3.656 \(\int \cos ^7(c+d x) \sin ^6(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=129 \[ -\frac{a \sin ^{14}(c+d x)}{14 d}-\frac{a \sin ^{13}(c+d x)}{13 d}+\frac{a \sin ^{12}(c+d x)}{4 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}-\frac{3 a \sin ^{10}(c+d x)}{10 d}-\frac{a \sin ^9(c+d x)}{3 d}+\frac{a \sin ^8(c+d x)}{8 d}+\frac{a \sin ^7(c+d x)}{7 d} \]

[Out]

(a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^8)/(8*d) - (a*Sin[c + d*x]^9)/(3*d) - (3*a*Sin[c + d*x]^10)/(10*d)
+ (3*a*Sin[c + d*x]^11)/(11*d) + (a*Sin[c + d*x]^12)/(4*d) - (a*Sin[c + d*x]^13)/(13*d) - (a*Sin[c + d*x]^14)/
(14*d)

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Rubi [A]  time = 0.100681, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ -\frac{a \sin ^{14}(c+d x)}{14 d}-\frac{a \sin ^{13}(c+d x)}{13 d}+\frac{a \sin ^{12}(c+d x)}{4 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}-\frac{3 a \sin ^{10}(c+d x)}{10 d}-\frac{a \sin ^9(c+d x)}{3 d}+\frac{a \sin ^8(c+d x)}{8 d}+\frac{a \sin ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*Sin[c + d*x]^6*(a + a*Sin[c + d*x]),x]

[Out]

(a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^8)/(8*d) - (a*Sin[c + d*x]^9)/(3*d) - (3*a*Sin[c + d*x]^10)/(10*d)
+ (3*a*Sin[c + d*x]^11)/(11*d) + (a*Sin[c + d*x]^12)/(4*d) - (a*Sin[c + d*x]^13)/(13*d) - (a*Sin[c + d*x]^14)/
(14*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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \cos ^7(c+d x) \sin ^6(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3 x^6 (a+x)^4}{a^6} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int (a-x)^3 x^6 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^{13} d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^7 x^6+a^6 x^7-3 a^5 x^8-3 a^4 x^9+3 a^3 x^{10}+3 a^2 x^{11}-a x^{12}-x^{13}\right ) \, dx,x,a \sin (c+d x)\right )}{a^{13} d}\\ &=\frac{a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^8(c+d x)}{8 d}-\frac{a \sin ^9(c+d x)}{3 d}-\frac{3 a \sin ^{10}(c+d x)}{10 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}+\frac{a \sin ^{12}(c+d x)}{4 d}-\frac{a \sin ^{13}(c+d x)}{13 d}-\frac{a \sin ^{14}(c+d x)}{14 d}\\ \end{align*}

Mathematica [A]  time = 1.03223, size = 117, normalized size = 0.91 \[ -\frac{a (-1201200 \sin (c+d x)+300300 \sin (3 (c+d x))+180180 \sin (5 (c+d x))-51480 \sin (7 (c+d x))-40040 \sin (9 (c+d x))+5460 \sin (11 (c+d x))+4620 \sin (13 (c+d x))+525525 \cos (2 (c+d x))-105105 \cos (6 (c+d x))+21021 \cos (10 (c+d x))-2145 \cos (14 (c+d x)))}{246005760 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*Sin[c + d*x]^6*(a + a*Sin[c + d*x]),x]

[Out]

-(a*(525525*Cos[2*(c + d*x)] - 105105*Cos[6*(c + d*x)] + 21021*Cos[10*(c + d*x)] - 2145*Cos[14*(c + d*x)] - 12
01200*Sin[c + d*x] + 300300*Sin[3*(c + d*x)] + 180180*Sin[5*(c + d*x)] - 51480*Sin[7*(c + d*x)] - 40040*Sin[9*
(c + d*x)] + 5460*Sin[11*(c + d*x)] + 4620*Sin[13*(c + d*x)]))/(246005760*d)

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Maple [A]  time = 0.035, size = 166, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{14}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{28}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{70}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{280}} \right ) +a \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{13}}-{\frac{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{143}}-{\frac{5\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{429}}+{\frac{5\,\sin \left ( dx+c \right ) }{3003} \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 ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*sin(d*x+c)^6*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/14*sin(d*x+c)^6*cos(d*x+c)^8-1/28*sin(d*x+c)^4*cos(d*x+c)^8-1/70*sin(d*x+c)^2*cos(d*x+c)^8-1/280*co
s(d*x+c)^8)+a*(-1/13*sin(d*x+c)^5*cos(d*x+c)^8-5/143*sin(d*x+c)^3*cos(d*x+c)^8-5/429*sin(d*x+c)*cos(d*x+c)^8+5
/3003*(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 = 1.05377, size = 127, normalized size = 0.98 \begin{align*} -\frac{8580 \, a \sin \left (d x + c\right )^{14} + 9240 \, a \sin \left (d x + c\right )^{13} - 30030 \, a \sin \left (d x + c\right )^{12} - 32760 \, a \sin \left (d x + c\right )^{11} + 36036 \, a \sin \left (d x + c\right )^{10} + 40040 \, a \sin \left (d x + c\right )^{9} - 15015 \, a \sin \left (d x + c\right )^{8} - 17160 \, a \sin \left (d x + c\right )^{7}}{120120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)^6*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/120120*(8580*a*sin(d*x + c)^14 + 9240*a*sin(d*x + c)^13 - 30030*a*sin(d*x + c)^12 - 32760*a*sin(d*x + c)^11
+ 36036*a*sin(d*x + c)^10 + 40040*a*sin(d*x + c)^9 - 15015*a*sin(d*x + c)^8 - 17160*a*sin(d*x + c)^7)/d

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Fricas [A]  time = 1.68238, size = 369, normalized size = 2.86 \begin{align*} \frac{8580 \, a \cos \left (d x + c\right )^{14} - 30030 \, a \cos \left (d x + c\right )^{12} + 36036 \, a \cos \left (d x + c\right )^{10} - 15015 \, a \cos \left (d x + c\right )^{8} - 40 \,{\left (231 \, a \cos \left (d x + c\right )^{12} - 567 \, a \cos \left (d x + c\right )^{10} + 371 \, a \cos \left (d x + c\right )^{8} - 5 \, a \cos \left (d x + c\right )^{6} - 6 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right )}{120120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)^6*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/120120*(8580*a*cos(d*x + c)^14 - 30030*a*cos(d*x + c)^12 + 36036*a*cos(d*x + c)^10 - 15015*a*cos(d*x + c)^8
- 40*(231*a*cos(d*x + c)^12 - 567*a*cos(d*x + c)^10 + 371*a*cos(d*x + c)^8 - 5*a*cos(d*x + c)^6 - 6*a*cos(d*x
+ c)^4 - 8*a*cos(d*x + c)^2 - 16*a)*sin(d*x + c))/d

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Sympy [A]  time = 175.577, size = 184, normalized size = 1.43 \begin{align*} \begin{cases} \frac{16 a \sin ^{13}{\left (c + d x \right )}}{3003 d} + \frac{8 a \sin ^{11}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{231 d} + \frac{2 a \sin ^{9}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{21 d} + \frac{a \sin ^{7}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{7 d} - \frac{a \sin ^{6}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac{3 a \sin ^{4}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{40 d} - \frac{a \sin ^{2}{\left (c + d x \right )} \cos ^{12}{\left (c + d x \right )}}{40 d} - \frac{a \cos ^{14}{\left (c + d x \right )}}{280 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \sin ^{6}{\left (c \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*sin(d*x+c)**6*(a+a*sin(d*x+c)),x)

[Out]

Piecewise((16*a*sin(c + d*x)**13/(3003*d) + 8*a*sin(c + d*x)**11*cos(c + d*x)**2/(231*d) + 2*a*sin(c + d*x)**9
*cos(c + d*x)**4/(21*d) + a*sin(c + d*x)**7*cos(c + d*x)**6/(7*d) - a*sin(c + d*x)**6*cos(c + d*x)**8/(8*d) -
3*a*sin(c + d*x)**4*cos(c + d*x)**10/(40*d) - a*sin(c + d*x)**2*cos(c + d*x)**12/(40*d) - a*cos(c + d*x)**14/(
280*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**6*cos(c)**7, True))

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Giac [A]  time = 1.38966, size = 220, normalized size = 1.71 \begin{align*} \frac{a \cos \left (14 \, d x + 14 \, c\right )}{114688 \, d} - \frac{7 \, a \cos \left (10 \, d x + 10 \, c\right )}{81920 \, d} + \frac{7 \, a \cos \left (6 \, d x + 6 \, c\right )}{16384 \, d} - \frac{35 \, a \cos \left (2 \, d x + 2 \, c\right )}{16384 \, d} - \frac{a \sin \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac{a \sin \left (11 \, d x + 11 \, c\right )}{45056 \, d} + \frac{a \sin \left (9 \, d x + 9 \, c\right )}{6144 \, d} + \frac{3 \, a \sin \left (7 \, d x + 7 \, c\right )}{14336 \, d} - \frac{3 \, a \sin \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac{5 \, a \sin \left (3 \, d x + 3 \, c\right )}{4096 \, d} + \frac{5 \, a \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)^7*sin(d*x+c)^6*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/114688*a*cos(14*d*x + 14*c)/d - 7/81920*a*cos(10*d*x + 10*c)/d + 7/16384*a*cos(6*d*x + 6*c)/d - 35/16384*a*c
os(2*d*x + 2*c)/d - 1/53248*a*sin(13*d*x + 13*c)/d - 1/45056*a*sin(11*d*x + 11*c)/d + 1/6144*a*sin(9*d*x + 9*c
)/d + 3/14336*a*sin(7*d*x + 7*c)/d - 3/4096*a*sin(5*d*x + 5*c)/d - 5/4096*a*sin(3*d*x + 3*c)/d + 5/1024*a*sin(
d*x + c)/d