3.42 \(\int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=286 \[ -\frac {4199 a^8 \cos ^5(c+d x)}{1920 d}-\frac {4199 \cos ^5(c+d x) \left (a^8 \sin (c+d x)+a^8\right )}{2688 d}+\frac {4199 a^8 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac {4199 a^8 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {4199 a^8 x}{1024}-\frac {4199 \cos ^5(c+d x) \left (a^4 \sin (c+d x)+a^4\right )^2}{4032 d}-\frac {323 a^3 \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{132 d}-\frac {4199 a^2 \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^3}{6336 d}-\frac {323 \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^4}{792 d}-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d} \]

[Out]

4199/1024*a^8*x-4199/1920*a^8*cos(d*x+c)^5/d+4199/1024*a^8*cos(d*x+c)*sin(d*x+c)/d+4199/1536*a^8*cos(d*x+c)^3*
sin(d*x+c)/d-323/1320*a^3*cos(d*x+c)^5*(a+a*sin(d*x+c))^5/d-19/132*a^2*cos(d*x+c)^5*(a+a*sin(d*x+c))^6/d-1/12*
a*cos(d*x+c)^5*(a+a*sin(d*x+c))^7/d-4199/6336*a^2*cos(d*x+c)^5*(a^2+a^2*sin(d*x+c))^3/d-323/792*cos(d*x+c)^5*(
a^2+a^2*sin(d*x+c))^4/d-4199/4032*cos(d*x+c)^5*(a^4+a^4*sin(d*x+c))^2/d-4199/2688*cos(d*x+c)^5*(a^8+a^8*sin(d*
x+c))/d

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Rubi [A]  time = 0.40, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2678, 2669, 2635, 8} \[ -\frac {4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac {4199 a^8 \sin (c+d x) \cos ^3(c+d x)}{1536 d}-\frac {323 a^3 \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{132 d}-\frac {4199 a^2 \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^3}{6336 d}-\frac {323 \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^4}{792 d}-\frac {4199 \cos ^5(c+d x) \left (a^4 \sin (c+d x)+a^4\right )^2}{4032 d}-\frac {4199 \cos ^5(c+d x) \left (a^8 \sin (c+d x)+a^8\right )}{2688 d}+\frac {4199 a^8 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {4199 a^8 x}{1024}-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^8,x]

[Out]

(4199*a^8*x)/1024 - (4199*a^8*Cos[c + d*x]^5)/(1920*d) + (4199*a^8*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (4199
*a^8*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) - (323*a^3*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^5)/(1320*d) - (19*a^
2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^6)/(132*d) - (a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^7)/(12*d) - (4199*a^
2*Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x])^3)/(6336*d) - (323*Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x])^4)/(792*d
) - (4199*Cos[c + d*x]^5*(a^4 + a^4*Sin[c + d*x])^2)/(4032*d) - (4199*Cos[c + d*x]^5*(a^8 + a^8*Sin[c + d*x]))
/(2688*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 ^4(c+d x) (a+a \sin (c+d x))^8 \, dx &=-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}+\frac {1}{12} (19 a) \int \cos ^4(c+d x) (a+a \sin (c+d x))^7 \, dx\\ &=-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}+\frac {1}{132} \left (323 a^2\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x))^6 \, dx\\ &=-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}+\frac {1}{88} \left (323 a^3\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x))^5 \, dx\\ &=-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}+\frac {1}{792} \left (4199 a^4\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x))^4 \, dx\\ &=-\frac {4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}+\frac {1}{576} \left (4199 a^5\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\\ &=-\frac {4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac {4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}+\frac {1}{448} \left (4199 a^6\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac {4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac {4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d}+\frac {1}{384} \left (4199 a^7\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {4199 a^8 \cos ^5(c+d x)}{1920 d}-\frac {4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac {4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac {4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d}+\frac {1}{384} \left (4199 a^8\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac {4199 a^8 \cos ^3(c+d x) \sin (c+d x)}{1536 d}-\frac {4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac {4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac {4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d}+\frac {1}{512} \left (4199 a^8\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac {4199 a^8 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {4199 a^8 \cos ^3(c+d x) \sin (c+d x)}{1536 d}-\frac {4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac {4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac {4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d}+\frac {\left (4199 a^8\right ) \int 1 \, dx}{1024}\\ &=\frac {4199 a^8 x}{1024}-\frac {4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac {4199 a^8 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {4199 a^8 \cos ^3(c+d x) \sin (c+d x)}{1536 d}-\frac {4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac {4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac {4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d}\\ \end {align*}

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Mathematica [A]  time = 3.20, size = 211, normalized size = 0.74 \[ -\frac {a^8 \left (\sqrt {\sin (c+d x)+1} \left (295680 \sin ^{12}(c+d x)+2284800 \sin ^{11}(c+d x)+6969984 \sin ^{10}(c+d x)+9086336 \sin ^9(c+d x)-1239728 \sin ^8(c+d x)-20428112 \sin ^7(c+d x)-26346616 \sin ^6(c+d x)-8321928 \sin ^5(c+d x)[phone] \sin ^4(c+d x)[phone] \sin ^3(c+d x)[phone] \sin ^2(c+d x)[phone] \sin (c+d x)-22470656\right )-29099070 \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}\right ) \cos ^5(c+d x)}{3548160 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^8,x]

[Out]

-1/3548160*(a^8*Cos[c + d*x]^5*(-29099070*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt
[1 + Sin[c + d*x]]*(-22470656 + 11469281*Sin[c + d*x] + 13958687*Sin[c + d*x]^2 + 20459158*Sin[c + d*x]^3 + 14
283114*Sin[c + d*x]^4 - 8321928*Sin[c + d*x]^5 - 26346616*Sin[c + d*x]^6 - 20428112*Sin[c + d*x]^7 - 1239728*S
in[c + d*x]^8 + 9086336*Sin[c + d*x]^9 + 6969984*Sin[c + d*x]^10 + 2284800*Sin[c + d*x]^11 + 295680*Sin[c + d*
x]^12)))/(d*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^(5/2))

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fricas [A]  time = 0.64, size = 150, normalized size = 0.52 \[ \frac {2580480 \, a^{8} \cos \left (d x + c\right )^{11} - 31539200 \, a^{8} \cos \left (d x + c\right )^{9} + 97320960 \, a^{8} \cos \left (d x + c\right )^{7} - 90832896 \, a^{8} \cos \left (d x + c\right )^{5} + 14549535 \, a^{8} d x + 231 \, {\left (1280 \, a^{8} \cos \left (d x + c\right )^{11} - 47744 \, a^{8} \cos \left (d x + c\right )^{9} + 253488 \, a^{8} \cos \left (d x + c\right )^{7} - 359624 \, a^{8} \cos \left (d x + c\right )^{5} + 41990 \, a^{8} \cos \left (d x + c\right )^{3} + 62985 \, a^{8} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/3548160*(2580480*a^8*cos(d*x + c)^11 - 31539200*a^8*cos(d*x + c)^9 + 97320960*a^8*cos(d*x + c)^7 - 90832896*
a^8*cos(d*x + c)^5 + 14549535*a^8*d*x + 231*(1280*a^8*cos(d*x + c)^11 - 47744*a^8*cos(d*x + c)^9 + 253488*a^8*
cos(d*x + c)^7 - 359624*a^8*cos(d*x + c)^5 + 41990*a^8*cos(d*x + c)^3 + 62985*a^8*cos(d*x + c))*sin(d*x + c))/
d

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giac [A]  time = 1.97, size = 208, normalized size = 0.73 \[ \frac {4199}{1024} \, a^{8} x + \frac {a^{8} \cos \left (11 \, d x + 11 \, c\right )}{1408 \, d} - \frac {31 \, a^{8} \cos \left (9 \, d x + 9 \, c\right )}{1152 \, d} + \frac {139 \, a^{8} \cos \left (7 \, d x + 7 \, c\right )}{896 \, d} + \frac {171 \, a^{8} \cos \left (5 \, d x + 5 \, c\right )}{640 \, d} - \frac {323 \, a^{8} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {323 \, a^{8} \cos \left (d x + c\right )}{64 \, d} + \frac {a^{8} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} - \frac {29 \, a^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {673 \, a^{8} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {361 \, a^{8} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {8721 \, a^{8} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac {323 \, a^{8} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

4199/1024*a^8*x + 1/1408*a^8*cos(11*d*x + 11*c)/d - 31/1152*a^8*cos(9*d*x + 9*c)/d + 139/896*a^8*cos(7*d*x + 7
*c)/d + 171/640*a^8*cos(5*d*x + 5*c)/d - 323/192*a^8*cos(3*d*x + 3*c)/d - 323/64*a^8*cos(d*x + c)/d + 1/24576*
a^8*sin(12*d*x + 12*c)/d - 29/5120*a^8*sin(10*d*x + 10*c)/d + 673/8192*a^8*sin(8*d*x + 8*c)/d - 361/3072*a^8*s
in(6*d*x + 6*c)/d - 8721/8192*a^8*sin(4*d*x + 4*c)/d + 323/512*a^8*sin(2*d*x + 2*c)/d

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maple [B]  time = 0.20, size = 535, normalized size = 1.87 \[ \frac {a^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{12}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{120}-\frac {7 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{192}-\frac {7 \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{384}+\frac {7 \left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{1536}+\frac {7 d x}{1024}+\frac {7 c}{1024}\right )+8 a^{8} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{11}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{33}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{231}-\frac {16 \left (\cos ^{5}\left (d x +c \right )\right )}{1155}\right )+28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+56 a^{8} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+56 a^{8} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right ) a^{8}}{5}+a^{8} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(a+a*sin(d*x+c))^8,x)

[Out]

1/d*(a^8*(-1/12*sin(d*x+c)^7*cos(d*x+c)^5-7/120*sin(d*x+c)^5*cos(d*x+c)^5-7/192*sin(d*x+c)^3*cos(d*x+c)^5-7/38
4*sin(d*x+c)*cos(d*x+c)^5+7/1536*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+7/1024*d*x+7/1024*c)+8*a^8*(-1/11*si
n(d*x+c)^6*cos(d*x+c)^5-2/33*sin(d*x+c)^4*cos(d*x+c)^5-8/231*sin(d*x+c)^2*cos(d*x+c)^5-16/1155*cos(d*x+c)^5)+2
8*a^8*(-1/10*sin(d*x+c)^5*cos(d*x+c)^5-1/16*sin(d*x+c)^3*cos(d*x+c)^5-1/32*sin(d*x+c)*cos(d*x+c)^5+1/128*(cos(
d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/256*d*x+3/256*c)+56*a^8*(-1/9*sin(d*x+c)^4*cos(d*x+c)^5-4/63*sin(d*x+c)^
2*cos(d*x+c)^5-8/315*cos(d*x+c)^5)+70*a^8*(-1/8*sin(d*x+c)^3*cos(d*x+c)^5-1/16*sin(d*x+c)*cos(d*x+c)^5+1/64*(c
os(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c)+56*a^8*(-1/7*sin(d*x+c)^2*cos(d*x+c)^5-2/35*cos(d*x+
c)^5)+28*a^8*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c)-8/5*
cos(d*x+c)^5*a^8+a^8*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c))

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maxima [A]  time = 0.66, size = 339, normalized size = 1.19 \[ -\frac {45416448 \, a^{8} \cos \left (d x + c\right )^{5} - 196608 \, {\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 5046272 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{8} - 45416448 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 231 \, {\left (384 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 20 \, \sin \left (4 \, d x + 4 \, c\right )^{3} - 840 \, d x - 840 \, c - 15 \, \sin \left (8 \, d x + 8 \, c\right ) + 240 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} + 77616 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 4139520 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 1940400 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 887040 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8}}{28385280 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/28385280*(45416448*a^8*cos(d*x + c)^5 - 196608*(105*cos(d*x + c)^11 - 385*cos(d*x + c)^9 + 495*cos(d*x + c)
^7 - 231*cos(d*x + c)^5)*a^8 + 5046272*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^8 - 45416
448*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^8 + 231*(384*sin(2*d*x + 2*c)^5 + 20*sin(4*d*x + 4*c)^3 - 840*d*x
- 840*c - 15*sin(8*d*x + 8*c) + 240*sin(4*d*x + 4*c))*a^8 + 77616*(32*sin(2*d*x + 2*c)^5 - 120*d*x - 120*c - 5
*sin(8*d*x + 8*c) + 40*sin(4*d*x + 4*c))*a^8 - 4139520*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4
*c))*a^8 - 1940400*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a^8 - 887040*(12*d*x + 12*c + sin(4
*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^8)/d

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mupad [B]  time = 7.05, size = 684, normalized size = 2.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^4*(a + a*sin(c + d*x))^8,x)

[Out]

(4199*a^8*x)/1024 - ((1543*a^8*tan(c/2 + (d*x)/2)^3)/512 - (1068767*a^8*tan(c/2 + (d*x)/2)^5)/2560 - (3297279*
a^8*tan(c/2 + (d*x)/2)^7)/2560 - (168283*a^8*tan(c/2 + (d*x)/2)^9)/3840 + (256139*a^8*tan(c/2 + (d*x)/2)^11)/2
56 - (256139*a^8*tan(c/2 + (d*x)/2)^13)/256 + (168283*a^8*tan(c/2 + (d*x)/2)^15)/3840 + (3297279*a^8*tan(c/2 +
(d*x)/2)^17)/2560 + (1068767*a^8*tan(c/2 + (d*x)/2)^19)/2560 - (1543*a^8*tan(c/2 + (d*x)/2)^21)/512 - (3175*a
^8*tan(c/2 + (d*x)/2)^23)/512 + a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((4199*c)/1024 + (4199*d*x)/1024 -
43888/3465) + tan(c/2 + (d*x)/2)^22*(12*a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((12597*c)/256 + (12597*d
*x)/256 - 16)) + tan(c/2 + (d*x)/2)^2*(12*a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((12597*c)/256 + (12597*
d*x)/256 - 157072/1155)) + tan(c/2 + (d*x)/2)^20*(66*a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((138567*c)/5
12 + (138567*d*x)/512 - 336)) + tan(c/2 + (d*x)/2)^4*(66*a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((138567*
c)/512 + (138567*d*x)/512 - 52496/105)) + tan(c/2 + (d*x)/2)^18*(220*a^8*((4199*c)/1024 + (4199*d*x)/1024) - a
^8*((230945*c)/256 + (230945*d*x)/256 - 5584/3)) + tan(c/2 + (d*x)/2)^6*(220*a^8*((4199*c)/1024 + (4199*d*x)/1
024) - a^8*((230945*c)/256 + (230945*d*x)/256 - 58288/63)) + tan(c/2 + (d*x)/2)^14*(792*a^8*((4199*c)/1024 + (
4199*d*x)/1024) - a^8*((415701*c)/128 + (415701*d*x)/128 - 17696/5)) + tan(c/2 + (d*x)/2)^10*(792*a^8*((4199*c
)/1024 + (4199*d*x)/1024) - a^8*((415701*c)/128 + (415701*d*x)/128 - 227232/35)) + tan(c/2 + (d*x)/2)^12*(924*
a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((969969*c)/256 + (969969*d*x)/256 - 87776/15)) + tan(c/2 + (d*x)/
2)^16*(495*a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((2078505*c)/1024 + (2078505*d*x)/1024 - 3504)) + tan(c
/2 + (d*x)/2)^8*(495*a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((2078505*c)/1024 + (2078505*d*x)/1024 - 1936
0/7)) + (3175*a^8*tan(c/2 + (d*x)/2))/512)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^12)

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sympy [A]  time = 94.80, size = 1280, normalized size = 4.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*(a+a*sin(d*x+c))**8,x)

[Out]

Piecewise((7*a**8*x*sin(c + d*x)**12/1024 + 21*a**8*x*sin(c + d*x)**10*cos(c + d*x)**2/512 + 21*a**8*x*sin(c +
d*x)**10/64 + 105*a**8*x*sin(c + d*x)**8*cos(c + d*x)**4/1024 + 105*a**8*x*sin(c + d*x)**8*cos(c + d*x)**2/64
+ 105*a**8*x*sin(c + d*x)**8/64 + 35*a**8*x*sin(c + d*x)**6*cos(c + d*x)**6/256 + 105*a**8*x*sin(c + d*x)**6*
cos(c + d*x)**4/32 + 105*a**8*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 7*a**8*x*sin(c + d*x)**6/4 + 105*a**8*x*s
in(c + d*x)**4*cos(c + d*x)**8/1024 + 105*a**8*x*sin(c + d*x)**4*cos(c + d*x)**6/32 + 315*a**8*x*sin(c + d*x)*
*4*cos(c + d*x)**4/32 + 21*a**8*x*sin(c + d*x)**4*cos(c + d*x)**2/4 + 3*a**8*x*sin(c + d*x)**4/8 + 21*a**8*x*s
in(c + d*x)**2*cos(c + d*x)**10/512 + 105*a**8*x*sin(c + d*x)**2*cos(c + d*x)**8/64 + 105*a**8*x*sin(c + d*x)*
*2*cos(c + d*x)**6/16 + 21*a**8*x*sin(c + d*x)**2*cos(c + d*x)**4/4 + 3*a**8*x*sin(c + d*x)**2*cos(c + d*x)**2
/4 + 7*a**8*x*cos(c + d*x)**12/1024 + 21*a**8*x*cos(c + d*x)**10/64 + 105*a**8*x*cos(c + d*x)**8/64 + 7*a**8*x
*cos(c + d*x)**6/4 + 3*a**8*x*cos(c + d*x)**4/8 + 7*a**8*sin(c + d*x)**11*cos(c + d*x)/(1024*d) + 119*a**8*sin
(c + d*x)**9*cos(c + d*x)**3/(3072*d) + 21*a**8*sin(c + d*x)**9*cos(c + d*x)/(64*d) - 281*a**8*sin(c + d*x)**7
*cos(c + d*x)**5/(2560*d) + 49*a**8*sin(c + d*x)**7*cos(c + d*x)**3/(32*d) + 105*a**8*sin(c + d*x)**7*cos(c +
d*x)/(64*d) - 8*a**8*sin(c + d*x)**6*cos(c + d*x)**5/(5*d) - 231*a**8*sin(c + d*x)**5*cos(c + d*x)**7/(2560*d)
- 14*a**8*sin(c + d*x)**5*cos(c + d*x)**5/(5*d) + 385*a**8*sin(c + d*x)**5*cos(c + d*x)**3/(64*d) + 7*a**8*si
n(c + d*x)**5*cos(c + d*x)/(4*d) - 48*a**8*sin(c + d*x)**4*cos(c + d*x)**7/(35*d) - 56*a**8*sin(c + d*x)**4*co
s(c + d*x)**5/(5*d) - 119*a**8*sin(c + d*x)**3*cos(c + d*x)**9/(3072*d) - 49*a**8*sin(c + d*x)**3*cos(c + d*x)
**7/(32*d) - 385*a**8*sin(c + d*x)**3*cos(c + d*x)**5/(64*d) + 14*a**8*sin(c + d*x)**3*cos(c + d*x)**3/(3*d) +
3*a**8*sin(c + d*x)**3*cos(c + d*x)/(8*d) - 64*a**8*sin(c + d*x)**2*cos(c + d*x)**9/(105*d) - 32*a**8*sin(c +
d*x)**2*cos(c + d*x)**7/(5*d) - 56*a**8*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 7*a**8*sin(c + d*x)*cos(c + d
*x)**11/(1024*d) - 21*a**8*sin(c + d*x)*cos(c + d*x)**9/(64*d) - 105*a**8*sin(c + d*x)*cos(c + d*x)**7/(64*d)
- 7*a**8*sin(c + d*x)*cos(c + d*x)**5/(4*d) + 5*a**8*sin(c + d*x)*cos(c + d*x)**3/(8*d) - 128*a**8*cos(c + d*x
)**11/(1155*d) - 64*a**8*cos(c + d*x)**9/(45*d) - 16*a**8*cos(c + d*x)**7/(5*d) - 8*a**8*cos(c + d*x)**5/(5*d)
, Ne(d, 0)), (x*(a*sin(c) + a)**8*cos(c)**4, True))

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