3.679 \(\int \frac{\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=109 \[ -\frac{\sin ^{10}(c+d x)}{10 a d}+\frac{\sin ^9(c+d x)}{9 a d}+\frac{\sin ^8(c+d x)}{4 a d}-\frac{2 \sin ^7(c+d x)}{7 a d}-\frac{\sin ^6(c+d x)}{6 a d}+\frac{\sin ^5(c+d x)}{5 a d} \]

[Out]

Sin[c + d*x]^5/(5*a*d) - Sin[c + d*x]^6/(6*a*d) - (2*Sin[c + d*x]^7)/(7*a*d) + Sin[c + d*x]^8/(4*a*d) + Sin[c
+ d*x]^9/(9*a*d) - Sin[c + d*x]^10/(10*a*d)

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Rubi [A]  time = 0.127047, 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} \[ -\frac{\sin ^{10}(c+d x)}{10 a d}+\frac{\sin ^9(c+d x)}{9 a d}+\frac{\sin ^8(c+d x)}{4 a d}-\frac{2 \sin ^7(c+d x)}{7 a d}-\frac{\sin ^6(c+d x)}{6 a d}+\frac{\sin ^5(c+d x)}{5 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.414949, size = 68, normalized size = 0.62 \[ \frac{\sin ^5(c+d x) \left (-126 \sin ^5(c+d x)+140 \sin ^4(c+d x)+315 \sin ^3(c+d x)-360 \sin ^2(c+d x)-210 \sin (c+d x)+252\right )}{1260 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sin[c + d*x]^5*(252 - 210*Sin[c + d*x] - 360*Sin[c + d*x]^2 + 315*Sin[c + d*x]^3 + 140*Sin[c + d*x]^4 - 126*S
in[c + d*x]^5))/(1260*a*d)

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Maple [A]  time = 0.1, size = 69, normalized size = 0.6 \begin{align*}{\frac{1}{da} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{10}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{9}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{4}}-{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}} \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*(-1/10*sin(d*x+c)^10+1/9*sin(d*x+c)^9+1/4*sin(d*x+c)^8-2/7*sin(d*x+c)^7-1/6*sin(d*x+c)^6+1/5*sin(d*x+c)^
5)

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Maxima [A]  time = 1.12277, size = 93, normalized size = 0.85 \begin{align*} -\frac{126 \, \sin \left (d x + c\right )^{10} - 140 \, \sin \left (d x + c\right )^{9} - 315 \, \sin \left (d x + c\right )^{8} + 360 \, \sin \left (d x + c\right )^{7} + 210 \, \sin \left (d x + c\right )^{6} - 252 \, \sin \left (d x + c\right )^{5}}{1260 \, 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)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/1260*(126*sin(d*x + c)^10 - 140*sin(d*x + c)^9 - 315*sin(d*x + c)^8 + 360*sin(d*x + c)^7 + 210*sin(d*x + c)
^6 - 252*sin(d*x + c)^5)/(a*d)

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Fricas [A]  time = 1.16745, size = 239, normalized size = 2.19 \begin{align*} \frac{126 \, \cos \left (d x + c\right )^{10} - 315 \, \cos \left (d x + c\right )^{8} + 210 \, \cos \left (d x + c\right )^{6} + 4 \,{\left (35 \, \cos \left (d x + c\right )^{8} - 50 \, \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 )}{1260 \, 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)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/1260*(126*cos(d*x + c)^10 - 315*cos(d*x + c)^8 + 210*cos(d*x + c)^6 + 4*(35*cos(d*x + c)^8 - 50*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)**4/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.29074, size = 93, normalized size = 0.85 \begin{align*} -\frac{126 \, \sin \left (d x + c\right )^{10} - 140 \, \sin \left (d x + c\right )^{9} - 315 \, \sin \left (d x + c\right )^{8} + 360 \, \sin \left (d x + c\right )^{7} + 210 \, \sin \left (d x + c\right )^{6} - 252 \, \sin \left (d x + c\right )^{5}}{1260 \, 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)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/1260*(126*sin(d*x + c)^10 - 140*sin(d*x + c)^9 - 315*sin(d*x + c)^8 + 360*sin(d*x + c)^7 + 210*sin(d*x + c)
^6 - 252*sin(d*x + c)^5)/(a*d)