3.677 \(\int \frac{\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=109 \[ -\frac{\sin ^{12}(c+d x)}{12 a d}+\frac{\sin ^{11}(c+d x)}{11 a d}+\frac{\sin ^{10}(c+d x)}{5 a d}-\frac{2 \sin ^9(c+d x)}{9 a d}-\frac{\sin ^8(c+d x)}{8 a d}+\frac{\sin ^7(c+d x)}{7 a d} \]

[Out]

Sin[c + d*x]^7/(7*a*d) - Sin[c + d*x]^8/(8*a*d) - (2*Sin[c + d*x]^9)/(9*a*d) + Sin[c + d*x]^10/(5*a*d) + Sin[c
+ d*x]^11/(11*a*d) - Sin[c + d*x]^12/(12*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.127646, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac{\sin ^{12}(c+d x)}{12 a d}+\frac{\sin ^{11}(c+d x)}{11 a d}+\frac{\sin ^{10}(c+d x)}{5 a d}-\frac{2 \sin ^9(c+d x)}{9 a d}-\frac{\sin ^8(c+d x)}{8 a d}+\frac{\sin ^7(c+d x)}{7 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.597994, size = 68, normalized size = 0.62 \[ \frac{\sin ^7(c+d x) \left (-2310 \sin ^5(c+d x)+2520 \sin ^4(c+d x)+5544 \sin ^3(c+d x)-6160 \sin ^2(c+d x)-3465 \sin (c+d x)+3960\right )}{27720 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sin[c + d*x]^7*(3960 - 3465*Sin[c + d*x] - 6160*Sin[c + d*x]^2 + 5544*Sin[c + d*x]^3 + 2520*Sin[c + d*x]^4 -
2310*Sin[c + d*x]^5))/(27720*a*d)

________________________________________________________________________________________

Maple [A]  time = 0.115, size = 69, normalized size = 0.6 \begin{align*}{\frac{1}{da} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{12}}{12}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{11}}{11}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{5}}-{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{9}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{8}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}} \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*(-1/12*sin(d*x+c)^12+1/11*sin(d*x+c)^11+1/5*sin(d*x+c)^10-2/9*sin(d*x+c)^9-1/8*sin(d*x+c)^8+1/7*sin(d*x+
c)^7)

________________________________________________________________________________________

Maxima [A]  time = 1.04859, size = 93, normalized size = 0.85 \begin{align*} -\frac{2310 \, \sin \left (d x + c\right )^{12} - 2520 \, \sin \left (d x + c\right )^{11} - 5544 \, \sin \left (d x + c\right )^{10} + 6160 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} - 3960 \, \sin \left (d x + c\right )^{7}}{27720 \, a 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]

-1/27720*(2310*sin(d*x + c)^12 - 2520*sin(d*x + c)^11 - 5544*sin(d*x + c)^10 + 6160*sin(d*x + c)^9 + 3465*sin(
d*x + c)^8 - 3960*sin(d*x + c)^7)/(a*d)

________________________________________________________________________________________

Fricas [A]  time = 1.21247, size = 311, normalized size = 2.85 \begin{align*} -\frac{2310 \, \cos \left (d x + c\right )^{12} - 8316 \, \cos \left (d x + c\right )^{10} + 10395 \, \cos \left (d x + c\right )^{8} - 4620 \, \cos \left (d x + c\right )^{6} + 40 \,{\left (63 \, \cos \left (d x + c\right )^{10} - 161 \, \cos \left (d x + c\right )^{8} + 113 \, \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right )}{27720 \, a 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]

-1/27720*(2310*cos(d*x + c)^12 - 8316*cos(d*x + c)^10 + 10395*cos(d*x + c)^8 - 4620*cos(d*x + c)^6 + 40*(63*co
s(d*x + c)^10 - 161*cos(d*x + c)^8 + 113*cos(d*x + c)^6 - 3*cos(d*x + c)^4 - 4*cos(d*x + c)^2 - 8)*sin(d*x + c
))/(a*d)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.3122, size = 93, normalized size = 0.85 \begin{align*} -\frac{2310 \, \sin \left (d x + c\right )^{12} - 2520 \, \sin \left (d x + c\right )^{11} - 5544 \, \sin \left (d x + c\right )^{10} + 6160 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} - 3960 \, \sin \left (d x + c\right )^{7}}{27720 \, a 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]

-1/27720*(2310*sin(d*x + c)^12 - 2520*sin(d*x + c)^11 - 5544*sin(d*x + c)^10 + 6160*sin(d*x + c)^9 + 3465*sin(
d*x + c)^8 - 3960*sin(d*x + c)^7)/(a*d)