3.678 \(\int \frac{\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=109 \[ -\frac{\sin ^{11}(c+d x)}{11 a d}+\frac{\sin ^{10}(c+d x)}{10 a d}+\frac{2 \sin ^9(c+d x)}{9 a d}-\frac{\sin ^8(c+d x)}{4 a d}-\frac{\sin ^7(c+d x)}{7 a d}+\frac{\sin ^6(c+d x)}{6 a d} \]

[Out]

Sin[c + d*x]^6/(6*a*d) - Sin[c + d*x]^7/(7*a*d) - Sin[c + d*x]^8/(4*a*d) + (2*Sin[c + d*x]^9)/(9*a*d) + Sin[c
+ d*x]^10/(10*a*d) - Sin[c + d*x]^11/(11*a*d)

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Rubi [A]  time = 0.126867, 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} \[ -\frac{\sin ^{11}(c+d x)}{11 a d}+\frac{\sin ^{10}(c+d x)}{10 a d}+\frac{2 \sin ^9(c+d x)}{9 a d}-\frac{\sin ^8(c+d x)}{4 a d}-\frac{\sin ^7(c+d x)}{7 a d}+\frac{\sin ^6(c+d x)}{6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.908169, size = 68, normalized size = 0.62 \[ \frac{\sin ^6(c+d x) \left (-1260 \sin ^5(c+d x)+1386 \sin ^4(c+d x)+3080 \sin ^3(c+d x)-3465 \sin ^2(c+d x)-1980 \sin (c+d x)+2310\right )}{13860 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sin[c + d*x]^6*(2310 - 1980*Sin[c + d*x] - 3465*Sin[c + d*x]^2 + 3080*Sin[c + d*x]^3 + 1386*Sin[c + d*x]^4 -
1260*Sin[c + d*x]^5))/(13860*a*d)

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Maple [A]  time = 0.105, size = 69, normalized size = 0.6 \begin{align*}{\frac{1}{da} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{11}}{11}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{10}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{9}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{4}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6}} \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*(-1/11*sin(d*x+c)^11+1/10*sin(d*x+c)^10+2/9*sin(d*x+c)^9-1/4*sin(d*x+c)^8-1/7*sin(d*x+c)^7+1/6*sin(d*x+c
)^6)

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Maxima [A]  time = 1.03119, size = 93, normalized size = 0.85 \begin{align*} -\frac{1260 \, \sin \left (d x + c\right )^{11} - 1386 \, \sin \left (d x + c\right )^{10} - 3080 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} + 1980 \, \sin \left (d x + c\right )^{7} - 2310 \, \sin \left (d x + c\right )^{6}}{13860 \, 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)^5/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/13860*(1260*sin(d*x + c)^11 - 1386*sin(d*x + c)^10 - 3080*sin(d*x + c)^9 + 3465*sin(d*x + c)^8 + 1980*sin(d
*x + c)^7 - 2310*sin(d*x + c)^6)/(a*d)

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Fricas [A]  time = 1.15045, size = 278, normalized size = 2.55 \begin{align*} -\frac{1386 \, \cos \left (d x + c\right )^{10} - 3465 \, \cos \left (d x + c\right )^{8} + 2310 \, \cos \left (d x + c\right )^{6} - 20 \,{\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 )}{13860 \, 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)^5/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/13860*(1386*cos(d*x + c)^10 - 3465*cos(d*x + c)^8 + 2310*cos(d*x + c)^6 - 20*(63*cos(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)

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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)**5/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.28024, size = 93, normalized size = 0.85 \begin{align*} -\frac{1260 \, \sin \left (d x + c\right )^{11} - 1386 \, \sin \left (d x + c\right )^{10} - 3080 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} + 1980 \, \sin \left (d x + c\right )^{7} - 2310 \, \sin \left (d x + c\right )^{6}}{13860 \, 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)^5/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/13860*(1260*sin(d*x + c)^11 - 1386*sin(d*x + c)^10 - 3080*sin(d*x + c)^9 + 3465*sin(d*x + c)^8 + 1980*sin(d
*x + c)^7 - 2310*sin(d*x + c)^6)/(a*d)