3.1200 \(\int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx\)

Optimal. Leaf size=81 \[ \frac {a \cos ^8(c+d x)}{8 d}-\frac {a \cos ^6(c+d x)}{6 d}+\frac {b \sin ^9(c+d x)}{9 d}-\frac {2 b \sin ^7(c+d x)}{7 d}+\frac {b \sin ^5(c+d x)}{5 d} \]

[Out]

-1/6*a*cos(d*x+c)^6/d+1/8*a*cos(d*x+c)^8/d+1/5*b*sin(d*x+c)^5/d-2/7*b*sin(d*x+c)^7/d+1/9*b*sin(d*x+c)^9/d

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Rubi [A]  time = 0.14, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2834, 2565, 14, 2564, 270} \[ \frac {a \cos ^8(c+d x)}{8 d}-\frac {a \cos ^6(c+d x)}{6 d}+\frac {b \sin ^9(c+d x)}{9 d}-\frac {2 b \sin ^7(c+d x)}{7 d}+\frac {b \sin ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + b*Sin[c + d*x]),x]

[Out]

-(a*Cos[c + d*x]^6)/(6*d) + (a*Cos[c + d*x]^8)/(8*d) + (b*Sin[c + d*x]^5)/(5*d) - (2*b*Sin[c + d*x]^7)/(7*d) +
(b*Sin[c + d*x]^9)/(9*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 2834

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]),
x_Symbol] :> Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[Cos[e + f*x]^p*(d*Sin[e +
f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[n] && ((LtQ[p, 0]
&& NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*p + 1])

Rubi steps

\begin {align*} \int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cos ^5(c+d x) \sin ^3(c+d x) \, dx+b \int \cos ^5(c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {a \operatorname {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {a \cos ^6(c+d x)}{6 d}+\frac {a \cos ^8(c+d x)}{8 d}+\frac {b \sin ^5(c+d x)}{5 d}-\frac {2 b \sin ^7(c+d x)}{7 d}+\frac {b \sin ^9(c+d x)}{9 d}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 105, normalized size = 1.30 \[ \frac {-7560 a \cos (2 (c+d x))-1260 a \cos (4 (c+d x))+840 a \cos (6 (c+d x))+315 a \cos (8 (c+d x))+7560 b \sin (c+d x)-1680 b \sin (3 (c+d x))-1008 b \sin (5 (c+d x))+180 b \sin (7 (c+d x))+140 b \sin (9 (c+d x))}{322560 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + b*Sin[c + d*x]),x]

[Out]

(-7560*a*Cos[2*(c + d*x)] - 1260*a*Cos[4*(c + d*x)] + 840*a*Cos[6*(c + d*x)] + 315*a*Cos[8*(c + d*x)] + 7560*b
*Sin[c + d*x] - 1680*b*Sin[3*(c + d*x)] - 1008*b*Sin[5*(c + d*x)] + 180*b*Sin[7*(c + d*x)] + 140*b*Sin[9*(c +
d*x)])/(322560*d)

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fricas [A]  time = 0.83, size = 84, normalized size = 1.04 \[ \frac {315 \, a \cos \left (d x + c\right )^{8} - 420 \, a \cos \left (d x + c\right )^{6} + 8 \, {\left (35 \, b \cos \left (d x + c\right )^{8} - 50 \, b \cos \left (d x + c\right )^{6} + 3 \, b \cos \left (d x + c\right )^{4} + 4 \, b \cos \left (d x + c\right )^{2} + 8 \, b\right )} \sin \left (d x + c\right )}{2520 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^3*(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/2520*(315*a*cos(d*x + c)^8 - 420*a*cos(d*x + c)^6 + 8*(35*b*cos(d*x + c)^8 - 50*b*cos(d*x + c)^6 + 3*b*cos(d
*x + c)^4 + 4*b*cos(d*x + c)^2 + 8*b)*sin(d*x + c))/d

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giac [A]  time = 0.25, size = 133, normalized size = 1.64 \[ \frac {a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {a \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {b \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {b \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {b \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {b \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {3 \, b \sin \left (d x + c\right )}{128 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^3*(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1024*a*cos(8*d*x + 8*c)/d + 1/384*a*cos(6*d*x + 6*c)/d - 1/256*a*cos(4*d*x + 4*c)/d - 3/128*a*cos(2*d*x + 2*
c)/d + 1/2304*b*sin(9*d*x + 9*c)/d + 1/1792*b*sin(7*d*x + 7*c)/d - 1/320*b*sin(5*d*x + 5*c)/d - 1/192*b*sin(3*
d*x + 3*c)/d + 3/128*b*sin(d*x + c)/d

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maple [A]  time = 0.26, size = 102, normalized size = 1.26 \[ \frac {a \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )+b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*sin(d*x+c)^3*(a+b*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+b*(-1/9*sin(d*x+c)^3*cos(d*x+c)^6-1/21*sin(d*x+c)*co
s(d*x+c)^6+1/105*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)))

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maxima [A]  time = 0.31, size = 72, normalized size = 0.89 \[ \frac {280 \, b \sin \left (d x + c\right )^{9} + 315 \, a \sin \left (d x + c\right )^{8} - 720 \, b \sin \left (d x + c\right )^{7} - 840 \, a \sin \left (d x + c\right )^{6} + 504 \, b \sin \left (d x + c\right )^{5} + 630 \, a \sin \left (d x + c\right )^{4}}{2520 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^3*(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/2520*(280*b*sin(d*x + c)^9 + 315*a*sin(d*x + c)^8 - 720*b*sin(d*x + c)^7 - 840*a*sin(d*x + c)^6 + 504*b*sin(
d*x + c)^5 + 630*a*sin(d*x + c)^4)/d

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mupad [B]  time = 0.06, size = 71, normalized size = 0.88 \[ \frac {\frac {b\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {2\,b\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {b\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5*sin(c + d*x)^3*(a + b*sin(c + d*x)),x)

[Out]

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

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sympy [A]  time = 13.68, size = 136, normalized size = 1.68 \[ \begin {cases} \frac {a \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac {a \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} + \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac {8 b \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 b \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {b \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right ) \sin ^{3}{\relax (c )} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*sin(d*x+c)**3*(a+b*sin(d*x+c)),x)

[Out]

Piecewise((a*sin(c + d*x)**8/(24*d) + a*sin(c + d*x)**6*cos(c + d*x)**2/(6*d) + a*sin(c + d*x)**4*cos(c + d*x)
**4/(4*d) + 8*b*sin(c + d*x)**9/(315*d) + 4*b*sin(c + d*x)**7*cos(c + d*x)**2/(35*d) + b*sin(c + d*x)**5*cos(c
+ d*x)**4/(5*d), Ne(d, 0)), (x*(a + b*sin(c))*sin(c)**3*cos(c)**5, True))

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