3.41 \(\int \cos ^5(c+d x) (a+a \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=67 \[ \frac {(a \sin (c+d x)+a)^{13}}{13 a^5 d}-\frac {(a \sin (c+d x)+a)^{12}}{3 a^4 d}+\frac {4 (a \sin (c+d x)+a)^{11}}{11 a^3 d} \]

[Out]

4/11*(a+a*sin(d*x+c))^11/a^3/d-1/3*(a+a*sin(d*x+c))^12/a^4/d+1/13*(a+a*sin(d*x+c))^13/a^5/d

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Rubi [A]  time = 0.08, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac {(a \sin (c+d x)+a)^{13}}{13 a^5 d}-\frac {(a \sin (c+d x)+a)^{12}}{3 a^4 d}+\frac {4 (a \sin (c+d x)+a)^{11}}{11 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^8,x]

[Out]

(4*(a + a*Sin[c + d*x])^11)/(11*a^3*d) - (a + a*Sin[c + d*x])^12/(3*a^4*d) + (a + a*Sin[c + d*x])^13/(13*a^5*d
)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2667

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 + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

\begin {align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^{10} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (a+x)^{10}-4 a (a+x)^{11}+(a+x)^{12}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {4 (a+a \sin (c+d x))^{11}}{11 a^3 d}-\frac {(a+a \sin (c+d x))^{12}}{3 a^4 d}+\frac {(a+a \sin (c+d x))^{13}}{13 a^5 d}\\ \end {align*}

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Mathematica [A]  time = 0.42, size = 58, normalized size = 0.87 \[ -\frac {a^8 (\sin (c+d x)+1)^8 \left (33 \sin ^2(c+d x)-77 \sin (c+d x)+46\right ) \cos ^6(c+d x)}{429 d (\sin (c+d x)-1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^8,x]

[Out]

-1/429*(a^8*Cos[c + d*x]^6*(1 + Sin[c + d*x])^8*(46 - 77*Sin[c + d*x] + 33*Sin[c + d*x]^2))/(d*(-1 + Sin[c + d
*x])^3)

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fricas [B]  time = 0.79, size = 149, normalized size = 2.22 \[ \frac {286 \, a^{8} \cos \left (d x + c\right )^{12} - 3432 \, a^{8} \cos \left (d x + c\right )^{10} + 10296 \, a^{8} \cos \left (d x + c\right )^{8} - 9152 \, a^{8} \cos \left (d x + c\right )^{6} + {\left (33 \, a^{8} \cos \left (d x + c\right )^{12} - 1212 \, a^{8} \cos \left (d x + c\right )^{10} + 6280 \, a^{8} \cos \left (d x + c\right )^{8} - 8512 \, a^{8} \cos \left (d x + c\right )^{6} + 768 \, a^{8} \cos \left (d x + c\right )^{4} + 1024 \, a^{8} \cos \left (d x + c\right )^{2} + 2048 \, a^{8}\right )} \sin \left (d x + c\right )}{429 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/429*(286*a^8*cos(d*x + c)^12 - 3432*a^8*cos(d*x + c)^10 + 10296*a^8*cos(d*x + c)^8 - 9152*a^8*cos(d*x + c)^6
+ (33*a^8*cos(d*x + c)^12 - 1212*a^8*cos(d*x + c)^10 + 6280*a^8*cos(d*x + c)^8 - 8512*a^8*cos(d*x + c)^6 + 76
8*a^8*cos(d*x + c)^4 + 1024*a^8*cos(d*x + c)^2 + 2048*a^8)*sin(d*x + c))/d

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giac [B]  time = 1.96, size = 219, normalized size = 3.27 \[ \frac {a^{8} \cos \left (12 \, d x + 12 \, c\right )}{3072 \, d} - \frac {3 \, a^{8} \cos \left (10 \, d x + 10 \, c\right )}{256 \, d} + \frac {27 \, a^{8} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} + \frac {155 \, a^{8} \cos \left (6 \, d x + 6 \, c\right )}{768 \, d} - \frac {475 \, a^{8} \cos \left (4 \, d x + 4 \, c\right )}{1024 \, d} - \frac {323 \, a^{8} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {a^{8} \sin \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac {115 \, a^{8} \sin \left (11 \, d x + 11 \, c\right )}{45056 \, d} + \frac {205 \, a^{8} \sin \left (9 \, d x + 9 \, c\right )}{6144 \, d} - \frac {7 \, a^{8} \sin \left (7 \, d x + 7 \, c\right )}{2048 \, d} - \frac {2033 \, a^{8} \sin \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac {6137 \, a^{8} \sin \left (3 \, d x + 3 \, c\right )}{12288 \, d} + \frac {4845 \, a^{8} \sin \left (d x + c\right )}{1024 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/3072*a^8*cos(12*d*x + 12*c)/d - 3/256*a^8*cos(10*d*x + 10*c)/d + 27/512*a^8*cos(8*d*x + 8*c)/d + 155/768*a^8
*cos(6*d*x + 6*c)/d - 475/1024*a^8*cos(4*d*x + 4*c)/d - 323/128*a^8*cos(2*d*x + 2*c)/d + 1/53248*a^8*sin(13*d*
x + 13*c)/d - 115/45056*a^8*sin(11*d*x + 11*c)/d + 205/6144*a^8*sin(9*d*x + 9*c)/d - 7/2048*a^8*sin(7*d*x + 7*
c)/d - 2033/4096*a^8*sin(5*d*x + 5*c)/d - 6137/12288*a^8*sin(3*d*x + 3*c)/d + 4845/1024*a^8*sin(d*x + c)/d

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maple [B]  time = 0.18, size = 513, normalized size = 7.66 \[ \frac {a^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{13}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{143}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{1287}-\frac {5 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{429}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{429}\right )+8 a^{8} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{20}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{40}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{120}\right )+28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{11}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{99}-\frac {5 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{231}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{231}\right )+56 a^{8} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{20}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{60}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )+56 a^{8} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )+28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {4 a^{8} \left (\cos ^{6}\left (d x +c \right )\right )}{3}+\frac {a^{8} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+a*sin(d*x+c))^8,x)

[Out]

1/d*(a^8*(-1/13*sin(d*x+c)^7*cos(d*x+c)^6-7/143*sin(d*x+c)^5*cos(d*x+c)^6-35/1287*sin(d*x+c)^3*cos(d*x+c)^6-5/
429*sin(d*x+c)*cos(d*x+c)^6+1/429*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+8*a^8*(-1/12*sin(d*x+c)^6*co
s(d*x+c)^6-1/20*sin(d*x+c)^4*cos(d*x+c)^6-1/40*sin(d*x+c)^2*cos(d*x+c)^6-1/120*cos(d*x+c)^6)+28*a^8*(-1/11*sin
(d*x+c)^5*cos(d*x+c)^6-5/99*sin(d*x+c)^3*cos(d*x+c)^6-5/231*sin(d*x+c)*cos(d*x+c)^6+1/231*(8/3+cos(d*x+c)^4+4/
3*cos(d*x+c)^2)*sin(d*x+c))+56*a^8*(-1/10*sin(d*x+c)^4*cos(d*x+c)^6-1/20*sin(d*x+c)^2*cos(d*x+c)^6-1/60*cos(d*
x+c)^6)+70*a^8*(-1/9*sin(d*x+c)^3*cos(d*x+c)^6-1/21*sin(d*x+c)*cos(d*x+c)^6+1/105*(8/3+cos(d*x+c)^4+4/3*cos(d*
x+c)^2)*sin(d*x+c))+56*a^8*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+28*a^8*(-1/7*sin(d*x+c)*cos(d*x+
c)^6+1/35*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-4/3*a^8*cos(d*x+c)^6+1/5*a^8*(8/3+cos(d*x+c)^4+4/3*c
os(d*x+c)^2)*sin(d*x+c))

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maxima [B]  time = 0.61, size = 173, normalized size = 2.58 \[ \frac {33 \, a^{8} \sin \left (d x + c\right )^{13} + 286 \, a^{8} \sin \left (d x + c\right )^{12} + 1014 \, a^{8} \sin \left (d x + c\right )^{11} + 1716 \, a^{8} \sin \left (d x + c\right )^{10} + 715 \, a^{8} \sin \left (d x + c\right )^{9} - 2574 \, a^{8} \sin \left (d x + c\right )^{8} - 5148 \, a^{8} \sin \left (d x + c\right )^{7} - 3432 \, a^{8} \sin \left (d x + c\right )^{6} + 1287 \, a^{8} \sin \left (d x + c\right )^{5} + 4290 \, a^{8} \sin \left (d x + c\right )^{4} + 3718 \, a^{8} \sin \left (d x + c\right )^{3} + 1716 \, a^{8} \sin \left (d x + c\right )^{2} + 429 \, a^{8} \sin \left (d x + c\right )}{429 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

1/429*(33*a^8*sin(d*x + c)^13 + 286*a^8*sin(d*x + c)^12 + 1014*a^8*sin(d*x + c)^11 + 1716*a^8*sin(d*x + c)^10
+ 715*a^8*sin(d*x + c)^9 - 2574*a^8*sin(d*x + c)^8 - 5148*a^8*sin(d*x + c)^7 - 3432*a^8*sin(d*x + c)^6 + 1287*
a^8*sin(d*x + c)^5 + 4290*a^8*sin(d*x + c)^4 + 3718*a^8*sin(d*x + c)^3 + 1716*a^8*sin(d*x + c)^2 + 429*a^8*sin
(d*x + c))/d

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mupad [B]  time = 0.18, size = 134, normalized size = 2.00 \[ \frac {a^8\,\sin \left (c+d\,x\right )\,\left (33\,{\sin \left (c+d\,x\right )}^{12}+286\,{\sin \left (c+d\,x\right )}^{11}+1014\,{\sin \left (c+d\,x\right )}^{10}+1716\,{\sin \left (c+d\,x\right )}^9+715\,{\sin \left (c+d\,x\right )}^8-2574\,{\sin \left (c+d\,x\right )}^7-5148\,{\sin \left (c+d\,x\right )}^6-3432\,{\sin \left (c+d\,x\right )}^5+1287\,{\sin \left (c+d\,x\right )}^4+4290\,{\sin \left (c+d\,x\right )}^3+3718\,{\sin \left (c+d\,x\right )}^2+1716\,\sin \left (c+d\,x\right )+429\right )}{429\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5*(a + a*sin(c + d*x))^8,x)

[Out]

(a^8*sin(c + d*x)*(1716*sin(c + d*x) + 3718*sin(c + d*x)^2 + 4290*sin(c + d*x)^3 + 1287*sin(c + d*x)^4 - 3432*
sin(c + d*x)^5 - 5148*sin(c + d*x)^6 - 2574*sin(c + d*x)^7 + 715*sin(c + d*x)^8 + 1716*sin(c + d*x)^9 + 1014*s
in(c + d*x)^10 + 286*sin(c + d*x)^11 + 33*sin(c + d*x)^12 + 429))/(429*d)

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sympy [A]  time = 127.89, size = 558, normalized size = 8.33 \[ \begin {cases} \frac {8 a^{8} \sin ^{13}{\left (c + d x \right )}}{1287 d} + \frac {4 a^{8} \sin ^{11}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{99 d} + \frac {32 a^{8} \sin ^{11}{\left (c + d x \right )}}{99 d} + \frac {a^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{9 d} + \frac {16 a^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{9 d} + \frac {16 a^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac {4 a^{8} \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {8 a^{8} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {32 a^{8} \sin ^{7}{\left (c + d x \right )}}{15 d} - \frac {4 a^{8} \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {14 a^{8} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {112 a^{8} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {8 a^{8} \sin ^{5}{\left (c + d x \right )}}{15 d} - \frac {a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {28 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {28 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} + \frac {4 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} - \frac {2 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{5 d} - \frac {14 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{3 d} - \frac {28 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {a^{8} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {a^{8} \cos ^{12}{\left (c + d x \right )}}{15 d} - \frac {14 a^{8} \cos ^{10}{\left (c + d x \right )}}{15 d} - \frac {7 a^{8} \cos ^{8}{\left (c + d x \right )}}{3 d} - \frac {4 a^{8} \cos ^{6}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{8} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*(a+a*sin(d*x+c))**8,x)

[Out]

Piecewise((8*a**8*sin(c + d*x)**13/(1287*d) + 4*a**8*sin(c + d*x)**11*cos(c + d*x)**2/(99*d) + 32*a**8*sin(c +
d*x)**11/(99*d) + a**8*sin(c + d*x)**9*cos(c + d*x)**4/(9*d) + 16*a**8*sin(c + d*x)**9*cos(c + d*x)**2/(9*d)
+ 16*a**8*sin(c + d*x)**9/(9*d) + 4*a**8*sin(c + d*x)**7*cos(c + d*x)**4/d + 8*a**8*sin(c + d*x)**7*cos(c + d*
x)**2/d + 32*a**8*sin(c + d*x)**7/(15*d) - 4*a**8*sin(c + d*x)**6*cos(c + d*x)**6/(3*d) + 14*a**8*sin(c + d*x)
**5*cos(c + d*x)**4/d + 112*a**8*sin(c + d*x)**5*cos(c + d*x)**2/(15*d) + 8*a**8*sin(c + d*x)**5/(15*d) - a**8
*sin(c + d*x)**4*cos(c + d*x)**8/d - 28*a**8*sin(c + d*x)**4*cos(c + d*x)**6/(3*d) + 28*a**8*sin(c + d*x)**3*c
os(c + d*x)**4/(3*d) + 4*a**8*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) - 2*a**8*sin(c + d*x)**2*cos(c + d*x)**10/
(5*d) - 14*a**8*sin(c + d*x)**2*cos(c + d*x)**8/(3*d) - 28*a**8*sin(c + d*x)**2*cos(c + d*x)**6/(3*d) + a**8*s
in(c + d*x)*cos(c + d*x)**4/d - a**8*cos(c + d*x)**12/(15*d) - 14*a**8*cos(c + d*x)**10/(15*d) - 7*a**8*cos(c
+ d*x)**8/(3*d) - 4*a**8*cos(c + d*x)**6/(3*d), Ne(d, 0)), (x*(a*sin(c) + a)**8*cos(c)**5, True))

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