3.44 \(\int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=262 \[ -\frac {2431 a^8 \cos ^3(c+d x)}{384 d}-\frac {2431 \cos ^3(c+d x) \left (a^8 \sin (c+d x)+a^8\right )}{640 d}+\frac {2431 a^8 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {2431 a^8 x}{256}-\frac {2431 \cos ^3(c+d x) \left (a^4 \sin (c+d x)+a^4\right )^2}{1120 d}-\frac {17 a^3 \cos ^3(c+d x) (a \sin (c+d x)+a)^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^6}{90 d}-\frac {2431 a^2 \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^3}{2016 d}-\frac {221 \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^4}{336 d}-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^7}{10 d} \]

[Out]

2431/256*a^8*x-2431/384*a^8*cos(d*x+c)^3/d+2431/256*a^8*cos(d*x+c)*sin(d*x+c)/d-17/48*a^3*cos(d*x+c)^3*(a+a*si
n(d*x+c))^5/d-17/90*a^2*cos(d*x+c)^3*(a+a*sin(d*x+c))^6/d-1/10*a*cos(d*x+c)^3*(a+a*sin(d*x+c))^7/d-2431/2016*a
^2*cos(d*x+c)^3*(a^2+a^2*sin(d*x+c))^3/d-221/336*cos(d*x+c)^3*(a^2+a^2*sin(d*x+c))^4/d-2431/1120*cos(d*x+c)^3*
(a^4+a^4*sin(d*x+c))^2/d-2431/640*cos(d*x+c)^3*(a^8+a^8*sin(d*x+c))/d

________________________________________________________________________________________

Rubi [A]  time = 0.37, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2678, 2669, 2635, 8} \[ -\frac {2431 a^8 \cos ^3(c+d x)}{384 d}-\frac {17 a^3 \cos ^3(c+d x) (a \sin (c+d x)+a)^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^6}{90 d}-\frac {2431 a^2 \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^3}{2016 d}-\frac {221 \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^4}{336 d}-\frac {2431 \cos ^3(c+d x) \left (a^4 \sin (c+d x)+a^4\right )^2}{1120 d}-\frac {2431 \cos ^3(c+d x) \left (a^8 \sin (c+d x)+a^8\right )}{640 d}+\frac {2431 a^8 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {2431 a^8 x}{256}-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^7}{10 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2431*a^8*x)/256 - (2431*a^8*Cos[c + d*x]^3)/(384*d) + (2431*a^8*Cos[c + d*x]*Sin[c + d*x])/(256*d) - (17*a^3*
Cos[c + d*x]^3*(a + a*Sin[c + d*x])^5)/(48*d) - (17*a^2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^6)/(90*d) - (a*Cos
[c + d*x]^3*(a + a*Sin[c + d*x])^7)/(10*d) - (2431*a^2*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^3)/(2016*d) - (
221*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^4)/(336*d) - (2431*Cos[c + d*x]^3*(a^4 + a^4*Sin[c + d*x])^2)/(112
0*d) - (2431*Cos[c + d*x]^3*(a^8 + a^8*Sin[c + d*x]))/(640*d)

Rule 8

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

Rule 2635

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] && Integer
Q[2*n]

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
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 2678

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[(a*(2*m + p - 1))/(m + p), Int[(
g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0]
&& GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rubi steps

\begin {align*} \int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx &=-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}+\frac {1}{10} (17 a) \int \cos ^2(c+d x) (a+a \sin (c+d x))^7 \, dx\\ &=-\frac {17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}+\frac {1}{6} \left (17 a^2\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^6 \, dx\\ &=-\frac {17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}+\frac {1}{48} \left (221 a^3\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^5 \, dx\\ &=-\frac {17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac {221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}+\frac {1}{336} \left (2431 a^4\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx\\ &=-\frac {2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac {17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac {221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}+\frac {1}{224} \left (2431 a^5\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\\ &=-\frac {2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac {17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac {221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac {2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}+\frac {1}{160} \left (2431 a^6\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac {17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac {221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac {2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}-\frac {2431 \cos ^3(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{640 d}+\frac {1}{128} \left (2431 a^7\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {2431 a^8 \cos ^3(c+d x)}{384 d}-\frac {2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac {17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac {221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac {2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}-\frac {2431 \cos ^3(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{640 d}+\frac {1}{128} \left (2431 a^8\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {2431 a^8 \cos ^3(c+d x)}{384 d}+\frac {2431 a^8 \cos (c+d x) \sin (c+d x)}{256 d}-\frac {2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac {17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac {221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac {2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}-\frac {2431 \cos ^3(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{640 d}+\frac {1}{256} \left (2431 a^8\right ) \int 1 \, dx\\ &=\frac {2431 a^8 x}{256}-\frac {2431 a^8 \cos ^3(c+d x)}{384 d}+\frac {2431 a^8 \cos (c+d x) \sin (c+d x)}{256 d}-\frac {2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac {17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac {221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac {2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}-\frac {2431 \cos ^3(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{640 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.49, size = 191, normalized size = 0.73 \[ -\frac {a^8 \left (1531530 \sqrt {1-\sin (c+d x)} \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )+\sqrt {\sin (c+d x)+1} \left (8064 \sin ^{10}(c+d x)+63616 \sin ^9(c+d x)+209552 \sin ^8(c+d x)+353648 \sin ^7(c+d x)+257704 \sin ^6(c+d x)-130728 \sin ^5(c+d x)-492846 \sin ^4(c+d x)-543442 \sin ^3(c+d x)-410693 \sin ^2(c+d x)-508859 \sin (c+d x)+1193984\right )\right ) \cos ^3(c+d x)}{80640 d (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/80640*(a^8*Cos[c + d*x]^3*(1531530*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 +
Sin[c + d*x]]*(1193984 - 508859*Sin[c + d*x] - 410693*Sin[c + d*x]^2 - 543442*Sin[c + d*x]^3 - 492846*Sin[c +
d*x]^4 - 130728*Sin[c + d*x]^5 + 257704*Sin[c + d*x]^6 + 353648*Sin[c + d*x]^7 + 209552*Sin[c + d*x]^8 + 6361
6*Sin[c + d*x]^9 + 8064*Sin[c + d*x]^10)))/(d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/2))

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 137, normalized size = 0.52 \[ \frac {71680 \, a^{8} \cos \left (d x + c\right )^{9} - 921600 \, a^{8} \cos \left (d x + c\right )^{7} + 3096576 \, a^{8} \cos \left (d x + c\right )^{5} - 3440640 \, a^{8} \cos \left (d x + c\right )^{3} + 765765 \, a^{8} d x + 63 \, {\left (128 \, a^{8} \cos \left (d x + c\right )^{9} - 4976 \, a^{8} \cos \left (d x + c\right )^{7} + 28328 \, a^{8} \cos \left (d x + c\right )^{5} - 46510 \, a^{8} \cos \left (d x + c\right )^{3} + 12155 \, a^{8} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80640*(71680*a^8*cos(d*x + c)^9 - 921600*a^8*cos(d*x + c)^7 + 3096576*a^8*cos(d*x + c)^5 - 3440640*a^8*cos(d
*x + c)^3 + 765765*a^8*d*x + 63*(128*a^8*cos(d*x + c)^9 - 4976*a^8*cos(d*x + c)^7 + 28328*a^8*cos(d*x + c)^5 -
46510*a^8*cos(d*x + c)^3 + 12155*a^8*cos(d*x + c))*sin(d*x + c))/d

________________________________________________________________________________________

giac [A]  time = 1.50, size = 174, normalized size = 0.66 \[ \frac {2431}{256} \, a^{8} x + \frac {a^{8} \cos \left (9 \, d x + 9 \, c\right )}{288 \, d} - \frac {33 \, a^{8} \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} + \frac {51 \, a^{8} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac {17 \, a^{8} \cos \left (3 \, d x + 3 \, c\right )}{8 \, d} - \frac {221 \, a^{8} \cos \left (d x + c\right )}{16 \, d} + \frac {a^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac {59 \, a^{8} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {527 \, a^{8} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {561 \, a^{8} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {663 \, a^{8} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2431/256*a^8*x + 1/288*a^8*cos(9*d*x + 9*c)/d - 33/224*a^8*cos(7*d*x + 7*c)/d + 51/40*a^8*cos(5*d*x + 5*c)/d -
17/8*a^8*cos(3*d*x + 3*c)/d - 221/16*a^8*cos(d*x + c)/d + 1/5120*a^8*sin(10*d*x + 10*c)/d - 59/2048*a^8*sin(8
*d*x + 8*c)/d + 527/1024*a^8*sin(6*d*x + 6*c)/d - 561/256*a^8*sin(4*d*x + 4*c)/d - 663/512*a^8*sin(2*d*x + 2*c
)/d

________________________________________________________________________________________

maple [A]  time = 0.14, size = 480, normalized size = 1.83 \[ \frac {a^{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 (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{96}-\frac {7 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\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 )+8 a^{8} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{9}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{21}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{105}-\frac {16 \left (\cos ^{3}\left (d x +c \right )\right )}{315}\right )+28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\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 )+56 a^{8} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\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^{8} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right ) a^{8}}{3}+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 )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^8*(-1/10*sin(d*x+c)^7*cos(d*x+c)^3-7/80*sin(d*x+c)^5*cos(d*x+c)^3-7/96*sin(d*x+c)^3*cos(d*x+c)^3-7/128*
sin(d*x+c)*cos(d*x+c)^3+7/256*cos(d*x+c)*sin(d*x+c)+7/256*d*x+7/256*c)+8*a^8*(-1/9*sin(d*x+c)^6*cos(d*x+c)^3-2
/21*sin(d*x+c)^4*cos(d*x+c)^3-8/105*sin(d*x+c)^2*cos(d*x+c)^3-16/315*cos(d*x+c)^3)+28*a^8*(-1/8*sin(d*x+c)^5*c
os(d*x+c)^3-5/48*sin(d*x+c)^3*cos(d*x+c)^3-5/64*sin(d*x+c)*cos(d*x+c)^3+5/128*cos(d*x+c)*sin(d*x+c)+5/128*d*x+
5/128*c)+56*a^8*(-1/7*sin(d*x+c)^4*cos(d*x+c)^3-4/35*sin(d*x+c)^2*cos(d*x+c)^3-8/105*cos(d*x+c)^3)+70*a^8*(-1/
6*sin(d*x+c)^3*cos(d*x+c)^3-1/8*sin(d*x+c)*cos(d*x+c)^3+1/16*cos(d*x+c)*sin(d*x+c)+1/16*d*x+1/16*c)+56*a^8*(-1
/5*sin(d*x+c)^2*cos(d*x+c)^3-2/15*cos(d*x+c)^3)+28*a^8*(-1/4*sin(d*x+c)*cos(d*x+c)^3+1/8*cos(d*x+c)*sin(d*x+c)
+1/8*d*x+1/8*c)-8/3*cos(d*x+c)^3*a^8+a^8*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 319, normalized size = 1.22 \[ -\frac {1720320 \, a^{8} \cos \left (d x + c\right )^{3} - 16384 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 344064 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{8} - 2408448 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{8} - 21 \, {\left (96 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c - 45 \, \sin \left (8 \, d x + 8 \, c\right ) - 120 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} + 5880 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 120 \, d x - 120 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} + 235200 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 564480 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 161280 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8}}{645120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/645120*(1720320*a^8*cos(d*x + c)^3 - 16384*(35*cos(d*x + c)^9 - 135*cos(d*x + c)^7 + 189*cos(d*x + c)^5 - 1
05*cos(d*x + c)^3)*a^8 + 344064*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*a^8 - 2408448*(3*c
os(d*x + c)^5 - 5*cos(d*x + c)^3)*a^8 - 21*(96*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))*a^8 + 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^8 + 235200*(4*sin(2*d*x + 2*c)^3 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*a^8 -
564480*(4*d*x + 4*c - sin(4*d*x + 4*c))*a^8 - 161280*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^8)/d

________________________________________________________________________________________

mupad [B]  time = 7.26, size = 572, normalized size = 2.18 \[ \frac {2431\,a^8\,x}{256}-\frac {\frac {11809\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}-\frac {23647\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {40749\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}-\frac {70499\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {70499\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {40749\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {23647\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}-\frac {11809\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {2175\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}\right )-a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}-\frac {9328}{315}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (10\,a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}\right )-a^8\,\left (\frac {12155\,c}{128}+\frac {12155\,d\,x}{128}-16\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (10\,a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}\right )-a^8\,\left (\frac {12155\,c}{128}+\frac {12155\,d\,x}{128}-\frac {17648}{63}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (120\,a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}\right )-a^8\,\left (\frac {36465\,c}{32}+\frac {36465\,d\,x}{32}-1984\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (120\,a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}\right )-a^8\,\left (\frac {36465\,c}{32}+\frac {36465\,d\,x}{32}-\frac {32960}{21}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (45\,a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}\right )-a^8\,\left (\frac {109395\,c}{256}+\frac {109395\,d\,x}{256}-336\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (45\,a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}\right )-a^8\,\left (\frac {109395\,c}{256}+\frac {109395\,d\,x}{256}-\frac {6976}{7}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (252\,a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}\right )-a^8\,\left (\frac {153153\,c}{64}+\frac {153153\,d\,x}{64}-\frac {18656}{5}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (210\,a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}\right )-a^8\,\left (\frac {255255\,c}{128}+\frac {255255\,d\,x}{128}-4288\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (210\,a^8\,\left (\frac {2431\,c}{256}+\frac {2431\,d\,x}{256}\right )-a^8\,\left (\frac {255255\,c}{128}+\frac {255255\,d\,x}{128}-\frac {5792}{3}\right )\right )+\frac {2175\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2431*a^8*x)/256 - ((11809*a^8*tan(c/2 + (d*x)/2)^3)/128 - (23647*a^8*tan(c/2 + (d*x)/2)^5)/160 - (40749*a^8*t
an(c/2 + (d*x)/2)^7)/32 - (70499*a^8*tan(c/2 + (d*x)/2)^9)/64 + (70499*a^8*tan(c/2 + (d*x)/2)^11)/64 + (40749*
a^8*tan(c/2 + (d*x)/2)^13)/32 + (23647*a^8*tan(c/2 + (d*x)/2)^15)/160 - (11809*a^8*tan(c/2 + (d*x)/2)^17)/128
- (2175*a^8*tan(c/2 + (d*x)/2)^19)/128 + a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((2431*c)/256 + (2431*d*x)/
256 - 9328/315) + tan(c/2 + (d*x)/2)^18*(10*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((12155*c)/128 + (12155*
d*x)/128 - 16)) + tan(c/2 + (d*x)/2)^2*(10*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((12155*c)/128 + (12155*d
*x)/128 - 17648/63)) + tan(c/2 + (d*x)/2)^14*(120*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((36465*c)/32 + (3
6465*d*x)/32 - 1984)) + tan(c/2 + (d*x)/2)^6*(120*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((36465*c)/32 + (3
6465*d*x)/32 - 32960/21)) + tan(c/2 + (d*x)/2)^16*(45*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((109395*c)/25
6 + (109395*d*x)/256 - 336)) + tan(c/2 + (d*x)/2)^4*(45*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((109395*c)/
256 + (109395*d*x)/256 - 6976/7)) + tan(c/2 + (d*x)/2)^10*(252*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((153
153*c)/64 + (153153*d*x)/64 - 18656/5)) + tan(c/2 + (d*x)/2)^12*(210*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8
*((255255*c)/128 + (255255*d*x)/128 - 4288)) + tan(c/2 + (d*x)/2)^8*(210*a^8*((2431*c)/256 + (2431*d*x)/256) -
a^8*((255255*c)/128 + (255255*d*x)/128 - 5792/3)) + (2175*a^8*tan(c/2 + (d*x)/2))/128)/(d*(tan(c/2 + (d*x)/2)
^2 + 1)^10)

________________________________________________________________________________________

sympy [A]  time = 37.27, size = 1018, normalized size = 3.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________