3.634 \(\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=135 \[ -\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{3 \cos ^5(c+d x)}{5 a^2 d}-\frac{2 \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{x}{8 a^2} \]

[Out]

-x/(8*a^2) - (2*Cos[c + d*x]^3)/(3*a^2*d) + (3*Cos[c + d*x]^5)/(5*a^2*d) - Cos[c + d*x]^7/(7*a^2*d) - (Cos[c +
d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(3*a
^2*d)

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Rubi [A]  time = 0.347352, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2875, 2873, 2565, 14, 2568, 2635, 8, 270} \[ -\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{3 \cos ^5(c+d x)}{5 a^2 d}-\frac{2 \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{x}{8 a^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]

[Out]

-x/(8*a^2) - (2*Cos[c + d*x]^3)/(3*a^2*d) + (3*Cos[c + d*x]^5)/(5*a^2*d) - Cos[c + d*x]^7/(7*a^2*d) - (Cos[c +
d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(3*a
^2*d)

Rule 2875

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[((g*Cos[e + f*x])^(2*m + p)*(d*Sin[e + f*x])^n)/(a - b*Sin[e +
f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]

Rule 2873

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 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]]

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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]

Rubi steps

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

Mathematica [B]  time = 3.0872, size = 418, normalized size = 3.1 \[ -\frac{1680 d x \sin \left (\frac{c}{2}\right )-1365 \sin \left (\frac{c}{2}+d x\right )+1365 \sin \left (\frac{3 c}{2}+d x\right )-210 \sin \left (\frac{3 c}{2}+2 d x\right )-210 \sin \left (\frac{5 c}{2}+2 d x\right )-175 \sin \left (\frac{5 c}{2}+3 d x\right )+175 \sin \left (\frac{7 c}{2}+3 d x\right )-210 \sin \left (\frac{7 c}{2}+4 d x\right )-210 \sin \left (\frac{9 c}{2}+4 d x\right )+147 \sin \left (\frac{9 c}{2}+5 d x\right )-147 \sin \left (\frac{11 c}{2}+5 d x\right )+70 \sin \left (\frac{11 c}{2}+6 d x\right )+70 \sin \left (\frac{13 c}{2}+6 d x\right )-15 \sin \left (\frac{13 c}{2}+7 d x\right )+15 \sin \left (\frac{15 c}{2}+7 d x\right )+210 \cos \left (\frac{c}{2}\right ) (8 d x+1)+1365 \cos \left (\frac{c}{2}+d x\right )+1365 \cos \left (\frac{3 c}{2}+d x\right )-210 \cos \left (\frac{3 c}{2}+2 d x\right )+210 \cos \left (\frac{5 c}{2}+2 d x\right )+175 \cos \left (\frac{5 c}{2}+3 d x\right )+175 \cos \left (\frac{7 c}{2}+3 d x\right )-210 \cos \left (\frac{7 c}{2}+4 d x\right )+210 \cos \left (\frac{9 c}{2}+4 d x\right )-147 \cos \left (\frac{9 c}{2}+5 d x\right )-147 \cos \left (\frac{11 c}{2}+5 d x\right )+70 \cos \left (\frac{11 c}{2}+6 d x\right )-70 \cos \left (\frac{13 c}{2}+6 d x\right )+15 \cos \left (\frac{13 c}{2}+7 d x\right )+15 \cos \left (\frac{15 c}{2}+7 d x\right )-210 \sin \left (\frac{c}{2}\right )}{13440 a^2 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]

[Out]

-(210*(1 + 8*d*x)*Cos[c/2] + 1365*Cos[c/2 + d*x] + 1365*Cos[(3*c)/2 + d*x] - 210*Cos[(3*c)/2 + 2*d*x] + 210*Co
s[(5*c)/2 + 2*d*x] + 175*Cos[(5*c)/2 + 3*d*x] + 175*Cos[(7*c)/2 + 3*d*x] - 210*Cos[(7*c)/2 + 4*d*x] + 210*Cos[
(9*c)/2 + 4*d*x] - 147*Cos[(9*c)/2 + 5*d*x] - 147*Cos[(11*c)/2 + 5*d*x] + 70*Cos[(11*c)/2 + 6*d*x] - 70*Cos[(1
3*c)/2 + 6*d*x] + 15*Cos[(13*c)/2 + 7*d*x] + 15*Cos[(15*c)/2 + 7*d*x] - 210*Sin[c/2] + 1680*d*x*Sin[c/2] - 136
5*Sin[c/2 + d*x] + 1365*Sin[(3*c)/2 + d*x] - 210*Sin[(3*c)/2 + 2*d*x] - 210*Sin[(5*c)/2 + 2*d*x] - 175*Sin[(5*
c)/2 + 3*d*x] + 175*Sin[(7*c)/2 + 3*d*x] - 210*Sin[(7*c)/2 + 4*d*x] - 210*Sin[(9*c)/2 + 4*d*x] + 147*Sin[(9*c)
/2 + 5*d*x] - 147*Sin[(11*c)/2 + 5*d*x] + 70*Sin[(11*c)/2 + 6*d*x] + 70*Sin[(13*c)/2 + 6*d*x] - 15*Sin[(13*c)/
2 + 7*d*x] + 15*Sin[(15*c)/2 + 7*d*x])/(13440*a^2*d*(Cos[c/2] + Sin[c/2]))

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Maple [B]  time = 0.098, size = 415, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x)

[Out]

-1/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13-5/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1
/2*c)^11-4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^10+97/12/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1
/2*d*x+1/2*c)^9-52/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^8+8/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^
7*tan(1/2*d*x+1/2*c)^6-97/12/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^5-24/5/d/a^2/(1+tan(1/2*d*x+1
/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4+5/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^3-44/15/d/a^2/(1+tan(1
/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2+1/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)-44/105/d/a^2/(
1+tan(1/2*d*x+1/2*c)^2)^7-1/4/d/a^2*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.56853, size = 562, normalized size = 4.16 \begin{align*} \frac{\frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1232 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{700 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{2016 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{3395 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{1120 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{7280 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{3395 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{1680 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac{700 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac{105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 176}{a^{2} + \frac{7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{21 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{35 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{35 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{21 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{7 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac{a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac{105 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/420*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 1232*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 700*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 - 2016*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3395*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 112
0*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 7280*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 3395*sin(d*x + c)^9/(cos(d*
x + c) + 1)^9 - 1680*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 700*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 105*s
in(d*x + c)^13/(cos(d*x + c) + 1)^13 - 176)/(a^2 + 7*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 21*a^2*sin(d*x
+ c)^4/(cos(d*x + c) + 1)^4 + 35*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 35*a^2*sin(d*x + c)^8/(cos(d*x + c)
+ 1)^8 + 21*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 7*a^2*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + a^2*sin
(d*x + c)^14/(cos(d*x + c) + 1)^14) - 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

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Fricas [A]  time = 1.10794, size = 220, normalized size = 1.63 \begin{align*} -\frac{120 \, \cos \left (d x + c\right )^{7} - 504 \, \cos \left (d x + c\right )^{5} + 560 \, \cos \left (d x + c\right )^{3} + 105 \, d x + 35 \,{\left (8 \, \cos \left (d x + c\right )^{5} - 14 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/840*(120*cos(d*x + c)^7 - 504*cos(d*x + c)^5 + 560*cos(d*x + c)^3 + 105*d*x + 35*(8*cos(d*x + c)^5 - 14*cos
(d*x + c)^3 + 3*cos(d*x + c))*sin(d*x + c))/(a^2*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)**6*sin(d*x+c)**3/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.32163, size = 242, normalized size = 1.79 \begin{align*} -\frac{\frac{105 \,{\left (d x + c\right )}}{a^{2}} + \frac{2 \,{\left (105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 700 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 1680 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 3395 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 7280 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 1120 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3395 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2016 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 700 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1232 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 176\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/840*(105*(d*x + c)/a^2 + 2*(105*tan(1/2*d*x + 1/2*c)^13 + 700*tan(1/2*d*x + 1/2*c)^11 + 1680*tan(1/2*d*x +
1/2*c)^10 - 3395*tan(1/2*d*x + 1/2*c)^9 + 7280*tan(1/2*d*x + 1/2*c)^8 - 1120*tan(1/2*d*x + 1/2*c)^6 + 3395*tan
(1/2*d*x + 1/2*c)^5 + 2016*tan(1/2*d*x + 1/2*c)^4 - 700*tan(1/2*d*x + 1/2*c)^3 + 1232*tan(1/2*d*x + 1/2*c)^2 -
105*tan(1/2*d*x + 1/2*c) + 176)/((tan(1/2*d*x + 1/2*c)^2 + 1)^7*a^2))/d