\(\int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx\) [415]

Optimal result

Integrand size = 19, antiderivative size = 22 \[ \int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {(a+b \sin (c+d x))^9}{9 b d} \]

[Out]

1/9*(a+b*sin(d*x+c))^9/b/d

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2747, 32} \[ \int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {(a+b \sin (c+d x))^9}{9 b d} \]

[In]

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

[Out]

(a + b*Sin[c + d*x])^9/(9*b*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x)^8 \, dx,x,b \sin (c+d x)\right )}{b d} \\ & = \frac {(a+b \sin (c+d x))^9}{9 b d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(137\) vs. \(2(22)=44\).

Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 6.23 \[ \int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\sin (c+d x) \left (9 a^8+36 a^7 b \sin (c+d x)+84 a^6 b^2 \sin ^2(c+d x)+126 a^5 b^3 \sin ^3(c+d x)+126 a^4 b^4 \sin ^4(c+d x)+84 a^3 b^5 \sin ^5(c+d x)+36 a^2 b^6 \sin ^6(c+d x)+9 a b^7 \sin ^7(c+d x)+b^8 \sin ^8(c+d x)\right )}{9 d} \]

[In]

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

[Out]

(Sin[c + d*x]*(9*a^8 + 36*a^7*b*Sin[c + d*x] + 84*a^6*b^2*Sin[c + d*x]^2 + 126*a^5*b^3*Sin[c + d*x]^3 + 126*a^
4*b^4*Sin[c + d*x]^4 + 84*a^3*b^5*Sin[c + d*x]^5 + 36*a^2*b^6*Sin[c + d*x]^6 + 9*a*b^7*Sin[c + d*x]^7 + b^8*Si
n[c + d*x]^8))/(9*d)

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\left (a +b \sin \left (d x +c \right )\right )^{9}}{9 b d}\)	\(21\)
default	\(\frac {\left (a +b \sin \left (d x +c \right )\right )^{9}}{9 b d}\)	\(21\)
parallelrisch	\(\frac {\left (4608 \sin \left (d x +c \right ) a^{7} b +35 b^{8}+12096 \sin \left (d x +c \right ) a^{5} b^{3}+6720 \sin \left (d x +c \right ) a^{3} b^{5}+630 \sin \left (d x +c \right ) a \,b^{7}+1152 a^{8}+5376 a^{6} b^{2}+6048 a^{4} b^{4}+1440 a^{2} b^{6}-56 b^{8} \cos \left (2 d x +2 c \right )+b^{8} \cos \left (8 d x +8 c \right )+28 b^{8} \cos \left (4 d x +4 c \right )-8 b^{8} \cos \left (6 d x +6 c \right )-378 a \,b^{7} \sin \left (3 d x +3 c \right )+126 a \,b^{7} \sin \left (5 d x +5 c \right )-18 a \,b^{7} \sin \left (7 d x +7 c \right )-5376 a^{6} b^{2} \cos \left (2 d x +2 c \right )-2160 a^{2} b^{6} \cos \left (2 d x +2 c \right )+864 a^{2} b^{6} \cos \left (4 d x +4 c \right )-144 a^{2} b^{6} \cos \left (6 d x +6 c \right )-8064 a^{4} b^{4} \cos \left (2 d x +2 c \right )+2016 a^{4} b^{4} \cos \left (4 d x +4 c \right )-4032 a^{5} b^{3} \sin \left (3 d x +3 c \right )-3360 a^{3} b^{5} \sin \left (3 d x +3 c \right )+672 a^{3} b^{5} \sin \left (5 d x +5 c \right )\right ) \sin \left (d x +c \right )}{1152 d}\)	\(352\)
risch	\(\frac {a^{8} \sin \left (d x +c \right )}{d}+\frac {7 \sin \left (d x +c \right ) b^{8}}{128 d}+\frac {\sin \left (9 d x +9 c \right ) b^{8}}{2304 d}-\frac {2 a^{7} b \cos \left (2 d x +2 c \right )}{d}-\frac {7 a^{5} b^{3} \cos \left (2 d x +2 c \right )}{d}+\frac {7 \sin \left (d x +c \right ) a^{6} b^{2}}{d}+\frac {7 a^{3} b^{5} \cos \left (4 d x +4 c \right )}{4 d}-\frac {7 a^{3} b^{5} \cos \left (6 d x +6 c \right )}{24 d}-\frac {a \,b^{7} \cos \left (6 d x +6 c \right )}{16 d}-\frac {35 a^{3} b^{5} \cos \left (2 d x +2 c \right )}{8 d}-\frac {7 a \,b^{7} \cos \left (2 d x +2 c \right )}{16 d}-\frac {\sin \left (7 d x +7 c \right ) a^{2} b^{6}}{16 d}-\frac {\sin \left (7 d x +7 c \right ) b^{8}}{256 d}+\frac {\sin \left (5 d x +5 c \right ) b^{8}}{64 d}-\frac {7 \sin \left (3 d x +3 c \right ) b^{8}}{192 d}+\frac {7 a^{5} b^{3} \cos \left (4 d x +4 c \right )}{4 d}+\frac {7 a \,b^{7} \cos \left (4 d x +4 c \right )}{32 d}+\frac {35 \sin \left (d x +c \right ) a^{4} b^{4}}{4 d}+\frac {35 \sin \left (d x +c \right ) a^{2} b^{6}}{16 d}-\frac {7 \sin \left (3 d x +3 c \right ) a^{6} b^{2}}{3 d}-\frac {35 \sin \left (3 d x +3 c \right ) a^{4} b^{4}}{8 d}-\frac {21 \sin \left (3 d x +3 c \right ) a^{2} b^{6}}{16 d}+\frac {7 \sin \left (5 d x +5 c \right ) a^{4} b^{4}}{8 d}+\frac {7 \sin \left (5 d x +5 c \right ) a^{2} b^{6}}{16 d}+\frac {a \,b^{7} \cos \left (8 d x +8 c \right )}{128 d}\)	\(458\)
norman	\(\frac {\frac {16 a^{7} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{7} b \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{8} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{8} \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 \left (28 a^{7} b +56 a^{5} b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 \left (28 a^{7} b +56 a^{5} b^{3}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {28 \left (36 a^{7} b +120 a^{5} b^{3}+64 a^{3} b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {28 \left (36 a^{7} b +120 a^{5} b^{3}+64 a^{3} b^{5}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 \left (315 a^{8}+3360 a^{6} b^{2}+6048 a^{4} b^{4}+2304 a^{2} b^{6}+128 b^{8}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9 d}+\frac {2 \left (280 a^{7} b +1120 a^{5} b^{3}+896 a^{3} b^{5}+128 a \,b^{7}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (280 a^{7} b +1120 a^{5} b^{3}+896 a^{3} b^{5}+128 a \,b^{7}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{2} \left (7 a^{6}+70 a^{4} b^{2}+112 a^{2} b^{4}+32 b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{2} \left (7 a^{6}+70 a^{4} b^{2}+112 a^{2} b^{4}+32 b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {56 a^{4} \left (a^{4}+8 a^{2} b^{2}+8 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {56 a^{4} \left (a^{4}+8 a^{2} b^{2}+8 b^{4}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{6} \left (3 a^{2}+14 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {16 a^{6} \left (3 a^{2}+14 b^{2}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\)	\(599\)

[In]

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

[Out]

1/9*(a+b*sin(d*x+c))^9/b/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (20) = 40\).

Time = 0.32 (sec) , antiderivative size = 257, normalized size of antiderivative = 11.68 \[ \int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {9 \, a b^{7} \cos \left (d x + c\right )^{8} - 12 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{6} + 18 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 36 \, {\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (b^{8} \cos \left (d x + c\right )^{8} + 9 \, a^{8} + 84 \, a^{6} b^{2} + 126 \, a^{4} b^{4} + 36 \, a^{2} b^{6} + b^{8} - 4 \, {\left (9 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (21 \, a^{4} b^{4} + 18 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (21 \, a^{6} b^{2} + 63 \, a^{4} b^{4} + 27 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{9 \, d} \]

[In]

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

[Out]

1/9*(9*a*b^7*cos(d*x + c)^8 - 12*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^6 + 18*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*
cos(d*x + c)^4 - 36*(a^7*b + 7*a^5*b^3 + 7*a^3*b^5 + a*b^7)*cos(d*x + c)^2 + (b^8*cos(d*x + c)^8 + 9*a^8 + 84*
a^6*b^2 + 126*a^4*b^4 + 36*a^2*b^6 + b^8 - 4*(9*a^2*b^6 + b^8)*cos(d*x + c)^6 + 6*(21*a^4*b^4 + 18*a^2*b^6 + b
^8)*cos(d*x + c)^4 - 4*(21*a^6*b^2 + 63*a^4*b^4 + 27*a^2*b^6 + b^8)*cos(d*x + c)^2)*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (15) = 30\).

Time = 0.94 (sec) , antiderivative size = 168, normalized size of antiderivative = 7.64 \[ \int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx=\begin {cases} \frac {a^{8} \sin {\left (c + d x \right )}}{d} + \frac {4 a^{7} b \sin ^{2}{\left (c + d x \right )}}{d} + \frac {28 a^{6} b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {14 a^{5} b^{3} \sin ^{4}{\left (c + d x \right )}}{d} + \frac {14 a^{4} b^{4} \sin ^{5}{\left (c + d x \right )}}{d} + \frac {28 a^{3} b^{5} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac {4 a^{2} b^{6} \sin ^{7}{\left (c + d x \right )}}{d} + \frac {a b^{7} \sin ^{8}{\left (c + d x \right )}}{d} + \frac {b^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{8} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**8*sin(c + d*x)/d + 4*a**7*b*sin(c + d*x)**2/d + 28*a**6*b**2*sin(c + d*x)**3/(3*d) + 14*a**5*b**
3*sin(c + d*x)**4/d + 14*a**4*b**4*sin(c + d*x)**5/d + 28*a**3*b**5*sin(c + d*x)**6/(3*d) + 4*a**2*b**6*sin(c
+ d*x)**7/d + a*b**7*sin(c + d*x)**8/d + b**8*sin(c + d*x)**9/(9*d), Ne(d, 0)), (x*(a + b*sin(c))**8*cos(c), T
rue))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, b d} \]

[In]

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

[Out]

1/9*(b*sin(d*x + c) + a)^9/(b*d)

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, b d} \]

[In]

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

[Out]

1/9*(b*sin(d*x + c) + a)^9/(b*d)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 6.14 \[ \int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {a^8\,\sin \left (c+d\,x\right )+4\,a^7\,b\,{\sin \left (c+d\,x\right )}^2+\frac {28\,a^6\,b^2\,{\sin \left (c+d\,x\right )}^3}{3}+14\,a^5\,b^3\,{\sin \left (c+d\,x\right )}^4+14\,a^4\,b^4\,{\sin \left (c+d\,x\right )}^5+\frac {28\,a^3\,b^5\,{\sin \left (c+d\,x\right )}^6}{3}+4\,a^2\,b^6\,{\sin \left (c+d\,x\right )}^7+a\,b^7\,{\sin \left (c+d\,x\right )}^8+\frac {b^8\,{\sin \left (c+d\,x\right )}^9}{9}}{d} \]

[In]

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

[Out]

(a^8*sin(c + d*x) + (b^8*sin(c + d*x)^9)/9 + 4*a^7*b*sin(c + d*x)^2 + a*b^7*sin(c + d*x)^8 + (28*a^6*b^2*sin(c
+ d*x)^3)/3 + 14*a^5*b^3*sin(c + d*x)^4 + 14*a^4*b^4*sin(c + d*x)^5 + (28*a^3*b^5*sin(c + d*x)^6)/3 + 4*a^2*b
^6*sin(c + d*x)^7)/d