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

Optimal. Leaf size=115 \[ -\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{24 a d}+\frac {\sin (c+d x) \cos (c+d x)}{16 a d}+\frac {x}{16 a} \]

[Out]

1/16*x/a+1/5*cos(d*x+c)^5/a/d-1/7*cos(d*x+c)^7/a/d+1/16*cos(d*x+c)*sin(d*x+c)/a/d+1/24*cos(d*x+c)^3*sin(d*x+c)
/a/d-1/6*cos(d*x+c)^5*sin(d*x+c)/a/d

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Rubi [A]  time = 0.18, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2839, 2568, 2635, 8, 2565, 14} \[ -\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{24 a d}+\frac {\sin (c+d x) \cos (c+d x)}{16 a d}+\frac {x}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(16*a) + Cos[c + d*x]^5/(5*a*d) - Cos[c + d*x]^7/(7*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (Cos[c + d
*x]^3*Sin[c + d*x])/(24*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(6*a*d)

Rule 8

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

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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{a}-\frac {\int \cos ^4(c+d x) \sin ^3(c+d x) \, dx}{a}\\ &=-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}+\frac {\int \cos ^4(c+d x) \, dx}{6 a}+\frac {\operatorname {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}+\frac {\int \cos ^2(c+d x) \, dx}{8 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 (c+d x) \sin (c+d x)}{16 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}+\frac {\int 1 \, dx}{16 a}\\ &=\frac {x}{16 a}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos (c+d x) \sin (c+d x)}{16 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}\\ \end {align*}

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Mathematica [B]  time = 11.39, size = 715, normalized size = 6.22 \[ \frac {5 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{64 d (a \sin (c+d x)+a)}-\frac {-\frac {50 \sin (c) \sin (d x)}{d}+\frac {10 \sin (3 c) \sin (3 d x)}{d}-\frac {2 \sin (5 c) \sin (5 d x)}{d}+\frac {50 \cos (c) \cos (d x)}{d}-\frac {10 \cos (3 c) \cos (3 d x)}{d}+\frac {2 \cos (5 c) \cos (5 d x)}{d}-\frac {20 \sin (2 c) \cos (2 d x)}{d}+\frac {5 \sin (4 c) \cos (4 d x)}{d}-\frac {20 \cos (2 c) \sin (2 d x)}{d}+\frac {5 \cos (4 c) \sin (4 d x)}{d}-\frac {10 \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+30 x}{160 a}+\frac {-\frac {\sin (c) \sin (d x)}{d}+\frac {\cos (c) \cos (d x)}{d}-\frac {\sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+x}{16 a}-\frac {\frac {9 \sin (c) \sin (d x)}{d}-\frac {\sin (3 c) \sin (3 d x)}{d}-\frac {9 \cos (c) \cos (d x)}{d}+\frac {\cos (3 c) \cos (3 d x)}{d}+\frac {3 \sin (2 c) \cos (2 d x)}{d}+\frac {3 \cos (2 c) \sin (2 d x)}{d}+\frac {3 \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}-6 x}{48 a}-\frac {\frac {735 \sin (c) \sin (d x)}{d}-\frac {175 \sin (3 c) \sin (3 d x)}{d}+\frac {63 \sin (5 c) \sin (5 d x)}{d}-\frac {15 \sin (7 c) \sin (7 d x)}{d}-\frac {735 \cos (c) \cos (d x)}{d}+\frac {175 \cos (3 c) \cos (3 d x)}{d}-\frac {63 \cos (5 c) \cos (5 d x)}{d}+\frac {15 \cos (7 c) \cos (7 d x)}{d}+\frac {315 \sin (2 c) \cos (2 d x)}{d}-\frac {105 \sin (4 c) \cos (4 d x)}{d}+\frac {35 \sin (6 c) \cos (6 d x)}{d}+\frac {315 \cos (2 c) \sin (2 d x)}{d}-\frac {105 \cos (4 c) \sin (4 d x)}{d}+\frac {35 \cos (6 c) \sin (6 d x)}{d}+\frac {105 \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}-420 x}{6720 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/160*(30*x + (50*Cos[c]*Cos[d*x])/d - (10*Cos[3*c]*Cos[3*d*x])/d + (2*Cos[5*c]*Cos[5*d*x])/d - (20*Cos[2*d*x
]*Sin[2*c])/d + (5*Cos[4*d*x]*Sin[4*c])/d - (50*Sin[c]*Sin[d*x])/d - (20*Cos[2*c]*Sin[2*d*x])/d + (10*Sin[3*c]
*Sin[3*d*x])/d + (5*Cos[4*c]*Sin[4*d*x])/d - (2*Sin[5*c]*Sin[5*d*x])/d - (10*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[
c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/a + (x + (Cos[c]*Cos[d*x])/d - (Sin[c]*Sin[d*x])/d - Sin[(d*x)/2
]/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(16*a) - (-6*x - (9*Cos[c]*Cos[d*x])/d + (C
os[3*c]*Cos[3*d*x])/d + (3*Cos[2*d*x]*Sin[2*c])/d + (9*Sin[c]*Sin[d*x])/d + (3*Cos[2*c]*Sin[2*d*x])/d - (Sin[3
*c]*Sin[3*d*x])/d + (3*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(48*a) -
(-420*x - (735*Cos[c]*Cos[d*x])/d + (175*Cos[3*c]*Cos[3*d*x])/d - (63*Cos[5*c]*Cos[5*d*x])/d + (15*Cos[7*c]*C
os[7*d*x])/d + (315*Cos[2*d*x]*Sin[2*c])/d - (105*Cos[4*d*x]*Sin[4*c])/d + (35*Cos[6*d*x]*Sin[6*c])/d + (735*S
in[c]*Sin[d*x])/d + (315*Cos[2*c]*Sin[2*d*x])/d - (175*Sin[3*c]*Sin[3*d*x])/d - (105*Cos[4*c]*Sin[4*d*x])/d +
(63*Sin[5*c]*Sin[5*d*x])/d + (35*Cos[6*c]*Sin[6*d*x])/d - (15*Sin[7*c]*Sin[7*d*x])/d + (105*Sin[(d*x)/2])/(d*(
Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(6720*a) + (5*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2]
+ Sin[(c + d*x)/2]))/(64*d*(a + a*Sin[c + d*x]))

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fricas [A]  time = 0.72, size = 70, normalized size = 0.61 \[ -\frac {240 \, \cos \left (d x + c\right )^{7} - 336 \, \cos \left (d x + c\right )^{5} - 105 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1680*(240*cos(d*x + c)^7 - 336*cos(d*x + c)^5 - 105*d*x + 35*(8*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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giac [A]  time = 0.16, size = 179, normalized size = 1.56 \[ \frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 1540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1085 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1085 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1344 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a}}{1680 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(105*(d*x + c)/a + 2*(105*tan(1/2*d*x + 1/2*c)^13 - 1540*tan(1/2*d*x + 1/2*c)^11 + 3360*tan(1/2*d*x + 1
/2*c)^10 + 1085*tan(1/2*d*x + 1/2*c)^9 - 3360*tan(1/2*d*x + 1/2*c)^8 + 6720*tan(1/2*d*x + 1/2*c)^6 - 1085*tan(
1/2*d*x + 1/2*c)^5 - 1344*tan(1/2*d*x + 1/2*c)^4 + 1540*tan(1/2*d*x + 1/2*c)^3 + 672*tan(1/2*d*x + 1/2*c)^2 -
105*tan(1/2*d*x + 1/2*c) + 96)/((tan(1/2*d*x + 1/2*c)^2 + 1)^7*a))/d

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maple [B]  time = 0.27, size = 415, normalized size = 3.61 \[ \frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {4 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {31 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {4 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {8 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {31 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {4}{35 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13-11/6/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c
)^11+4/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^10+31/24/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1
/2*c)^9-4/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^8+8/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2
*c)^6-31/24/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^5-8/5/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x
+1/2*c)^4+11/6/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^3+4/5/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*
d*x+1/2*c)^2-1/8/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)+4/35/a/d/(1+tan(1/2*d*x+1/2*c)^2)^7+1/8/a/d
*arctan(tan(1/2*d*x+1/2*c))

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maxima [B]  time = 0.56, size = 400, normalized size = 3.48 \[ -\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {672 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1344 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1085 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {6720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {3360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {1085 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {3360 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {1540 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 96}{a + \frac {7 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {21 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {35 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {21 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {7 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/840*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 672*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1540*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 + 1344*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1085*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 67
20*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 3360*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 1085*sin(d*x + c)^9/(cos(d
*x + c) + 1)^9 - 3360*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 1540*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 105
*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 96)/(a + 7*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 21*a*sin(d*x + c)^
4/(cos(d*x + c) + 1)^4 + 35*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 35*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 +
21*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 7*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + a*sin(d*x + c)^14/(c
os(d*x + c) + 1)^14) - 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

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mupad [B]  time = 12.75, size = 172, normalized size = 1.50 \[ \frac {x}{16\,a}+\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{6}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{24}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {4}{35}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(16*a) + ((4*tan(c/2 + (d*x)/2)^2)/5 - tan(c/2 + (d*x)/2)/8 + (11*tan(c/2 + (d*x)/2)^3)/6 - (8*tan(c/2 + (d*
x)/2)^4)/5 - (31*tan(c/2 + (d*x)/2)^5)/24 + 8*tan(c/2 + (d*x)/2)^6 - 4*tan(c/2 + (d*x)/2)^8 + (31*tan(c/2 + (d
*x)/2)^9)/24 + 4*tan(c/2 + (d*x)/2)^10 - (11*tan(c/2 + (d*x)/2)^11)/6 + tan(c/2 + (d*x)/2)^13/8 + 4/35)/(a*d*(
tan(c/2 + (d*x)/2)^2 + 1)^7)

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sympy [A]  time = 78.79, size = 2773, normalized size = 24.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**2/(a+a*sin(d*x+c)),x)

[Out]

Piecewise((105*d*x*tan(c/2 + d*x/2)**14/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 3528
0*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2
+ d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 735*d*x*tan(c/2 + d*x/2)**12/(1680*a*d*tan(c/2 + d*
x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 5
8800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 220
5*d*x*tan(c/2 + d*x/2)**10/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2
+ d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4
+ 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 3675*d*x*tan(c/2 + d*x/2)**8/(1680*a*d*tan(c/2 + d*x/2)**14 + 11
760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(
c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 3675*d*x*tan(c/2
+ d*x/2)**6/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10
+ 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*ta
n(c/2 + d*x/2)**2 + 1680*a*d) + 2205*d*x*tan(c/2 + d*x/2)**4/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/
2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**
6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 735*d*x*tan(c/2 + d*x/2)**2/(1
680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan
(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)*
*2 + 1680*a*d) + 105*d*x/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 +
d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 +
11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 210*tan(c/2 + d*x/2)**13/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*
d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 +
d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 3080*tan(c/2 + d*x/2)*
*11/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a
*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d
*x/2)**2 + 1680*a*d) + 6720*tan(c/2 + d*x/2)**10/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**
12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*
d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 2170*tan(c/2 + d*x/2)**9/(1680*a*d*tan(c/2
+ d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**
8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d)
- 6720*tan(c/2 + d*x/2)**8/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2
+ d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4
+ 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 13440*tan(c/2 + d*x/2)**6/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760
*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2
+ d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 2170*tan(c/2 + d*x/
2)**5/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800
*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 +
d*x/2)**2 + 1680*a*d) - 2688*tan(c/2 + d*x/2)**4/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)*
*12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a
*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 3080*tan(c/2 + d*x/2)**3/(1680*a*d*tan(c/
2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)*
*8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d)
+ 1344*tan(c/2 + d*x/2)**2/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/
2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4
+ 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 210*tan(c/2 + d*x/2)/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d
*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d
*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 192/(1680*a*d*tan(c/2 +
d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8
+ 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d), N
e(d, 0)), (x*sin(c)**2*cos(c)**6/(a*sin(c) + a), True))

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