3.953 \(\int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx\)

Optimal. Leaf size=134 \[ -\frac{(A-7 B) (a \sin (c+d x)+a)^8}{8 a^7 d}+\frac{6 (A-3 B) (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac{2 (3 A-5 B) (a \sin (c+d x)+a)^6}{3 a^5 d}+\frac{8 (A-B) (a \sin (c+d x)+a)^5}{5 a^4 d}-\frac{B (a \sin (c+d x)+a)^9}{9 a^8 d} \]

[Out]

(8*(A - B)*(a + a*Sin[c + d*x])^5)/(5*a^4*d) - (2*(3*A - 5*B)*(a + a*Sin[c + d*x])^6)/(3*a^5*d) + (6*(A - 3*B)
*(a + a*Sin[c + d*x])^7)/(7*a^6*d) - ((A - 7*B)*(a + a*Sin[c + d*x])^8)/(8*a^7*d) - (B*(a + a*Sin[c + d*x])^9)
/(9*a^8*d)

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Rubi [A]  time = 0.142071, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2836, 77} \[ -\frac{(A-7 B) (a \sin (c+d x)+a)^8}{8 a^7 d}+\frac{6 (A-3 B) (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac{2 (3 A-5 B) (a \sin (c+d x)+a)^6}{3 a^5 d}+\frac{8 (A-B) (a \sin (c+d x)+a)^5}{5 a^4 d}-\frac{B (a \sin (c+d x)+a)^9}{9 a^8 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

(8*(A - B)*(a + a*Sin[c + d*x])^5)/(5*a^4*d) - (2*(3*A - 5*B)*(a + a*Sin[c + d*x])^6)/(3*a^5*d) + (6*(A - 3*B)
*(a + a*Sin[c + d*x])^7)/(7*a^6*d) - ((A - 7*B)*(a + a*Sin[c + d*x])^8)/(8*a^7*d) - (B*(a + a*Sin[c + d*x])^9)
/(9*a^8*d)

Rule 2836

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.792265, size = 194, normalized size = 1.45 \[ \frac{a (\sin (c+d x)+1) (-17640 (A+B) \cos (2 (c+d x))-8820 (A+B) \cos (4 (c+d x))+176400 A \sin (c+d x)+35280 A \sin (3 (c+d x))+7056 A \sin (5 (c+d x))+720 A \sin (7 (c+d x))-2520 A \cos (6 (c+d x))-315 A \cos (8 (c+d x))+17640 B \sin (c+d x)-2016 B \sin (5 (c+d x))-900 B \sin (7 (c+d x))-140 B \sin (9 (c+d x))-2520 B \cos (6 (c+d x))-315 B \cos (8 (c+d x)))}{322560 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

(a*(1 + Sin[c + d*x])*(-17640*(A + B)*Cos[2*(c + d*x)] - 8820*(A + B)*Cos[4*(c + d*x)] - 2520*A*Cos[6*(c + d*x
)] - 2520*B*Cos[6*(c + d*x)] - 315*A*Cos[8*(c + d*x)] - 315*B*Cos[8*(c + d*x)] + 176400*A*Sin[c + d*x] + 17640
*B*Sin[c + d*x] + 35280*A*Sin[3*(c + d*x)] + 7056*A*Sin[5*(c + d*x)] - 2016*B*Sin[5*(c + d*x)] + 720*A*Sin[7*(
c + d*x)] - 900*B*Sin[7*(c + d*x)] - 140*B*Sin[9*(c + d*x)]))/(322560*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^
2)

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Maple [A]  time = 0.058, size = 128, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( aB \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}\sin \left ( dx+c \right ) }{9}}+{\frac{\sin \left ( dx+c \right ) }{63} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) -{\frac{aA \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{8}}-{\frac{aB \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{8}}+{\frac{aA\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

1/d*(a*B*(-1/9*cos(d*x+c)^8*sin(d*x+c)+1/63*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))-
1/8*a*A*cos(d*x+c)^8-1/8*a*B*cos(d*x+c)^8+1/7*a*A*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*
x+c))

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Maxima [A]  time = 1.02365, size = 181, normalized size = 1.35 \begin{align*} -\frac{280 \, B a \sin \left (d x + c\right )^{9} + 315 \,{\left (A + B\right )} a \sin \left (d x + c\right )^{8} + 360 \,{\left (A - 3 \, B\right )} a \sin \left (d x + c\right )^{7} - 1260 \,{\left (A + B\right )} a \sin \left (d x + c\right )^{6} - 1512 \,{\left (A - B\right )} a \sin \left (d x + c\right )^{5} + 1890 \,{\left (A + B\right )} a \sin \left (d x + c\right )^{4} + 840 \,{\left (3 \, A - B\right )} a \sin \left (d x + c\right )^{3} - 1260 \,{\left (A + B\right )} a \sin \left (d x + c\right )^{2} - 2520 \, A a \sin \left (d x + c\right )}{2520 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2520*(280*B*a*sin(d*x + c)^9 + 315*(A + B)*a*sin(d*x + c)^8 + 360*(A - 3*B)*a*sin(d*x + c)^7 - 1260*(A + B)
*a*sin(d*x + c)^6 - 1512*(A - B)*a*sin(d*x + c)^5 + 1890*(A + B)*a*sin(d*x + c)^4 + 840*(3*A - B)*a*sin(d*x +
c)^3 - 1260*(A + B)*a*sin(d*x + c)^2 - 2520*A*a*sin(d*x + c))/d

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Fricas [A]  time = 1.87929, size = 261, normalized size = 1.95 \begin{align*} -\frac{315 \,{\left (A + B\right )} a \cos \left (d x + c\right )^{8} + 8 \,{\left (35 \, B a \cos \left (d x + c\right )^{8} - 5 \,{\left (9 \, A + B\right )} a \cos \left (d x + c\right )^{6} - 6 \,{\left (9 \, A + B\right )} a \cos \left (d x + c\right )^{4} - 8 \,{\left (9 \, A + B\right )} a \cos \left (d x + c\right )^{2} - 16 \,{\left (9 \, A + B\right )} a\right )} \sin \left (d x + c\right )}{2520 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2520*(315*(A + B)*a*cos(d*x + c)^8 + 8*(35*B*a*cos(d*x + c)^8 - 5*(9*A + B)*a*cos(d*x + c)^6 - 6*(9*A + B)*
a*cos(d*x + c)^4 - 8*(9*A + B)*a*cos(d*x + c)^2 - 16*(9*A + B)*a)*sin(d*x + c))/d

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Sympy [A]  time = 22.8752, size = 228, normalized size = 1.7 \begin{align*} \begin{cases} \frac{16 A a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{8 A a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{2 A a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{A a \sin{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac{A a \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac{16 B a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac{8 B a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac{2 B a \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac{B a \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac{B a \cos ^{8}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (A + B \sin{\left (c \right )}\right ) \left (a \sin{\left (c \right )} + a\right ) \cos ^{7}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

Piecewise((16*A*a*sin(c + d*x)**7/(35*d) + 8*A*a*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*A*a*sin(c + d*x)**3
*cos(c + d*x)**4/d + A*a*sin(c + d*x)*cos(c + d*x)**6/d - A*a*cos(c + d*x)**8/(8*d) + 16*B*a*sin(c + d*x)**9/(
315*d) + 8*B*a*sin(c + d*x)**7*cos(c + d*x)**2/(35*d) + 2*B*a*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) + B*a*sin(
c + d*x)**3*cos(c + d*x)**6/(3*d) - B*a*cos(c + d*x)**8/(8*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c) + a)*cos
(c)**7, True))

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Giac [A]  time = 1.32345, size = 246, normalized size = 1.84 \begin{align*} -\frac{B a \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac{7 \, A a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac{{\left (A a + B a\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{{\left (A a + B a\right )} \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac{7 \,{\left (A a + B a\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac{7 \,{\left (A a + B a\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac{{\left (4 \, A a - 5 \, B a\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac{{\left (7 \, A a - 2 \, B a\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{7 \,{\left (10 \, A a + B a\right )} \sin \left (d x + c\right )}{128 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/2304*B*a*sin(9*d*x + 9*c)/d + 7/64*A*a*sin(3*d*x + 3*c)/d - 1/1024*(A*a + B*a)*cos(8*d*x + 8*c)/d - 1/128*(
A*a + B*a)*cos(6*d*x + 6*c)/d - 7/256*(A*a + B*a)*cos(4*d*x + 4*c)/d - 7/128*(A*a + B*a)*cos(2*d*x + 2*c)/d +
1/1792*(4*A*a - 5*B*a)*sin(7*d*x + 7*c)/d + 1/320*(7*A*a - 2*B*a)*sin(5*d*x + 5*c)/d + 7/128*(10*A*a + B*a)*si
n(d*x + c)/d