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

Optimal. Leaf size=144 \[ \frac{2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac{2 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9}{9 b^5 d}+\frac{(a+b \sin (c+d x))^{13}}{13 b^5 d}-\frac{a (a+b \sin (c+d x))^{12}}{3 b^5 d} \]

[Out]

((a^2 - b^2)^2*(a + b*Sin[c + d*x])^9)/(9*b^5*d) - (2*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^10)/(5*b^5*d) + (2*(3
*a^2 - b^2)*(a + b*Sin[c + d*x])^11)/(11*b^5*d) - (a*(a + b*Sin[c + d*x])^12)/(3*b^5*d) + (a + b*Sin[c + d*x])
^13/(13*b^5*d)

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Rubi [A]  time = 0.221048, antiderivative size = 144, normalized size of antiderivative = 1., 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 = {2668, 697} \[ \frac{2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac{2 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9}{9 b^5 d}+\frac{(a+b \sin (c+d x))^{13}}{13 b^5 d}-\frac{a (a+b \sin (c+d x))^{12}}{3 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2 - b^2)^2*(a + b*Sin[c + d*x])^9)/(9*b^5*d) - (2*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^10)/(5*b^5*d) + (2*(3
*a^2 - b^2)*(a + b*Sin[c + d*x])^11)/(11*b^5*d) - (a*(a + b*Sin[c + d*x])^12)/(3*b^5*d) + (a + b*Sin[c + d*x])
^13/(13*b^5*d)

Rule 2668

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]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^8 \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\left (a^2-b^2\right )^2 (a+x)^8-4 \left (a^3-a b^2\right ) (a+x)^9+2 \left (3 a^2-b^2\right ) (a+x)^{10}-4 a (a+x)^{11}+(a+x)^{12}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9}{9 b^5 d}-\frac{2 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac{2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac{a (a+b \sin (c+d x))^{12}}{3 b^5 d}+\frac{(a+b \sin (c+d x))^{13}}{13 b^5 d}\\ \end{align*}

Mathematica [A]  time = 2.06117, size = 120, normalized size = 0.83 \[ \frac{\frac{2}{11} \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}+\frac{1}{9} \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9+\frac{1}{13} (a+b \sin (c+d x))^{13}-\frac{1}{3} a (a+b \sin (c+d x))^{12}-\frac{2}{5} a (a-b) (a+b) (a+b \sin (c+d x))^{10}}{b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((a^2 - b^2)^2*(a + b*Sin[c + d*x])^9)/9 - (2*a*(a - b)*(a + b)*(a + b*Sin[c + d*x])^10)/5 + (2*(3*a^2 - b^2)
*(a + b*Sin[c + d*x])^11)/11 - (a*(a + b*Sin[c + d*x])^12)/3 + (a + b*Sin[c + d*x])^13/13)/(b^5*d)

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Maple [B]  time = 0.086, size = 530, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^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*b^7*(-1/12*sin(d*x+c)^6*
cos(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^2*b^6*(-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^3*b^5*(-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^4*b^4*(-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^5*b^3*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+28*a^6*b^2*(-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^7*b*cos(d*x+c)^6+1/5*a^8
*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))

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Maxima [B]  time = 0.957436, size = 420, normalized size = 2.92 \begin{align*} \frac{495 \, b^{8} \sin \left (d x + c\right )^{13} + 4290 \, a b^{7} \sin \left (d x + c\right )^{12} + 1170 \,{\left (14 \, a^{2} b^{6} - b^{8}\right )} \sin \left (d x + c\right )^{11} + 5148 \,{\left (7 \, a^{3} b^{5} - 2 \, a b^{7}\right )} \sin \left (d x + c\right )^{10} + 25740 \, a^{7} b \sin \left (d x + c\right )^{2} + 715 \,{\left (70 \, a^{4} b^{4} - 56 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{9} + 6435 \, a^{8} \sin \left (d x + c\right ) + 6435 \,{\left (7 \, a^{5} b^{3} - 14 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{8} + 25740 \,{\left (a^{6} b^{2} - 5 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sin \left (d x + c\right )^{7} + 8580 \,{\left (a^{7} b - 14 \, a^{5} b^{3} + 7 \, a^{3} b^{5}\right )} \sin \left (d x + c\right )^{6} + 1287 \,{\left (a^{8} - 56 \, a^{6} b^{2} + 70 \, a^{4} b^{4}\right )} \sin \left (d x + c\right )^{5} - 12870 \,{\left (2 \, a^{7} b - 7 \, a^{5} b^{3}\right )} \sin \left (d x + c\right )^{4} - 4290 \,{\left (a^{8} - 14 \, a^{6} b^{2}\right )} \sin \left (d x + c\right )^{3}}{6435 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6435*(495*b^8*sin(d*x + c)^13 + 4290*a*b^7*sin(d*x + c)^12 + 1170*(14*a^2*b^6 - b^8)*sin(d*x + c)^11 + 5148*
(7*a^3*b^5 - 2*a*b^7)*sin(d*x + c)^10 + 25740*a^7*b*sin(d*x + c)^2 + 715*(70*a^4*b^4 - 56*a^2*b^6 + b^8)*sin(d
*x + c)^9 + 6435*a^8*sin(d*x + c) + 6435*(7*a^5*b^3 - 14*a^3*b^5 + a*b^7)*sin(d*x + c)^8 + 25740*(a^6*b^2 - 5*
a^4*b^4 + a^2*b^6)*sin(d*x + c)^7 + 8580*(a^7*b - 14*a^5*b^3 + 7*a^3*b^5)*sin(d*x + c)^6 + 1287*(a^8 - 56*a^6*
b^2 + 70*a^4*b^4)*sin(d*x + c)^5 - 12870*(2*a^7*b - 7*a^5*b^3)*sin(d*x + c)^4 - 4290*(a^8 - 14*a^6*b^2)*sin(d*
x + c)^3)/d

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Fricas [B]  time = 3.13386, size = 879, normalized size = 6.1 \begin{align*} \frac{4290 \, a b^{7} \cos \left (d x + c\right )^{12} - 5148 \,{\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{10} + 6435 \,{\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{8} - 8580 \,{\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{6} +{\left (495 \, b^{8} \cos \left (d x + c\right )^{12} - 180 \,{\left (91 \, a^{2} b^{6} + 10 \, b^{8}\right )} \cos \left (d x + c\right )^{10} + 10 \,{\left (5005 \, a^{4} b^{4} + 4186 \, a^{2} b^{6} + 229 \, b^{8}\right )} \cos \left (d x + c\right )^{8} + 3432 \, a^{8} + 13728 \, a^{6} b^{2} + 11440 \, a^{4} b^{4} + 2080 \, a^{2} b^{6} + 40 \, b^{8} - 20 \,{\left (1287 \, a^{6} b^{2} + 3575 \, a^{4} b^{4} + 1469 \, a^{2} b^{6} + 53 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 3 \,{\left (429 \, a^{8} + 1716 \, a^{6} b^{2} + 1430 \, a^{4} b^{4} + 260 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (429 \, a^{8} + 1716 \, a^{6} b^{2} + 1430 \, a^{4} b^{4} + 260 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6435 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6435*(4290*a*b^7*cos(d*x + c)^12 - 5148*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^10 + 6435*(7*a^5*b^3 + 14*a^3*b^5
+ 3*a*b^7)*cos(d*x + c)^8 - 8580*(a^7*b + 7*a^5*b^3 + 7*a^3*b^5 + a*b^7)*cos(d*x + c)^6 + (495*b^8*cos(d*x +
c)^12 - 180*(91*a^2*b^6 + 10*b^8)*cos(d*x + c)^10 + 10*(5005*a^4*b^4 + 4186*a^2*b^6 + 229*b^8)*cos(d*x + c)^8
+ 3432*a^8 + 13728*a^6*b^2 + 11440*a^4*b^4 + 2080*a^2*b^6 + 40*b^8 - 20*(1287*a^6*b^2 + 3575*a^4*b^4 + 1469*a^
2*b^6 + 53*b^8)*cos(d*x + c)^6 + 3*(429*a^8 + 1716*a^6*b^2 + 1430*a^4*b^4 + 260*a^2*b^6 + 5*b^8)*cos(d*x + c)^
4 + 4*(429*a^8 + 1716*a^6*b^2 + 1430*a^4*b^4 + 260*a^2*b^6 + 5*b^8)*cos(d*x + c)^2)*sin(d*x + c))/d

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Sympy [A]  time = 122.701, size = 614, normalized size = 4.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.2091, size = 626, normalized size = 4.35 \begin{align*} \frac{a b^{7} \cos \left (12 \, d x + 12 \, c\right )}{3072 \, d} + \frac{b^{8} \sin \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac{{\left (14 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (10 \, d x + 10 \, c\right )}{1280 \, d} + \frac{{\left (28 \, a^{5} b^{3} - a b^{7}\right )} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac{{\left (32 \, a^{7} b - 112 \, a^{5} b^{3} - 70 \, a^{3} b^{5} - 5 \, a b^{7}\right )} \cos \left (6 \, d x + 6 \, c\right )}{768 \, d} - \frac{{\left (256 \, a^{7} b + 224 \, a^{5} b^{3} - 5 \, a b^{7}\right )} \cos \left (4 \, d x + 4 \, c\right )}{1024 \, d} - \frac{{\left (80 \, a^{7} b + 168 \, a^{5} b^{3} + 70 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac{{\left (112 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (11 \, d x + 11 \, c\right )}{45056 \, d} + \frac{{\left (560 \, a^{4} b^{4} + 56 \, a^{2} b^{6} - b^{8}\right )} \sin \left (9 \, d x + 9 \, c\right )}{18432 \, d} - \frac{{\left (128 \, a^{6} b^{2} - 80 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - b^{8}\right )} \sin \left (7 \, d x + 7 \, c\right )}{2048 \, d} + \frac{{\left (256 \, a^{8} - 5376 \, a^{6} b^{2} - 4480 \, a^{4} b^{4} - 560 \, a^{2} b^{6} - 5 \, b^{8}\right )} \sin \left (5 \, d x + 5 \, c\right )}{20480 \, d} + \frac{{\left (1280 \, a^{8} - 1792 \, a^{6} b^{2} - 4480 \, a^{4} b^{4} - 1120 \, a^{2} b^{6} - 25 \, b^{8}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12288 \, d} + \frac{5 \,{\left (128 \, a^{8} + 448 \, a^{6} b^{2} + 336 \, a^{4} b^{4} + 56 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )}{1024 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3072*a*b^7*cos(12*d*x + 12*c)/d + 1/53248*b^8*sin(13*d*x + 13*c)/d - 1/1280*(14*a^3*b^5 + a*b^7)*cos(10*d*x
+ 10*c)/d + 1/512*(28*a^5*b^3 - a*b^7)*cos(8*d*x + 8*c)/d - 1/768*(32*a^7*b - 112*a^5*b^3 - 70*a^3*b^5 - 5*a*b
^7)*cos(6*d*x + 6*c)/d - 1/1024*(256*a^7*b + 224*a^5*b^3 - 5*a*b^7)*cos(4*d*x + 4*c)/d - 1/128*(80*a^7*b + 168
*a^5*b^3 + 70*a^3*b^5 + 5*a*b^7)*cos(2*d*x + 2*c)/d - 1/45056*(112*a^2*b^6 + 3*b^8)*sin(11*d*x + 11*c)/d + 1/1
8432*(560*a^4*b^4 + 56*a^2*b^6 - b^8)*sin(9*d*x + 9*c)/d - 1/2048*(128*a^6*b^2 - 80*a^4*b^4 - 40*a^2*b^6 - b^8
)*sin(7*d*x + 7*c)/d + 1/20480*(256*a^8 - 5376*a^6*b^2 - 4480*a^4*b^4 - 560*a^2*b^6 - 5*b^8)*sin(5*d*x + 5*c)/
d + 1/12288*(1280*a^8 - 1792*a^6*b^2 - 4480*a^4*b^4 - 1120*a^2*b^6 - 25*b^8)*sin(3*d*x + 3*c)/d + 5/1024*(128*
a^8 + 448*a^6*b^2 + 336*a^4*b^4 + 56*a^2*b^6 + b^8)*sin(d*x + c)/d