*
d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] &&
GtQ[m, 0] &&  !LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])

Rubi steps \begin{align*} \text {integral}& = -\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {1}{10} \int \cos ^2(c+d x) (a+b \sin (c+d x))^6 \left (10 a^2+7 b^2+17 a b \sin (c+d x)\right ) \, dx \\ & = -\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {1}{90} \int \cos ^2(c+d x) (a+b \sin (c+d x))^5 \left (15 a \left (6 a^2+11 b^2\right )+3 b \left (64 a^2+21 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {1}{720} \int \cos ^2(c+d x) (a+b \sin (c+d x))^4 \left (15 \left (48 a^4+152 a^2 b^2+21 b^4\right )+15 a b \left (112 a^2+109 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {\int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \left (15 a \left (336 a^4+1512 a^2 b^2+583 b^4\right )+15 b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \sin (c+d x)\right ) \, dx}{5040} \\ & = -\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {\int \cos ^2(c+d x) (a+b \sin (c+d x))^2 \left (45 \left (672 a^6+3808 a^4 b^2+2666 a^2 b^4+147 b^6\right )+585 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \sin (c+d x)\right ) \, dx}{30240} \\ & = -\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {\int \cos ^2(c+d x) (a+b \sin (c+d x)) \left (45 a \left (3360 a^6+21952 a^4 b^2+22378 a^2 b^4+3361 b^6\right )+45 b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \sin (c+d x)\right ) \, dx}{151200} \\ & = -\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {\int \cos ^2(c+d x) \left (4725 \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right )+495 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \sin (c+d x)\right ) \, dx}{604800} \\ & = -\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {1}{128} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}+\frac {\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \cos (c+d x) \sin (c+d x)}{256 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {1}{256} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \int 1 \, dx \\ & = \frac {1}{256} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) x-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}+\frac {\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \cos (c+d x) \sin (c+d x)}{256 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.06 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.08 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {322560 a^8 c+2257920 a^6 b^2 c+2822400 a^4 b^4 c+705600 a^2 b^6 c+17640 b^8 c+322560 a^8 d x+2257920 a^6 b^2 d x+2822400 a^4 b^4 d x+705600 a^2 b^6 d x+17640 b^8 d x-40320 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right ) \cos (c+d x)-26880 \left (16 a^7 b+28 a^5 b^3+7 a^3 b^5\right ) \cos (3 (c+d x))+451584 a^5 b^3 \cos (5 (c+d x))+338688 a^3 b^5 \cos (5 (c+d x))+32256 a b^7 \cos (5 (c+d x))-80640 a^3 b^5 \cos (7 (c+d x))-14400 a b^7 \cos (7 (c+d x))+2240 a b^7 \cos (9 (c+d x))+161280 a^8 \sin (2 (c+d x))-705600 a^4 b^4 \sin (2 (c+d x))-282240 a^2 b^6 \sin (2 (c+d x))-8820 b^8 \sin (2 (c+d x))-564480 a^6 b^2 \sin (4 (c+d x))-705600 a^4 b^4 \sin (4 (c+d x))-141120 a^2 b^6 \sin (4 (c+d x))-2520 b^8 \sin (4 (c+d x))+235200 a^4 b^4 \sin (6 (c+d x))+94080 a^2 b^6 \sin (6 (c+d x))+2730 b^8 \sin (6 (c+d x))-17640 a^2 b^6 \sin (8 (c+d x))-945 b^8 \sin (8 (c+d x))+126 b^8 \sin (10 (c+d x))}{645120 d} \]

[In]

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

[Out]

(322560*a^8*c + 2257920*a^6*b^2*c + 2822400*a^4*b^4*c + 705600*a^2*b^6*c + 17640*b^8*c + 322560*a^8*d*x + 2257
920*a^6*b^2*d*x + 2822400*a^4*b^4*d*x + 705600*a^2*b^6*d*x + 17640*b^8*d*x - 40320*a*b*(32*a^6 + 112*a^4*b^2 +
70*a^2*b^4 + 7*b^6)*Cos[c + d*x] - 26880*(16*a^7*b + 28*a^5*b^3 + 7*a^3*b^5)*Cos[3*(c + d*x)] + 451584*a^5*b^
3*Cos[5*(c + d*x)] + 338688*a^3*b^5*Cos[5*(c + d*x)] + 32256*a*b^7*Cos[5*(c + d*x)] - 80640*a^3*b^5*Cos[7*(c +
d*x)] - 14400*a*b^7*Cos[7*(c + d*x)] + 2240*a*b^7*Cos[9*(c + d*x)] + 161280*a^8*Sin[2*(c + d*x)] - 705600*a^4
*b^4*Sin[2*(c + d*x)] - 282240*a^2*b^6*Sin[2*(c + d*x)] - 8820*b^8*Sin[2*(c + d*x)] - 564480*a^6*b^2*Sin[4*(c
+ d*x)] - 705600*a^4*b^4*Sin[4*(c + d*x)] - 141120*a^2*b^6*Sin[4*(c + d*x)] - 2520*b^8*Sin[4*(c + d*x)] + 2352
00*a^4*b^4*Sin[6*(c + d*x)] + 94080*a^2*b^6*Sin[6*(c + d*x)] + 2730*b^8*Sin[6*(c + d*x)] - 17640*a^2*b^6*Sin[8
*(c + d*x)] - 945*b^8*Sin[8*(c + d*x)] + 126*b^8*Sin[10*(c + d*x)])/(645120*d)

Maple [A] (verified)

Time = 2.15 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.87

method	result	size
parallelrisch	\(\frac {\left (161280 a^{8}-705600 a^{4} b^{4}-282240 a^{2} b^{6}-8820 b^{8}\right ) \sin \left (2 d x +2 c \right )+\left (-564480 a^{6} b^{2}-705600 a^{4} b^{4}-141120 a^{2} b^{6}-2520 b^{8}\right ) \sin \left (4 d x +4 c \right )+\left (-430080 a^{7} b -752640 a^{5} b^{3}-188160 a^{3} b^{5}\right ) \cos \left (3 d x +3 c \right )+\left (451584 a^{5} b^{3}+338688 a^{3} b^{5}+32256 a \,b^{7}\right ) \cos \left (5 d x +5 c \right )+\left (235200 a^{4} b^{4}+94080 a^{2} b^{6}+2730 b^{8}\right ) \sin \left (6 d x +6 c \right )+\left (-80640 a^{3} b^{5}-14400 a \,b^{7}\right ) \cos \left (7 d x +7 c \right )+\left (-17640 a^{2} b^{6}-945 b^{8}\right ) \sin \left (8 d x +8 c \right )+126 b^{8} \sin \left (10 d x +10 c \right )+2240 a \,b^{7} \cos \left (9 d x +9 c \right )-1290240 \left (\frac {7}{32} b^{6}+a^{6}+\frac {7}{2} a^{4} b^{2}+\frac {35}{16} a^{2} b^{4}\right ) a b \cos \left (d x +c \right )+322560 a^{8} d x +2257920 a^{6} b^{2} d x +2822400 a^{4} b^{4} d x +705600 a^{2} b^{6} d x +17640 b^{8} d x -1720320 a^{7} b -4816896 a^{5} b^{3}-2752512 a^{3} b^{5}-262144 a \,b^{7}}{645120 d}\)	\(367\)
derivativedivides	\(\frac {a^{8} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 a^{7} b \left (\cos ^{3}\left (d x +c \right )\right )}{3}+28 a^{6} b^{2} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+56 a^{5} b^{3} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+70 a^{4} b^{4} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+56 a^{3} b^{5} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{7}-\frac {4 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+28 a^{2} b^{6} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {5 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{128}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+8 a \,b^{7} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{21}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{105}-\frac {16 \left (\cos ^{3}\left (d x +c \right )\right )}{315}\right )+b^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{10}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{80}-\frac {7 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{96}-\frac {7 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{128}+\frac {7 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{256}+\frac {7 d x}{256}+\frac {7 c}{256}\right )}{d}\)	\(497\)
default	\(\frac {a^{8} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 a^{7} b \left (\cos ^{3}\left (d x +c \right )\right )}{3}+28 a^{6} b^{2} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+56 a^{5} b^{3} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+70 a^{4} b^{4} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+56 a^{3} b^{5} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{7}-\frac {4 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+28 a^{2} b^{6} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {5 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{128}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+8 a \,b^{7} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{21}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{105}-\frac {16 \left (\cos ^{3}\left (d x +c \right )\right )}{315}\right )+b^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{10}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{80}-\frac {7 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{96}-\frac {7 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{128}+\frac {7 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{256}+\frac {7 d x}{256}+\frac {7 c}{256}\right )}{d}\)	\(497\)
risch	\(\frac {a \,b^{7} \cos \left (9 d x +9 c \right )}{288 d}-\frac {35 \sin \left (2 d x +2 c \right ) a^{4} b^{4}}{32 d}-\frac {7 \sin \left (2 d x +2 c \right ) a^{2} b^{6}}{16 d}-\frac {3 \sin \left (8 d x +8 c \right ) b^{8}}{2048 d}+\frac {13 \sin \left (6 d x +6 c \right ) b^{8}}{3072 d}-\frac {\sin \left (4 d x +4 c \right ) b^{8}}{256 d}+\frac {\sin \left (2 d x +2 c \right ) a^{8}}{4 d}-\frac {7 \sin \left (2 d x +2 c \right ) b^{8}}{512 d}+\frac {a^{8} x}{2}+\frac {7 b^{8} x}{256}+\frac {b^{8} \sin \left (10 d x +10 c \right )}{5120 d}+\frac {7 x \,a^{6} b^{2}}{2}+\frac {35 x \,a^{4} b^{4}}{8}+\frac {35 x \,a^{2} b^{6}}{32}-\frac {2 a^{7} b \cos \left (d x +c \right )}{d}-\frac {7 a^{5} b^{3} \cos \left (d x +c \right )}{d}-\frac {35 a^{3} b^{5} \cos \left (d x +c \right )}{8 d}-\frac {7 a \,b^{7} \cos \left (d x +c \right )}{16 d}-\frac {7 \sin \left (8 d x +8 c \right ) a^{2} b^{6}}{256 d}-\frac {a^{3} b^{5} \cos \left (7 d x +7 c \right )}{8 d}-\frac {5 a \,b^{7} \cos \left (7 d x +7 c \right )}{224 d}+\frac {35 \sin \left (6 d x +6 c \right ) a^{4} b^{4}}{96 d}+\frac {7 \sin \left (6 d x +6 c \right ) a^{2} b^{6}}{48 d}+\frac {7 a^{5} b^{3} \cos \left (5 d x +5 c \right )}{10 d}+\frac {21 a^{3} b^{5} \cos \left (5 d x +5 c \right )}{40 d}+\frac {a \,b^{7} \cos \left (5 d x +5 c \right )}{20 d}-\frac {7 \sin \left (4 d x +4 c \right ) a^{6} b^{2}}{8 d}-\frac {35 \sin \left (4 d x +4 c \right ) a^{4} b^{4}}{32 d}-\frac {7 \sin \left (4 d x +4 c \right ) a^{2} b^{6}}{32 d}-\frac {2 a^{7} b \cos \left (3 d x +3 c \right )}{3 d}-\frac {7 a^{5} b^{3} \cos \left (3 d x +3 c \right )}{6 d}-\frac {7 a^{3} b^{5} \cos \left (3 d x +3 c \right )}{24 d}\)	\(539\)

[In]

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

[Out]

1/645120*((161280*a^8-705600*a^4*b^4-282240*a^2*b^6-8820*b^8)*sin(2*d*x+2*c)+(-564480*a^6*b^2-705600*a^4*b^4-1
41120*a^2*b^6-2520*b^8)*sin(4*d*x+4*c)+(-430080*a^7*b-752640*a^5*b^3-188160*a^3*b^5)*cos(3*d*x+3*c)+(451584*a^
5*b^3+338688*a^3*b^5+32256*a*b^7)*cos(5*d*x+5*c)+(235200*a^4*b^4+94080*a^2*b^6+2730*b^8)*sin(6*d*x+6*c)+(-8064
0*a^3*b^5-14400*a*b^7)*cos(7*d*x+7*c)+(-17640*a^2*b^6-945*b^8)*sin(8*d*x+8*c)+126*b^8*sin(10*d*x+10*c)+2240*a*
b^7*cos(9*d*x+9*c)-1290240*(7/32*b^6+a^6+7/2*a^4*b^2+35/16*a^2*b^4)*a*b*cos(d*x+c)+322560*a^8*d*x+2257920*a^6*
b^2*d*x+2822400*a^4*b^4*d*x+705600*a^2*b^6*d*x+17640*b^8*d*x-1720320*a^7*b-4816896*a^5*b^3-2752512*a^3*b^5-262
144*a*b^7)/d

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.74 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {71680 \, a b^{7} \cos \left (d x + c\right )^{9} - 92160 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{7} + 129024 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{5} - 215040 \, {\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{3} + 315 \, {\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} d x + 21 \, {\left (384 \, b^{8} \cos \left (d x + c\right )^{9} - 48 \, {\left (280 \, a^{2} b^{6} + 31 \, b^{8}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (5600 \, a^{4} b^{4} + 4760 \, a^{2} b^{6} + 263 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (2688 \, a^{6} b^{2} + 7840 \, a^{4} b^{4} + 3304 \, a^{2} b^{6} + 121 \, b^{8}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]

[In]

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

[Out]

1/80640*(71680*a*b^7*cos(d*x + c)^9 - 92160*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^7 + 129024*(7*a^5*b^3 + 14*a^3*
b^5 + 3*a*b^7)*cos(d*x + c)^5 - 215040*(a^7*b + 7*a^5*b^3 + 7*a^3*b^5 + a*b^7)*cos(d*x + c)^3 + 315*(128*a^8 +
896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*d*x + 21*(384*b^8*cos(d*x + c)^9 - 48*(280*a^2*b^6 + 31*b^8
)*cos(d*x + c)^7 + 8*(5600*a^4*b^4 + 4760*a^2*b^6 + 263*b^8)*cos(d*x + c)^5 - 10*(2688*a^6*b^2 + 7840*a^4*b^4
+ 3304*a^2*b^6 + 121*b^8)*cos(d*x + c)^3 + 15*(128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*cos
(d*x + c))*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1115 vs. \(2 (415) = 830\).

Time = 1.48 (sec) , antiderivative size = 1115, normalized size of antiderivative = 2.64 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((a**8*x*sin(c + d*x)**2/2 + a**8*x*cos(c + d*x)**2/2 + a**8*sin(c + d*x)*cos(c + d*x)/(2*d) - 8*a**7
*b*cos(c + d*x)**3/(3*d) + 7*a**6*b**2*x*sin(c + d*x)**4/2 + 7*a**6*b**2*x*sin(c + d*x)**2*cos(c + d*x)**2 + 7
*a**6*b**2*x*cos(c + d*x)**4/2 + 7*a**6*b**2*sin(c + d*x)**3*cos(c + d*x)/(2*d) - 7*a**6*b**2*sin(c + d*x)*cos
(c + d*x)**3/(2*d) - 56*a**5*b**3*sin(c + d*x)**2*cos(c + d*x)**3/(3*d) - 112*a**5*b**3*cos(c + d*x)**5/(15*d)
+ 35*a**4*b**4*x*sin(c + d*x)**6/8 + 105*a**4*b**4*x*sin(c + d*x)**4*cos(c + d*x)**2/8 + 105*a**4*b**4*x*sin(
c + d*x)**2*cos(c + d*x)**4/8 + 35*a**4*b**4*x*cos(c + d*x)**6/8 + 35*a**4*b**4*sin(c + d*x)**5*cos(c + d*x)/(
8*d) - 35*a**4*b**4*sin(c + d*x)**3*cos(c + d*x)**3/(3*d) - 35*a**4*b**4*sin(c + d*x)*cos(c + d*x)**5/(8*d) -
56*a**3*b**5*sin(c + d*x)**4*cos(c + d*x)**3/(3*d) - 224*a**3*b**5*sin(c + d*x)**2*cos(c + d*x)**5/(15*d) - 64
*a**3*b**5*cos(c + d*x)**7/(15*d) + 35*a**2*b**6*x*sin(c + d*x)**8/32 + 35*a**2*b**6*x*sin(c + d*x)**6*cos(c +
d*x)**2/8 + 105*a**2*b**6*x*sin(c + d*x)**4*cos(c + d*x)**4/16 + 35*a**2*b**6*x*sin(c + d*x)**2*cos(c + d*x)*
*6/8 + 35*a**2*b**6*x*cos(c + d*x)**8/32 + 35*a**2*b**6*sin(c + d*x)**7*cos(c + d*x)/(32*d) - 511*a**2*b**6*si
n(c + d*x)**5*cos(c + d*x)**3/(96*d) - 385*a**2*b**6*sin(c + d*x)**3*cos(c + d*x)**5/(96*d) - 35*a**2*b**6*sin
(c + d*x)*cos(c + d*x)**7/(32*d) - 8*a*b**7*sin(c + d*x)**6*cos(c + d*x)**3/(3*d) - 16*a*b**7*sin(c + d*x)**4*
cos(c + d*x)**5/(5*d) - 64*a*b**7*sin(c + d*x)**2*cos(c + d*x)**7/(35*d) - 128*a*b**7*cos(c + d*x)**9/(315*d)
+ 7*b**8*x*sin(c + d*x)**10/256 + 35*b**8*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 35*b**8*x*sin(c + d*x)**6*co
s(c + d*x)**4/128 + 35*b**8*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 35*b**8*x*sin(c + d*x)**2*cos(c + d*x)**8/
256 + 7*b**8*x*cos(c + d*x)**10/256 + 7*b**8*sin(c + d*x)**9*cos(c + d*x)/(256*d) - 79*b**8*sin(c + d*x)**7*co
s(c + d*x)**3/(384*d) - 7*b**8*sin(c + d*x)**5*cos(c + d*x)**5/(30*d) - 49*b**8*sin(c + d*x)**3*cos(c + d*x)**
7/(384*d) - 7*b**8*sin(c + d*x)*cos(c + d*x)**9/(256*d), Ne(d, 0)), (x*(a + b*sin(c))**8*cos(c)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.79 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {1720320 \, a^{7} b \cos \left (d x + c\right )^{3} - 161280 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8} - 564480 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{6} b^{2} - 2408448 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{5} b^{3} + 235200 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} b^{4} + 344064 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{3} b^{5} + 5880 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 120 \, d x - 120 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{6} - 16384 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a b^{7} - 21 \, {\left (96 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c - 45 \, \sin \left (8 \, d x + 8 \, c\right ) - 120 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{8}}{645120 \, d} \]

[In]

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

[Out]

-1/645120*(1720320*a^7*b*cos(d*x + c)^3 - 161280*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^8 - 564480*(4*d*x + 4*c -
sin(4*d*x + 4*c))*a^6*b^2 - 2408448*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a^5*b^3 + 235200*(4*sin(2*d*x + 2*c)
^3 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*a^4*b^4 + 344064*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x
+ c)^3)*a^3*b^5 + 5880*(64*sin(2*d*x + 2*c)^3 - 120*d*x - 120*c + 3*sin(8*d*x + 8*c) + 24*sin(4*d*x + 4*c))*a^
2*b^6 - 16384*(35*cos(d*x + c)^9 - 135*cos(d*x + c)^7 + 189*cos(d*x + c)^5 - 105*cos(d*x + c)^3)*a*b^7 - 21*(9
6*sin(2*d*x + 2*c)^5 - 640*sin(2*d*x + 2*c)^3 + 840*d*x + 840*c - 45*sin(8*d*x + 8*c) - 120*sin(4*d*x + 4*c))*
b^8)/d

Giac [A] (verification not implemented)

none

Time = 0.55 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {a b^{7} \cos \left (9 \, d x + 9 \, c\right )}{288 \, d} + \frac {b^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {1}{256} \, {\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} x - \frac {{\left (28 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} + \frac {{\left (28 \, a^{5} b^{3} + 21 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac {{\left (16 \, a^{7} b + 28 \, a^{5} b^{3} + 7 \, a^{3} b^{5}\right )} \cos \left (3 \, d x + 3 \, c\right )}{24 \, d} - \frac {{\left (32 \, a^{7} b + 112 \, a^{5} b^{3} + 70 \, a^{3} b^{5} + 7 \, a b^{7}\right )} \cos \left (d x + c\right )}{16 \, d} - \frac {{\left (56 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {{\left (1120 \, a^{4} b^{4} + 448 \, a^{2} b^{6} + 13 \, b^{8}\right )} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {{\left (224 \, a^{6} b^{2} + 280 \, a^{4} b^{4} + 56 \, a^{2} b^{6} + b^{8}\right )} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {{\left (128 \, a^{8} - 560 \, a^{4} b^{4} - 224 \, a^{2} b^{6} - 7 \, b^{8}\right )} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

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

[Out]

1/288*a*b^7*cos(9*d*x + 9*c)/d + 1/5120*b^8*sin(10*d*x + 10*c)/d + 1/256*(128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4
+ 280*a^2*b^6 + 7*b^8)*x - 1/224*(28*a^3*b^5 + 5*a*b^7)*cos(7*d*x + 7*c)/d + 1/40*(28*a^5*b^3 + 21*a^3*b^5 +
2*a*b^7)*cos(5*d*x + 5*c)/d - 1/24*(16*a^7*b + 28*a^5*b^3 + 7*a^3*b^5)*cos(3*d*x + 3*c)/d - 1/16*(32*a^7*b + 1
12*a^5*b^3 + 70*a^3*b^5 + 7*a*b^7)*cos(d*x + c)/d - 1/2048*(56*a^2*b^6 + 3*b^8)*sin(8*d*x + 8*c)/d + 1/3072*(1
120*a^4*b^4 + 448*a^2*b^6 + 13*b^8)*sin(6*d*x + 6*c)/d - 1/256*(224*a^6*b^2 + 280*a^4*b^4 + 56*a^2*b^6 + b^8)*
sin(4*d*x + 4*c)/d + 1/512*(128*a^8 - 560*a^4*b^4 - 224*a^2*b^6 - 7*b^8)*sin(2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 6.88 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.10 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {\frac {2205\,b^8\,\sin \left (2\,c+2\,d\,x\right )}{2}-20160\,a^8\,\sin \left (2\,c+2\,d\,x\right )+315\,b^8\,\sin \left (4\,c+4\,d\,x\right )-\frac {1365\,b^8\,\sin \left (6\,c+6\,d\,x\right )}{4}+\frac {945\,b^8\,\sin \left (8\,c+8\,d\,x\right )}{8}-\frac {63\,b^8\,\sin \left (10\,c+10\,d\,x\right )}{4}+53760\,a^7\,b\,\cos \left (3\,c+3\,d\,x\right )-4032\,a\,b^7\,\cos \left (5\,c+5\,d\,x\right )+1800\,a\,b^7\,\cos \left (7\,c+7\,d\,x\right )-280\,a\,b^7\,\cos \left (9\,c+9\,d\,x\right )+352800\,a^3\,b^5\,\cos \left (c+d\,x\right )+564480\,a^5\,b^3\,\cos \left (c+d\,x\right )+23520\,a^3\,b^5\,\cos \left (3\,c+3\,d\,x\right )+94080\,a^5\,b^3\,\cos \left (3\,c+3\,d\,x\right )-42336\,a^3\,b^5\,\cos \left (5\,c+5\,d\,x\right )-56448\,a^5\,b^3\,\cos \left (5\,c+5\,d\,x\right )+10080\,a^3\,b^5\,\cos \left (7\,c+7\,d\,x\right )+35280\,a^2\,b^6\,\sin \left (2\,c+2\,d\,x\right )+88200\,a^4\,b^4\,\sin \left (2\,c+2\,d\,x\right )+17640\,a^2\,b^6\,\sin \left (4\,c+4\,d\,x\right )+88200\,a^4\,b^4\,\sin \left (4\,c+4\,d\,x\right )+70560\,a^6\,b^2\,\sin \left (4\,c+4\,d\,x\right )-11760\,a^2\,b^6\,\sin \left (6\,c+6\,d\,x\right )-29400\,a^4\,b^4\,\sin \left (6\,c+6\,d\,x\right )+2205\,a^2\,b^6\,\sin \left (8\,c+8\,d\,x\right )+35280\,a\,b^7\,\cos \left (c+d\,x\right )+161280\,a^7\,b\,\cos \left (c+d\,x\right )-40320\,a^8\,d\,x-2205\,b^8\,d\,x-88200\,a^2\,b^6\,d\,x-352800\,a^4\,b^4\,d\,x-282240\,a^6\,b^2\,d\,x}{80640\,d} \]

[In]

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

[Out]

-((2205*b^8*sin(2*c + 2*d*x))/2 - 20160*a^8*sin(2*c + 2*d*x) + 315*b^8*sin(4*c + 4*d*x) - (1365*b^8*sin(6*c +
6*d*x))/4 + (945*b^8*sin(8*c + 8*d*x))/8 - (63*b^8*sin(10*c + 10*d*x))/4 + 53760*a^7*b*cos(3*c + 3*d*x) - 4032
*a*b^7*cos(5*c + 5*d*x) + 1800*a*b^7*cos(7*c + 7*d*x) - 280*a*b^7*cos(9*c + 9*d*x) + 352800*a^3*b^5*cos(c + d*
x) + 564480*a^5*b^3*cos(c + d*x) + 23520*a^3*b^5*cos(3*c + 3*d*x) + 94080*a^5*b^3*cos(3*c + 3*d*x) - 42336*a^3
*b^5*cos(5*c + 5*d*x) - 56448*a^5*b^3*cos(5*c + 5*d*x) + 10080*a^3*b^5*cos(7*c + 7*d*x) + 35280*a^2*b^6*sin(2*
c + 2*d*x) + 88200*a^4*b^4*sin(2*c + 2*d*x) + 17640*a^2*b^6*sin(4*c + 4*d*x) + 88200*a^4*b^4*sin(4*c + 4*d*x)
+ 70560*a^6*b^2*sin(4*c + 4*d*x) - 11760*a^2*b^6*sin(6*c + 6*d*x) - 29400*a^4*b^4*sin(6*c + 6*d*x) + 2205*a^2*
b^6*sin(8*c + 8*d*x) + 35280*a*b^7*cos(c + d*x) + 161280*a^7*b*cos(c + d*x) - 40320*a^8*d*x - 2205*b^8*d*x - 8
8200*a^2*b^6*d*x - 352800*a^4*b^4*d*x - 282240*a^6*b^2*d*x)/(80640*d)