3.681 \(\int \frac{\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=91 \[ \frac{\sin ^7(c+d x)}{7 a d}-\frac{2 \sin ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x)}{3 a d}-\frac{\cos ^8(c+d x)}{8 a d}+\frac{\cos ^6(c+d x)}{6 a d} \]

[Out]

Cos[c + d*x]^6/(6*a*d) - Cos[c + d*x]^8/(8*a*d) + Sin[c + d*x]^3/(3*a*d) - (2*Sin[c + d*x]^5)/(5*a*d) + Sin[c
+ d*x]^7/(7*a*d)

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Rubi [A]  time = 0.161681, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2835, 2564, 270, 2565, 14} \[ \frac{\sin ^7(c+d x)}{7 a d}-\frac{2 \sin ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x)}{3 a d}-\frac{\cos ^8(c+d x)}{8 a d}+\frac{\cos ^6(c+d x)}{6 a d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^7*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]

[Out]

Cos[c + d*x]^6/(6*a*d) - Cos[c + d*x]^8/(8*a*d) + Sin[c + d*x]^3/(3*a*d) - (2*Sin[c + d*x]^5)/(5*a*d) + Sin[c
+ d*x]^7/(7*a*d)

Rule 2835

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.291162, size = 68, normalized size = 0.75 \[ \frac{\sin ^3(c+d x) \left (-105 \sin ^5(c+d x)+120 \sin ^4(c+d x)+280 \sin ^3(c+d x)-336 \sin ^2(c+d x)-210 \sin (c+d x)+280\right )}{840 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]

[Out]

(Sin[c + d*x]^3*(280 - 210*Sin[c + d*x] - 336*Sin[c + d*x]^2 + 280*Sin[c + d*x]^3 + 120*Sin[c + d*x]^4 - 105*S
in[c + d*x]^5))/(840*a*d)

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Maple [A]  time = 0.082, size = 69, normalized size = 0.8 \begin{align*}{\frac{1}{da} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{8}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{3}}-{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*sin(d*x+c)^2/(a+a*sin(d*x+c)),x)

[Out]

1/d/a*(-1/8*sin(d*x+c)^8+1/7*sin(d*x+c)^7+1/3*sin(d*x+c)^6-2/5*sin(d*x+c)^5-1/4*sin(d*x+c)^4+1/3*sin(d*x+c)^3)

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Maxima [A]  time = 1.01142, size = 93, normalized size = 1.02 \begin{align*} -\frac{105 \, \sin \left (d x + c\right )^{8} - 120 \, \sin \left (d x + c\right )^{7} - 280 \, \sin \left (d x + c\right )^{6} + 336 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} - 280 \, \sin \left (d x + c\right )^{3}}{840 \, 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)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/840*(105*sin(d*x + c)^8 - 120*sin(d*x + c)^7 - 280*sin(d*x + c)^6 + 336*sin(d*x + c)^5 + 210*sin(d*x + c)^4
- 280*sin(d*x + c)^3)/(a*d)

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Fricas [A]  time = 1.10554, size = 182, normalized size = 2. \begin{align*} -\frac{105 \, \cos \left (d x + c\right )^{8} - 140 \, \cos \left (d x + c\right )^{6} + 8 \,{\left (15 \, \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 )}{840 \, 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)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/840*(105*cos(d*x + c)^8 - 140*cos(d*x + c)^6 + 8*(15*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)**2/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.22116, size = 93, normalized size = 1.02 \begin{align*} -\frac{105 \, \sin \left (d x + c\right )^{8} - 120 \, \sin \left (d x + c\right )^{7} - 280 \, \sin \left (d x + c\right )^{6} + 336 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} - 280 \, \sin \left (d x + c\right )^{3}}{840 \, 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)^2/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/840*(105*sin(d*x + c)^8 - 120*sin(d*x + c)^7 - 280*sin(d*x + c)^6 + 336*sin(d*x + c)^5 + 210*sin(d*x + c)^4
- 280*sin(d*x + c)^3)/(a*d)