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

Optimal. Leaf size=141 \[ \frac{\cos ^7(c+d x)}{7 a d}-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{3 x}{128 a} \]

[Out]

(-3*x)/(128*a) - Cos[c + d*x]^5/(5*a*d) + Cos[c + d*x]^7/(7*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) - (
Cos[c + d*x]^3*Sin[c + d*x])/(64*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^5*Sin[c + d*x]^
3)/(8*a*d)

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Rubi [A]  time = 0.214673, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 2565, 14, 2568, 2635, 8} \[ \frac{\cos ^7(c+d x)}{7 a d}-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{3 x}{128 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x)/(128*a) - Cos[c + d*x]^5/(5*a*d) + Cos[c + d*x]^7/(7*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) - (
Cos[c + d*x]^3*Sin[c + d*x])/(64*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^5*Sin[c + d*x]^
3)/(8*a*d)

Rule 2839

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

Rubi steps

\begin{align*} \int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \cos ^4(c+d x) \sin ^3(c+d x) \, dx}{a}-\frac{\int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{a}\\ &=\frac{\cos ^5(c+d x) \sin ^3(c+d x)}{8 a d}-\frac{3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{8 a}-\frac{\operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac{\cos ^5(c+d x) \sin (c+d x)}{16 a d}+\frac{\cos ^5(c+d x) \sin ^3(c+d x)}{8 a d}-\frac{\int \cos ^4(c+d x) \, dx}{16 a}-\frac{\operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac{\cos ^5(c+d x) \sin (c+d x)}{16 a d}+\frac{\cos ^5(c+d x) \sin ^3(c+d x)}{8 a d}-\frac{3 \int \cos ^2(c+d x) \, dx}{64 a}\\ &=-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{3 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac{\cos ^5(c+d x) \sin (c+d x)}{16 a d}+\frac{\cos ^5(c+d x) \sin ^3(c+d x)}{8 a d}-\frac{3 \int 1 \, dx}{128 a}\\ &=-\frac{3 x}{128 a}-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{3 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac{\cos ^5(c+d x) \sin (c+d x)}{16 a d}+\frac{\cos ^5(c+d x) \sin ^3(c+d x)}{8 a d}\\ \end{align*}

Mathematica [B]  time = 8.7552, size = 375, normalized size = 2.66 \[ \frac{-1680 d x \sin \left (\frac{c}{2}\right )+1680 \sin \left (\frac{c}{2}+d x\right )-1680 \sin \left (\frac{3 c}{2}+d x\right )+560 \sin \left (\frac{5 c}{2}+3 d x\right )-560 \sin \left (\frac{7 c}{2}+3 d x\right )+280 \sin \left (\frac{7 c}{2}+4 d x\right )+280 \sin \left (\frac{9 c}{2}+4 d x\right )-112 \sin \left (\frac{9 c}{2}+5 d x\right )+112 \sin \left (\frac{11 c}{2}+5 d x\right )-80 \sin \left (\frac{13 c}{2}+7 d x\right )+80 \sin \left (\frac{15 c}{2}+7 d x\right )-35 \sin \left (\frac{15 c}{2}+8 d x\right )-35 \sin \left (\frac{17 c}{2}+8 d x\right )+1680 \cos \left (\frac{c}{2}\right ) (c-d x)-1680 \cos \left (\frac{c}{2}+d x\right )-1680 \cos \left (\frac{3 c}{2}+d x\right )-560 \cos \left (\frac{5 c}{2}+3 d x\right )-560 \cos \left (\frac{7 c}{2}+3 d x\right )+280 \cos \left (\frac{7 c}{2}+4 d x\right )-280 \cos \left (\frac{9 c}{2}+4 d x\right )+112 \cos \left (\frac{9 c}{2}+5 d x\right )+112 \cos \left (\frac{11 c}{2}+5 d x\right )+80 \cos \left (\frac{13 c}{2}+7 d x\right )+80 \cos \left (\frac{15 c}{2}+7 d x\right )-35 \cos \left (\frac{15 c}{2}+8 d x\right )+35 \cos \left (\frac{17 c}{2}+8 d x\right )+1680 c \sin \left (\frac{c}{2}\right )-3360 \sin \left (\frac{c}{2}\right )}{71680 a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1680*(c - d*x)*Cos[c/2] - 1680*Cos[c/2 + d*x] - 1680*Cos[(3*c)/2 + d*x] - 560*Cos[(5*c)/2 + 3*d*x] - 560*Cos[
(7*c)/2 + 3*d*x] + 280*Cos[(7*c)/2 + 4*d*x] - 280*Cos[(9*c)/2 + 4*d*x] + 112*Cos[(9*c)/2 + 5*d*x] + 112*Cos[(1
1*c)/2 + 5*d*x] + 80*Cos[(13*c)/2 + 7*d*x] + 80*Cos[(15*c)/2 + 7*d*x] - 35*Cos[(15*c)/2 + 8*d*x] + 35*Cos[(17*
c)/2 + 8*d*x] - 3360*Sin[c/2] + 1680*c*Sin[c/2] - 1680*d*x*Sin[c/2] + 1680*Sin[c/2 + d*x] - 1680*Sin[(3*c)/2 +
d*x] + 560*Sin[(5*c)/2 + 3*d*x] - 560*Sin[(7*c)/2 + 3*d*x] + 280*Sin[(7*c)/2 + 4*d*x] + 280*Sin[(9*c)/2 + 4*d
*x] - 112*Sin[(9*c)/2 + 5*d*x] + 112*Sin[(11*c)/2 + 5*d*x] - 80*Sin[(13*c)/2 + 7*d*x] + 80*Sin[(15*c)/2 + 7*d*
x] - 35*Sin[(15*c)/2 + 8*d*x] - 35*Sin[(17*c)/2 + 8*d*x])/(71680*a*d*(Cos[c/2] + Sin[c/2]))

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Maple [B]  time = 0.09, size = 483, normalized size = 3.4 \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)),x)

[Out]

-4/35/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8+3/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)-32/35/d/a/(1+tan(1
/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^2+23/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^3+4/5/d/a/(1+t
an(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^4-333/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^5-32/5/d/
a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^6+671/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^7-4
/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^8-671/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^
9+333/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^11-4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2
*c)^12-23/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^13-3/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*
d*x+1/2*c)^15-3/64/a/d*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.58512, size = 622, normalized size = 4.41 \begin{align*} \text{result too large to display} \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)),x, algorithm="maxima")

[Out]

1/2240*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 2048*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 805*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 + 1792*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 11655*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1
4336*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 23485*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 8960*sin(d*x + c)^8/(co
s(d*x + c) + 1)^8 - 23485*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 11655*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 -
8960*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 805*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 105*sin(d*x + c)^15/(
cos(d*x + c) + 1)^15 - 256)/(a + 8*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 28*a*sin(d*x + c)^4/(cos(d*x + c) +
1)^4 + 56*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 70*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 56*a*sin(d*x + c
)^10/(cos(d*x + c) + 1)^10 + 28*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 8*a*sin(d*x + c)^14/(cos(d*x + c) +
1)^14 + a*sin(d*x + c)^16/(cos(d*x + c) + 1)^16) - 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

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Fricas [A]  time = 1.15118, size = 216, normalized size = 1.53 \begin{align*} \frac{640 \, \cos \left (d x + c\right )^{7} - 896 \, \cos \left (d x + c\right )^{5} - 105 \, d x - 35 \,{\left (16 \, \cos \left (d x + c\right )^{7} - 24 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4480 \, a 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)),x, algorithm="fricas")

[Out]

1/4480*(640*cos(d*x + c)^7 - 896*cos(d*x + c)^5 - 105*d*x - 35*(16*cos(d*x + c)^7 - 24*cos(d*x + c)^5 + 2*cos(
d*x + c)^3 + 3*cos(d*x + c))*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)**6*sin(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.29971, size = 277, normalized size = 1.96 \begin{align*} -\frac{\frac{105 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 805 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 8960 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 11655 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 23485 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 8960 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 23485 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 14336 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 11655 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1792 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 805 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2048 \, \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 ) + 256\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{8} a}}{4480 \, 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)),x, algorithm="giac")

[Out]

-1/4480*(105*(d*x + c)/a + 2*(105*tan(1/2*d*x + 1/2*c)^15 + 805*tan(1/2*d*x + 1/2*c)^13 + 8960*tan(1/2*d*x + 1
/2*c)^12 - 11655*tan(1/2*d*x + 1/2*c)^11 + 23485*tan(1/2*d*x + 1/2*c)^9 + 8960*tan(1/2*d*x + 1/2*c)^8 - 23485*
tan(1/2*d*x + 1/2*c)^7 + 14336*tan(1/2*d*x + 1/2*c)^6 + 11655*tan(1/2*d*x + 1/2*c)^5 - 1792*tan(1/2*d*x + 1/2*
c)^4 - 805*tan(1/2*d*x + 1/2*c)^3 + 2048*tan(1/2*d*x + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 256)/((tan(1/2*d*
x + 1/2*c)^2 + 1)^8*a))/d