3.657 \(\int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=113 \[ -\frac{a \sin ^{13}(c+d x)}{13 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}-\frac{a \sin ^9(c+d x)}{3 d}+\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \cos ^{12}(c+d x)}{12 d}+\frac{a \cos ^{10}(c+d x)}{5 d}-\frac{a \cos ^8(c+d x)}{8 d} \]

[Out]

-(a*Cos[c + d*x]^8)/(8*d) + (a*Cos[c + d*x]^10)/(5*d) - (a*Cos[c + d*x]^12)/(12*d) + (a*Sin[c + d*x]^7)/(7*d)
- (a*Sin[c + d*x]^9)/(3*d) + (3*a*Sin[c + d*x]^11)/(11*d) - (a*Sin[c + d*x]^13)/(13*d)

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Rubi [A]  time = 0.139859, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2834, 2565, 266, 43, 2564, 270} \[ -\frac{a \sin ^{13}(c+d x)}{13 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}-\frac{a \sin ^9(c+d x)}{3 d}+\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \cos ^{12}(c+d x)}{12 d}+\frac{a \cos ^{10}(c+d x)}{5 d}-\frac{a \cos ^8(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*Sin[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Cos[c + d*x]^8)/(8*d) + (a*Cos[c + d*x]^10)/(5*d) - (a*Cos[c + d*x]^12)/(12*d) + (a*Sin[c + d*x]^7)/(7*d)
- (a*Sin[c + d*x]^9)/(3*d) + (3*a*Sin[c + d*x]^11)/(11*d) - (a*Sin[c + d*x]^13)/(13*d)

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])

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 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]

Rubi steps

\begin{align*} \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^7(c+d x) \sin ^5(c+d x) \, dx+a \int \cos ^7(c+d x) \sin ^6(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int x^7 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}+\frac{a \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^3 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{a \operatorname{Subst}\left (\int (1-x)^2 x^3 \, dx,x,\cos ^2(c+d x)\right )}{2 d}+\frac{a \operatorname{Subst}\left (\int \left (x^6-3 x^8+3 x^{10}-x^{12}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^9(c+d x)}{3 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}-\frac{a \sin ^{13}(c+d x)}{13 d}-\frac{a \operatorname{Subst}\left (\int \left (x^3-2 x^4+x^5\right ) \, dx,x,\cos ^2(c+d x)\right )}{2 d}\\ &=-\frac{a \cos ^8(c+d x)}{8 d}+\frac{a \cos ^{10}(c+d x)}{5 d}-\frac{a \cos ^{12}(c+d x)}{12 d}+\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^9(c+d x)}{3 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}-\frac{a \sin ^{13}(c+d x)}{13 d}\\ \end{align*}

Mathematica [A]  time = 0.724221, size = 137, normalized size = 1.21 \[ -\frac{a (-600600 \sin (c+d x)+150150 \sin (3 (c+d x))+90090 \sin (5 (c+d x))-25740 \sin (7 (c+d x))-20020 \sin (9 (c+d x))+2730 \sin (11 (c+d x))+2310 \sin (13 (c+d x))+600600 \cos (2 (c+d x))+75075 \cos (4 (c+d x))-100100 \cos (6 (c+d x))-30030 \cos (8 (c+d x))+12012 \cos (10 (c+d x))+5005 \cos (12 (c+d x)))}{123002880 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*Sin[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

-(a*(600600*Cos[2*(c + d*x)] + 75075*Cos[4*(c + d*x)] - 100100*Cos[6*(c + d*x)] - 30030*Cos[8*(c + d*x)] + 120
12*Cos[10*(c + d*x)] + 5005*Cos[12*(c + d*x)] - 600600*Sin[c + d*x] + 150150*Sin[3*(c + d*x)] + 90090*Sin[5*(c
+ d*x)] - 25740*Sin[7*(c + d*x)] - 20020*Sin[9*(c + d*x)] + 2730*Sin[11*(c + d*x)] + 2310*Sin[13*(c + d*x)]))
/(123002880*d)

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Maple [A]  time = 0.034, size = 148, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ( 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 ) +a \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{12}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{30}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{120}} \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)^5*(a+a*sin(d*x+c)),x)

[Out]

1/d*(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))+a*(-1/12*sin(d*x+c)^4*cos(d*x+c)^8-1/30*sin(d
*x+c)^2*cos(d*x+c)^8-1/120*cos(d*x+c)^8))

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Maxima [A]  time = 1.04414, size = 127, normalized size = 1.12 \begin{align*} -\frac{9240 \, a \sin \left (d x + c\right )^{13} + 10010 \, 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} + 45045 \, a \sin \left (d x + c\right )^{8} - 17160 \, a \sin \left (d x + c\right )^{7} - 20020 \, a \sin \left (d x + c\right )^{6}}{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)^5*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/120120*(9240*a*sin(d*x + c)^13 + 10010*a*sin(d*x + c)^12 - 32760*a*sin(d*x + c)^11 - 36036*a*sin(d*x + c)^1
0 + 40040*a*sin(d*x + c)^9 + 45045*a*sin(d*x + c)^8 - 17160*a*sin(d*x + c)^7 - 20020*a*sin(d*x + c)^6)/d

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Fricas [A]  time = 1.64133, size = 336, normalized size = 2.97 \begin{align*} -\frac{10010 \, a \cos \left (d x + c\right )^{12} - 24024 \, 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)^5*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/120120*(10010*a*cos(d*x + c)^12 - 24024*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 = 117.722, size = 182, normalized size = 1.61 \begin{align*} \begin{cases} \frac{16 a \sin ^{13}{\left (c + d x \right )}}{3003 d} + \frac{a \sin ^{12}{\left (c + d x \right )}}{120 d} + \frac{8 a \sin ^{11}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{231 d} + \frac{a \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{20 d} + \frac{2 a \sin ^{9}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{21 d} + \frac{a \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8 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 ^{6}{\left (c + d x \right )}}{6 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \sin ^{5}{\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)**5*(a+a*sin(d*x+c)),x)

[Out]

Piecewise((16*a*sin(c + d*x)**13/(3003*d) + a*sin(c + d*x)**12/(120*d) + 8*a*sin(c + d*x)**11*cos(c + d*x)**2/
(231*d) + a*sin(c + d*x)**10*cos(c + d*x)**2/(20*d) + 2*a*sin(c + d*x)**9*cos(c + d*x)**4/(21*d) + a*sin(c + d
*x)**8*cos(c + d*x)**4/(8*d) + a*sin(c + d*x)**7*cos(c + d*x)**6/(7*d) + a*sin(c + d*x)**6*cos(c + d*x)**6/(6*
d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**5*cos(c)**7, True))

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Giac [A]  time = 1.26667, size = 261, normalized size = 2.31 \begin{align*} -\frac{a \cos \left (12 \, d x + 12 \, c\right )}{24576 \, d} - \frac{a \cos \left (10 \, d x + 10 \, c\right )}{10240 \, d} + \frac{a \cos \left (8 \, d x + 8 \, c\right )}{4096 \, d} + \frac{5 \, a \cos \left (6 \, d x + 6 \, c\right )}{6144 \, d} - \frac{5 \, a \cos \left (4 \, d x + 4 \, c\right )}{8192 \, d} - \frac{5 \, a \cos \left (2 \, d x + 2 \, c\right )}{1024 \, 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)^5*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/24576*a*cos(12*d*x + 12*c)/d - 1/10240*a*cos(10*d*x + 10*c)/d + 1/4096*a*cos(8*d*x + 8*c)/d + 5/6144*a*cos(
6*d*x + 6*c)/d - 5/8192*a*cos(4*d*x + 4*c)/d - 5/1024*a*cos(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