3.414 \(\int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=77 \[ -\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^9}{9 b^3 d}-\frac {(a+b \sin (c+d x))^{11}}{11 b^3 d}+\frac {a (a+b \sin (c+d x))^{10}}{5 b^3 d} \]

[Out]

-1/9*(a^2-b^2)*(a+b*sin(d*x+c))^9/b^3/d+1/5*a*(a+b*sin(d*x+c))^10/b^3/d-1/11*(a+b*sin(d*x+c))^11/b^3/d

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Rubi [A]  time = 0.15, antiderivative size = 77, 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 = {2668, 697} \[ -\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^9}{9 b^3 d}-\frac {(a+b \sin (c+d x))^{11}}{11 b^3 d}+\frac {a (a+b \sin (c+d x))^{10}}{5 b^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^8,x]

[Out]

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

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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.85, size = 56, normalized size = 0.73 \[ \frac {(a+b \sin (c+d x))^9 \left (-2 a^2+18 a b \sin (c+d x)+45 b^2 \cos (2 (c+d x))+65 b^2\right )}{990 b^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^8,x]

[Out]

((a + b*Sin[c + d*x])^9*(-2*a^2 + 65*b^2 + 45*b^2*Cos[2*(c + d*x)] + 18*a*b*Sin[c + d*x]))/(990*b^3*d)

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fricas [B]  time = 0.52, size = 310, normalized size = 4.03 \[ \frac {396 \, a b^{7} \cos \left (d x + c\right )^{10} - 495 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{8} + 660 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{6} - 990 \, {\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (45 \, b^{8} \cos \left (d x + c\right )^{10} - 10 \, {\left (154 \, a^{2} b^{6} + 17 \, b^{8}\right )} \cos \left (d x + c\right )^{8} + 330 \, a^{8} + 1848 \, a^{6} b^{2} + 1980 \, a^{4} b^{4} + 440 \, a^{2} b^{6} + 10 \, b^{8} + 10 \, {\left (495 \, a^{4} b^{4} + 418 \, a^{2} b^{6} + 23 \, b^{8}\right )} \cos \left (d x + c\right )^{6} - 12 \, {\left (231 \, a^{6} b^{2} + 660 \, a^{4} b^{4} + 275 \, a^{2} b^{6} + 10 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + {\left (165 \, a^{8} + 924 \, a^{6} b^{2} + 990 \, a^{4} b^{4} + 220 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{495 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/495*(396*a*b^7*cos(d*x + c)^10 - 495*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^8 + 660*(7*a^5*b^3 + 14*a^3*b^5 + 3*
a*b^7)*cos(d*x + c)^6 - 990*(a^7*b + 7*a^5*b^3 + 7*a^3*b^5 + a*b^7)*cos(d*x + c)^4 + (45*b^8*cos(d*x + c)^10 -
10*(154*a^2*b^6 + 17*b^8)*cos(d*x + c)^8 + 330*a^8 + 1848*a^6*b^2 + 1980*a^4*b^4 + 440*a^2*b^6 + 10*b^8 + 10*
(495*a^4*b^4 + 418*a^2*b^6 + 23*b^8)*cos(d*x + c)^6 - 12*(231*a^6*b^2 + 660*a^4*b^4 + 275*a^2*b^6 + 10*b^8)*co
s(d*x + c)^4 + (165*a^8 + 924*a^6*b^2 + 990*a^4*b^4 + 220*a^2*b^6 + 5*b^8)*cos(d*x + c)^2)*sin(d*x + c))/d

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giac [B]  time = 1.97, size = 272, normalized size = 3.53 \[ -\frac {45 \, b^{8} \sin \left (d x + c\right )^{11} + 396 \, a b^{7} \sin \left (d x + c\right )^{10} + 1540 \, a^{2} b^{6} \sin \left (d x + c\right )^{9} - 55 \, b^{8} \sin \left (d x + c\right )^{9} + 3465 \, a^{3} b^{5} \sin \left (d x + c\right )^{8} - 495 \, a b^{7} \sin \left (d x + c\right )^{8} + 4950 \, a^{4} b^{4} \sin \left (d x + c\right )^{7} - 1980 \, a^{2} b^{6} \sin \left (d x + c\right )^{7} + 4620 \, a^{5} b^{3} \sin \left (d x + c\right )^{6} - 4620 \, a^{3} b^{5} \sin \left (d x + c\right )^{6} + 2772 \, a^{6} b^{2} \sin \left (d x + c\right )^{5} - 6930 \, a^{4} b^{4} \sin \left (d x + c\right )^{5} + 990 \, a^{7} b \sin \left (d x + c\right )^{4} - 6930 \, a^{5} b^{3} \sin \left (d x + c\right )^{4} + 165 \, a^{8} \sin \left (d x + c\right )^{3} - 4620 \, a^{6} b^{2} \sin \left (d x + c\right )^{3} - 1980 \, a^{7} b \sin \left (d x + c\right )^{2} - 495 \, a^{8} \sin \left (d x + c\right )}{495 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-1/495*(45*b^8*sin(d*x + c)^11 + 396*a*b^7*sin(d*x + c)^10 + 1540*a^2*b^6*sin(d*x + c)^9 - 55*b^8*sin(d*x + c)
^9 + 3465*a^3*b^5*sin(d*x + c)^8 - 495*a*b^7*sin(d*x + c)^8 + 4950*a^4*b^4*sin(d*x + c)^7 - 1980*a^2*b^6*sin(d
*x + c)^7 + 4620*a^5*b^3*sin(d*x + c)^6 - 4620*a^3*b^5*sin(d*x + c)^6 + 2772*a^6*b^2*sin(d*x + c)^5 - 6930*a^4
*b^4*sin(d*x + c)^5 + 990*a^7*b*sin(d*x + c)^4 - 6930*a^5*b^3*sin(d*x + c)^4 + 165*a^8*sin(d*x + c)^3 - 4620*a
^6*b^2*sin(d*x + c)^3 - 1980*a^7*b*sin(d*x + c)^2 - 495*a^8*sin(d*x + c))/d

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maple [B]  time = 0.26, size = 480, normalized size = 6.23 \[ \frac {b^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{11}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{99}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{99}-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{33}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{99}\right )+8 a \,b^{7} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{10}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{40}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{20}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{40}\right )+28 a^{2} b^{6} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{9}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{63}-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{21}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{63}\right )+56 a^{3} b^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{12}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{24}\right )+70 a^{4} b^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{7}-\frac {3 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35}\right )+56 a^{5} b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+28 a^{6} b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15}\right )-2 a^{7} b \left (\cos ^{4}\left (d x +c \right )\right )+\frac {a^{8} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(a+b*sin(d*x+c))^8,x)

[Out]

1/d*(b^8*(-1/11*sin(d*x+c)^7*cos(d*x+c)^4-7/99*sin(d*x+c)^5*cos(d*x+c)^4-5/99*sin(d*x+c)^3*cos(d*x+c)^4-1/33*s
in(d*x+c)*cos(d*x+c)^4+1/99*(2+cos(d*x+c)^2)*sin(d*x+c))+8*a*b^7*(-1/10*sin(d*x+c)^6*cos(d*x+c)^4-3/40*sin(d*x
+c)^4*cos(d*x+c)^4-1/20*cos(d*x+c)^4*sin(d*x+c)^2-1/40*cos(d*x+c)^4)+28*a^2*b^6*(-1/9*sin(d*x+c)^5*cos(d*x+c)^
4-5/63*sin(d*x+c)^3*cos(d*x+c)^4-1/21*sin(d*x+c)*cos(d*x+c)^4+1/63*(2+cos(d*x+c)^2)*sin(d*x+c))+56*a^3*b^5*(-1
/8*sin(d*x+c)^4*cos(d*x+c)^4-1/12*cos(d*x+c)^4*sin(d*x+c)^2-1/24*cos(d*x+c)^4)+70*a^4*b^4*(-1/7*sin(d*x+c)^3*c
os(d*x+c)^4-3/35*sin(d*x+c)*cos(d*x+c)^4+1/35*(2+cos(d*x+c)^2)*sin(d*x+c))+56*a^5*b^3*(-1/6*cos(d*x+c)^4*sin(d
*x+c)^2-1/12*cos(d*x+c)^4)+28*a^6*b^2*(-1/5*sin(d*x+c)*cos(d*x+c)^4+1/15*(2+cos(d*x+c)^2)*sin(d*x+c))-2*a^7*b*
cos(d*x+c)^4+1/3*a^8*(2+cos(d*x+c)^2)*sin(d*x+c))

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maxima [B]  time = 0.33, size = 233, normalized size = 3.03 \[ -\frac {45 \, b^{8} \sin \left (d x + c\right )^{11} + 396 \, a b^{7} \sin \left (d x + c\right )^{10} - 1980 \, a^{7} b \sin \left (d x + c\right )^{2} + 55 \, {\left (28 \, a^{2} b^{6} - b^{8}\right )} \sin \left (d x + c\right )^{9} - 495 \, a^{8} \sin \left (d x + c\right ) + 495 \, {\left (7 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (d x + c\right )^{8} + 990 \, {\left (5 \, a^{4} b^{4} - 2 \, a^{2} b^{6}\right )} \sin \left (d x + c\right )^{7} + 4620 \, {\left (a^{5} b^{3} - a^{3} b^{5}\right )} \sin \left (d x + c\right )^{6} + 1386 \, {\left (2 \, a^{6} b^{2} - 5 \, a^{4} b^{4}\right )} \sin \left (d x + c\right )^{5} + 990 \, {\left (a^{7} b - 7 \, a^{5} b^{3}\right )} \sin \left (d x + c\right )^{4} + 165 \, {\left (a^{8} - 28 \, a^{6} b^{2}\right )} \sin \left (d x + c\right )^{3}}{495 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/495*(45*b^8*sin(d*x + c)^11 + 396*a*b^7*sin(d*x + c)^10 - 1980*a^7*b*sin(d*x + c)^2 + 55*(28*a^2*b^6 - b^8)
*sin(d*x + c)^9 - 495*a^8*sin(d*x + c) + 495*(7*a^3*b^5 - a*b^7)*sin(d*x + c)^8 + 990*(5*a^4*b^4 - 2*a^2*b^6)*
sin(d*x + c)^7 + 4620*(a^5*b^3 - a^3*b^5)*sin(d*x + c)^6 + 1386*(2*a^6*b^2 - 5*a^4*b^4)*sin(d*x + c)^5 + 990*(
a^7*b - 7*a^5*b^3)*sin(d*x + c)^4 + 165*(a^8 - 28*a^6*b^2)*sin(d*x + c)^3)/d

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mupad [B]  time = 5.37, size = 231, normalized size = 3.00 \[ -\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {a^8}{3}-\frac {28\,a^6\,b^2}{3}\right )-{\sin \left (c+d\,x\right )}^5\,\left (14\,a^4\,b^4-\frac {28\,a^6\,b^2}{5}\right )-{\sin \left (c+d\,x\right )}^7\,\left (4\,a^2\,b^6-10\,a^4\,b^4\right )-a^8\,\sin \left (c+d\,x\right )-{\sin \left (c+d\,x\right )}^9\,\left (\frac {b^8}{9}-\frac {28\,a^2\,b^6}{9}\right )+\frac {b^8\,{\sin \left (c+d\,x\right )}^{11}}{11}-4\,a^7\,b\,{\sin \left (c+d\,x\right )}^2+\frac {4\,a\,b^7\,{\sin \left (c+d\,x\right )}^{10}}{5}+2\,a^5\,b\,{\sin \left (c+d\,x\right )}^4\,\left (a^2-7\,b^2\right )+a\,b^5\,{\sin \left (c+d\,x\right )}^8\,\left (7\,a^2-b^2\right )+\frac {28\,a^3\,b^3\,{\sin \left (c+d\,x\right )}^6\,\left (a^2-b^2\right )}{3}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^3*(a + b*sin(c + d*x))^8,x)

[Out]

-(sin(c + d*x)^3*(a^8/3 - (28*a^6*b^2)/3) - sin(c + d*x)^5*(14*a^4*b^4 - (28*a^6*b^2)/5) - sin(c + d*x)^7*(4*a
^2*b^6 - 10*a^4*b^4) - a^8*sin(c + d*x) - sin(c + d*x)^9*(b^8/9 - (28*a^2*b^6)/9) + (b^8*sin(c + d*x)^11)/11 -
4*a^7*b*sin(c + d*x)^2 + (4*a*b^7*sin(c + d*x)^10)/5 + 2*a^5*b*sin(c + d*x)^4*(a^2 - 7*b^2) + a*b^5*sin(c + d
*x)^8*(7*a^2 - b^2) + (28*a^3*b^3*sin(c + d*x)^6*(a^2 - b^2))/3)/d

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sympy [A]  time = 54.46, size = 468, normalized size = 6.08 \[ \begin {cases} \frac {2 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{8} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {2 a^{7} b \cos ^{4}{\left (c + d x \right )}}{d} + \frac {56 a^{6} b^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {28 a^{6} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} - \frac {14 a^{5} b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {14 a^{5} b^{3} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {4 a^{4} b^{4} \sin ^{7}{\left (c + d x \right )}}{d} + \frac {14 a^{4} b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {14 a^{3} b^{5} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {28 a^{3} b^{5} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {7 a^{3} b^{5} \cos ^{8}{\left (c + d x \right )}}{3 d} + \frac {8 a^{2} b^{6} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac {4 a^{2} b^{6} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {2 a b^{7} \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {2 a b^{7} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {a b^{7} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {a b^{7} \cos ^{10}{\left (c + d x \right )}}{5 d} + \frac {2 b^{8} \sin ^{11}{\left (c + d x \right )}}{99 d} + \frac {b^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{9 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{8} \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(a+b*sin(d*x+c))**8,x)

[Out]

Piecewise((2*a**8*sin(c + d*x)**3/(3*d) + a**8*sin(c + d*x)*cos(c + d*x)**2/d - 2*a**7*b*cos(c + d*x)**4/d + 5
6*a**6*b**2*sin(c + d*x)**5/(15*d) + 28*a**6*b**2*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) - 14*a**5*b**3*sin(c +
d*x)**2*cos(c + d*x)**4/d - 14*a**5*b**3*cos(c + d*x)**6/(3*d) + 4*a**4*b**4*sin(c + d*x)**7/d + 14*a**4*b**4
*sin(c + d*x)**5*cos(c + d*x)**2/d - 14*a**3*b**5*sin(c + d*x)**4*cos(c + d*x)**4/d - 28*a**3*b**5*sin(c + d*x
)**2*cos(c + d*x)**6/(3*d) - 7*a**3*b**5*cos(c + d*x)**8/(3*d) + 8*a**2*b**6*sin(c + d*x)**9/(9*d) + 4*a**2*b*
*6*sin(c + d*x)**7*cos(c + d*x)**2/d - 2*a*b**7*sin(c + d*x)**6*cos(c + d*x)**4/d - 2*a*b**7*sin(c + d*x)**4*c
os(c + d*x)**6/d - a*b**7*sin(c + d*x)**2*cos(c + d*x)**8/d - a*b**7*cos(c + d*x)**10/(5*d) + 2*b**8*sin(c + d
*x)**11/(99*d) + b**8*sin(c + d*x)**9*cos(c + d*x)**2/(9*d), Ne(d, 0)), (x*(a + b*sin(c))**8*cos(c)**3, True))

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