3.658 \(\int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx\)

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

[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]^5)/(5*d)
- (3*a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^9)/(3*d) - (a*Sin[c + d*x]^11)/(11*d)

________________________________________________________________________________________

Rubi [A]  time = 0.136612, 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, 2564, 270, 2565, 266, 43} \[ -\frac{a \sin ^{11}(c+d x)}{11 d}+\frac{a \sin ^9(c+d x)}{3 d}-\frac{3 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \cos ^{12}(c+d x)}{12 d}+\frac{a \cos ^{10}(c+d x)}{5 d}-\frac{a \cos ^8(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*Sin[c + d*x]^4*(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]^5)/(5*d)
- (3*a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^9)/(3*d) - (a*Sin[c + d*x]^11)/(11*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 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]

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])

Rubi steps

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

Mathematica [A]  time = 0.558947, size = 127, normalized size = 1.12 \[ -\frac{a (-129360 \sin (c+d x)+18480 \sin (3 (c+d x))+20328 \sin (5 (c+d x))+1320 \sin (7 (c+d x))-3080 \sin (9 (c+d x))-840 \sin (11 (c+d x))+46200 \cos (2 (c+d x))+5775 \cos (4 (c+d x))-7700 \cos (6 (c+d x))-2310 \cos (8 (c+d x))+924 \cos (10 (c+d x))+385 \cos (12 (c+d x)))}{9461760 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

-(a*(46200*Cos[2*(c + d*x)] + 5775*Cos[4*(c + d*x)] - 7700*Cos[6*(c + d*x)] - 2310*Cos[8*(c + d*x)] + 924*Cos[
10*(c + d*x)] + 385*Cos[12*(c + d*x)] - 129360*Sin[c + d*x] + 18480*Sin[3*(c + d*x)] + 20328*Sin[5*(c + d*x)]
+ 1320*Sin[7*(c + d*x)] - 3080*Sin[9*(c + d*x)] - 840*Sin[11*(c + d*x)]))/(9461760*d)

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 130, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( 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 ) +a \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{11}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{33}}+{\frac{\sin \left ( dx+c \right ) }{231} \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)^4*(a+a*sin(d*x+c)),x)

[Out]

1/d*(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)+a*(-1/11*sin(d*x+c)
^3*cos(d*x+c)^8-1/33*sin(d*x+c)*cos(d*x+c)^8+1/231*(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)))

________________________________________________________________________________________

Maxima [A]  time = 0.998697, size = 127, normalized size = 1.12 \begin{align*} -\frac{770 \, a \sin \left (d x + c\right )^{12} + 840 \, a \sin \left (d x + c\right )^{11} - 2772 \, a \sin \left (d x + c\right )^{10} - 3080 \, a \sin \left (d x + c\right )^{9} + 3465 \, a \sin \left (d x + c\right )^{8} + 3960 \, a \sin \left (d x + c\right )^{7} - 1540 \, a \sin \left (d x + c\right )^{6} - 1848 \, a \sin \left (d x + c\right )^{5}}{9240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/9240*(770*a*sin(d*x + c)^12 + 840*a*sin(d*x + c)^11 - 2772*a*sin(d*x + c)^10 - 3080*a*sin(d*x + c)^9 + 3465
*a*sin(d*x + c)^8 + 3960*a*sin(d*x + c)^7 - 1540*a*sin(d*x + c)^6 - 1848*a*sin(d*x + c)^5)/d

________________________________________________________________________________________

Fricas [A]  time = 1.60779, size = 294, normalized size = 2.6 \begin{align*} -\frac{770 \, a \cos \left (d x + c\right )^{12} - 1848 \, a \cos \left (d x + c\right )^{10} + 1155 \, a \cos \left (d x + c\right )^{8} - 8 \,{\left (105 \, a \cos \left (d x + c\right )^{10} - 140 \, 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 )}{9240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/9240*(770*a*cos(d*x + c)^12 - 1848*a*cos(d*x + c)^10 + 1155*a*cos(d*x + c)^8 - 8*(105*a*cos(d*x + c)^10 - 1
40*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

________________________________________________________________________________________

Sympy [A]  time = 82.1131, size = 182, normalized size = 1.61 \begin{align*} \begin{cases} \frac{a \sin ^{12}{\left (c + d x \right )}}{120 d} + \frac{16 a \sin ^{11}{\left (c + d x \right )}}{1155 d} + \frac{a \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{20 d} + \frac{8 a \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{105 d} + \frac{a \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8 d} + \frac{6 a \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{35 d} + \frac{a \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac{a \sin ^{5}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{5 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \sin ^{4}{\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)**4*(a+a*sin(d*x+c)),x)

[Out]

Piecewise((a*sin(c + d*x)**12/(120*d) + 16*a*sin(c + d*x)**11/(1155*d) + a*sin(c + d*x)**10*cos(c + d*x)**2/(2
0*d) + 8*a*sin(c + d*x)**9*cos(c + d*x)**2/(105*d) + a*sin(c + d*x)**8*cos(c + d*x)**4/(8*d) + 6*a*sin(c + d*x
)**7*cos(c + d*x)**4/(35*d) + a*sin(c + d*x)**6*cos(c + d*x)**6/(6*d) + a*sin(c + d*x)**5*cos(c + d*x)**6/(5*d
), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**4*cos(c)**7, True))

________________________________________________________________________________________

Giac [A]  time = 1.31886, size = 240, normalized size = 2.12 \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 (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac{a \sin \left (9 \, d x + 9 \, c\right )}{3072 \, d} - \frac{a \sin \left (7 \, d x + 7 \, c\right )}{7168 \, d} - \frac{11 \, a \sin \left (5 \, d x + 5 \, c\right )}{5120 \, d} - \frac{a \sin \left (3 \, d x + 3 \, c\right )}{512 \, d} + \frac{7 \, a \sin \left (d x + c\right )}{512 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)^4*(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 + 1/11264*a*sin(11*d*x + 11*c)/d +
1/3072*a*sin(9*d*x + 9*c)/d - 1/7168*a*sin(7*d*x + 7*c)/d - 11/5120*a*sin(5*d*x + 5*c)/d - 1/512*a*sin(3*d*x +
3*c)/d + 7/512*a*sin(d*x + c)/d