3.709 \(\int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=121 \[ -\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{48 a d}-\frac {5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}-\frac {5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac {5 x}{128 a} \]

[Out]

-5/128*x/a-1/7*cos(d*x+c)^7/a/d-5/128*cos(d*x+c)*sin(d*x+c)/a/d-5/192*cos(d*x+c)^3*sin(d*x+c)/a/d-1/48*cos(d*x
+c)^5*sin(d*x+c)/a/d+1/8*cos(d*x+c)^7*sin(d*x+c)/a/d

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Rubi [A]  time = 0.15, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2839, 2565, 30, 2568, 2635, 8} \[ -\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{48 a d}-\frac {5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}-\frac {5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac {5 x}{128 a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

(-5*x)/(128*a) - Cos[c + d*x]^7/(7*a*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) - (5*Cos[c + d*x]^3*Sin[c +
d*x])/(192*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(48*a*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(8*a*d)

Rule 8

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos ^6(c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{a}\\ &=\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac {\int \cos ^6(c+d x) \, dx}{8 a}-\frac {\operatorname {Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\cos ^7(c+d x)}{7 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac {5 \int \cos ^4(c+d x) \, dx}{48 a}\\ &=-\frac {\cos ^7(c+d x)}{7 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac {5 \int \cos ^2(c+d x) \, dx}{64 a}\\ &=-\frac {\cos ^7(c+d x)}{7 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac {5 \int 1 \, dx}{128 a}\\ &=-\frac {5 x}{128 a}-\frac {\cos ^7(c+d x)}{7 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}\\ \end {align*}

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Mathematica [B]  time = 10.35, size = 481, normalized size = 3.98 \[ -\frac {1680 d x \sin \left (\frac {c}{2}\right )-1680 \sin \left (\frac {c}{2}+d x\right )+1680 \sin \left (\frac {3 c}{2}+d x\right )+336 \sin \left (\frac {3 c}{2}+2 d x\right )+336 \sin \left (\frac {5 c}{2}+2 d x\right )-1008 \sin \left (\frac {5 c}{2}+3 d x\right )+1008 \sin \left (\frac {7 c}{2}+3 d x\right )-168 \sin \left (\frac {7 c}{2}+4 d x\right )-168 \sin \left (\frac {9 c}{2}+4 d x\right )-336 \sin \left (\frac {9 c}{2}+5 d x\right )+336 \sin \left (\frac {11 c}{2}+5 d x\right )-112 \sin \left (\frac {11 c}{2}+6 d x\right )-112 \sin \left (\frac {13 c}{2}+6 d x\right )-48 \sin \left (\frac {13 c}{2}+7 d x\right )+48 \sin \left (\frac {15 c}{2}+7 d x\right )-21 \sin \left (\frac {15 c}{2}+8 d x\right )-21 \sin \left (\frac {17 c}{2}+8 d x\right )-336 \cos \left (\frac {c}{2}\right ) (7 c-5 d x)+1680 \cos \left (\frac {c}{2}+d x\right )+1680 \cos \left (\frac {3 c}{2}+d x\right )+336 \cos \left (\frac {3 c}{2}+2 d x\right )-336 \cos \left (\frac {5 c}{2}+2 d x\right )+1008 \cos \left (\frac {5 c}{2}+3 d x\right )+1008 \cos \left (\frac {7 c}{2}+3 d x\right )-168 \cos \left (\frac {7 c}{2}+4 d x\right )+168 \cos \left (\frac {9 c}{2}+4 d x\right )+336 \cos \left (\frac {9 c}{2}+5 d x\right )+336 \cos \left (\frac {11 c}{2}+5 d x\right )-112 \cos \left (\frac {11 c}{2}+6 d x\right )+112 \cos \left (\frac {13 c}{2}+6 d x\right )+48 \cos \left (\frac {13 c}{2}+7 d x\right )+48 \cos \left (\frac {15 c}{2}+7 d x\right )-21 \cos \left (\frac {15 c}{2}+8 d x\right )+21 \cos \left (\frac {17 c}{2}+8 d x\right )-2352 c \sin \left (\frac {c}{2}\right )+4704 \sin \left (\frac {c}{2}\right )}{43008 a d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

-1/43008*(-336*(7*c - 5*d*x)*Cos[c/2] + 1680*Cos[c/2 + d*x] + 1680*Cos[(3*c)/2 + d*x] + 336*Cos[(3*c)/2 + 2*d*
x] - 336*Cos[(5*c)/2 + 2*d*x] + 1008*Cos[(5*c)/2 + 3*d*x] + 1008*Cos[(7*c)/2 + 3*d*x] - 168*Cos[(7*c)/2 + 4*d*
x] + 168*Cos[(9*c)/2 + 4*d*x] + 336*Cos[(9*c)/2 + 5*d*x] + 336*Cos[(11*c)/2 + 5*d*x] - 112*Cos[(11*c)/2 + 6*d*
x] + 112*Cos[(13*c)/2 + 6*d*x] + 48*Cos[(13*c)/2 + 7*d*x] + 48*Cos[(15*c)/2 + 7*d*x] - 21*Cos[(15*c)/2 + 8*d*x
] + 21*Cos[(17*c)/2 + 8*d*x] + 4704*Sin[c/2] - 2352*c*Sin[c/2] + 1680*d*x*Sin[c/2] - 1680*Sin[c/2 + d*x] + 168
0*Sin[(3*c)/2 + d*x] + 336*Sin[(3*c)/2 + 2*d*x] + 336*Sin[(5*c)/2 + 2*d*x] - 1008*Sin[(5*c)/2 + 3*d*x] + 1008*
Sin[(7*c)/2 + 3*d*x] - 168*Sin[(7*c)/2 + 4*d*x] - 168*Sin[(9*c)/2 + 4*d*x] - 336*Sin[(9*c)/2 + 5*d*x] + 336*Si
n[(11*c)/2 + 5*d*x] - 112*Sin[(11*c)/2 + 6*d*x] - 112*Sin[(13*c)/2 + 6*d*x] - 48*Sin[(13*c)/2 + 7*d*x] + 48*Si
n[(15*c)/2 + 7*d*x] - 21*Sin[(15*c)/2 + 8*d*x] - 21*Sin[(17*c)/2 + 8*d*x])/(a*d*(Cos[c/2] + Sin[c/2]))

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fricas [A]  time = 0.47, size = 70, normalized size = 0.58 \[ -\frac {384 \, \cos \left (d x + c\right )^{7} + 105 \, d x - 7 \, {\left (48 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2688*(384*cos(d*x + c)^7 + 105*d*x - 7*(48*cos(d*x + c)^7 - 8*cos(d*x + c)^5 - 10*cos(d*x + c)^3 - 15*cos(d
*x + c))*sin(d*x + c))/(a*d)

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giac [B]  time = 0.16, size = 231, normalized size = 1.91 \[ -\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 2688 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 2779 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2688 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 6265 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 13440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 12355 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 12355 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 6265 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2779 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 384\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a}}{2688 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/2688*(105*(d*x + c)/a + 2*(105*tan(1/2*d*x + 1/2*c)^15 + 2688*tan(1/2*d*x + 1/2*c)^14 - 2779*tan(1/2*d*x +
1/2*c)^13 + 2688*tan(1/2*d*x + 1/2*c)^12 + 6265*tan(1/2*d*x + 1/2*c)^11 + 13440*tan(1/2*d*x + 1/2*c)^10 - 1235
5*tan(1/2*d*x + 1/2*c)^9 + 13440*tan(1/2*d*x + 1/2*c)^8 + 12355*tan(1/2*d*x + 1/2*c)^7 + 8064*tan(1/2*d*x + 1/
2*c)^6 - 6265*tan(1/2*d*x + 1/2*c)^5 + 8064*tan(1/2*d*x + 1/2*c)^4 + 2779*tan(1/2*d*x + 1/2*c)^3 + 384*tan(1/2
*d*x + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 384)/((tan(1/2*d*x + 1/2*c)^2 + 1)^8*a))/d

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maple [B]  time = 0.21, size = 551, normalized size = 4.55 \[ -\frac {2}{7 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {397 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {6 \left (\tan ^{4}\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 )^{8}}+\frac {895 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {6 \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 )^{8}}-\frac {1765 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {10 \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 )^{8}}+\frac {1765 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {10 \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 )^{8}}-\frac {895 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {2 \left (\tan ^{12}\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 )^{8}}+\frac {397 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {2 \left (\tan ^{14}\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 )^{8}}-\frac {5 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

-2/7/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8+5/64/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)-2/7/a/d/(1+tan(1/2*
d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^2-397/192/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^3-6/a/d/(1+tan(
1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^4+895/192/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^5-6/a/d/(1+
tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^6-1765/192/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^7-10/a
/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^8+1765/192/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^
9-10/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^10-895/192/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1
/2*c)^11-2/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^12+397/192/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2
*d*x+1/2*c)^13-2/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^14-5/64/a/d/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(
1/2*d*x+1/2*c)^15-5/64/a/d*arctan(tan(1/2*d*x+1/2*c))

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maxima [B]  time = 0.43, size = 501, normalized size = 4.14 \[ \frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {384 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2779 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {8064 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6265 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {8064 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {12355 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {13440 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {12355 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {13440 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6265 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {2688 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {2779 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {2688 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {105 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - 384}{a + \frac {8 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {56 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {56 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {8 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{1344 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/1344*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 384*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2779*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 - 8064*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 6265*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 80
64*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 12355*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 13440*sin(d*x + c)^8/(cos
(d*x + c) + 1)^8 + 12355*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 13440*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 6
265*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 2688*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 2779*sin(d*x + c)^13/
(cos(d*x + c) + 1)^13 - 2688*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 105*sin(d*x + c)^15/(cos(d*x + c) + 1)^15
- 384)/(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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mupad [B]  time = 11.68, size = 225, normalized size = 1.86 \[ -\frac {5\,x}{128\,a}-\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {397\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\frac {895\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {1765\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {1765\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{192}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {895\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {397\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {2}{7}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^8*sin(c + d*x))/(a + a*sin(c + d*x)),x)

[Out]

- (5*x)/(128*a) - ((2*tan(c/2 + (d*x)/2)^2)/7 - (5*tan(c/2 + (d*x)/2))/64 + (397*tan(c/2 + (d*x)/2)^3)/192 + 6
*tan(c/2 + (d*x)/2)^4 - (895*tan(c/2 + (d*x)/2)^5)/192 + 6*tan(c/2 + (d*x)/2)^6 + (1765*tan(c/2 + (d*x)/2)^7)/
192 + 10*tan(c/2 + (d*x)/2)^8 - (1765*tan(c/2 + (d*x)/2)^9)/192 + 10*tan(c/2 + (d*x)/2)^10 + (895*tan(c/2 + (d
*x)/2)^11)/192 + 2*tan(c/2 + (d*x)/2)^12 - (397*tan(c/2 + (d*x)/2)^13)/192 + 2*tan(c/2 + (d*x)/2)^14 + (5*tan(
c/2 + (d*x)/2)^15)/64 + 2/7)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^8)

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sympy [A]  time = 111.64, size = 3888, normalized size = 32.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**8*sin(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

Piecewise((-105*d*x*tan(c/2 + d*x/2)**16/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 752
64*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*ta
n(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 840*d*x*tan(c/
2 + d*x/2)**14/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**1
2 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*
a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 2940*d*x*tan(c/2 + d*x/2)**12/(2688*a*d*
tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 +
d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 +
21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 5880*d*x*tan(c/2 + d*x/2)**10/(2688*a*d*tan(c/2 + d*x/2)**16 + 21
504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*t
an(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/
2)**2 + 2688*a*d) - 7350*d*x*tan(c/2 + d*x/2)**8/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**
14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 15052
8*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 5880*d
*x*tan(c/2 + d*x/2)**6/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d
*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6
+ 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 2940*d*x*tan(c/2 + d*x/2)**4/(26
88*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan
(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/
2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 840*d*x*tan(c/2 + d*x/2)**2/(2688*a*d*tan(c/2 + d*x/2)**16
+ 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*
a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 +
d*x/2)**2 + 2688*a*d) - 105*d*x/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*t
an(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 +
d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 210*tan(c/2 + d*x/2)**
15/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a
*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2
+ d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 5376*tan(c/2 + d*x/2)**14/(2688*a*d*tan(c/2 + d*x/2)
**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188
160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c
/2 + d*x/2)**2 + 2688*a*d) + 5558*tan(c/2 + d*x/2)**13/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*
x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 +
150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) -
5376*tan(c/2 + d*x/2)**12/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2
+ d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)*
*6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 12530*tan(c/2 + d*x/2)**11/(2
688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*ta
n(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x
/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 26880*tan(c/2 + d*x/2)**10/(2688*a*d*tan(c/2 + d*x/2)**16
+ 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*
a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 +
d*x/2)**2 + 2688*a*d) + 24710*tan(c/2 + d*x/2)**9/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)
**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150
528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 2688
0*tan(c/2 + d*x/2)**8/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*
x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 +
75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 24710*tan(c/2 + d*x/2)**7/(2688*a
*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2
+ d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**
4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 16128*tan(c/2 + d*x/2)**6/(2688*a*d*tan(c/2 + d*x/2)**16 + 215
04*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*ta
n(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2
)**2 + 2688*a*d) + 12530*tan(c/2 + d*x/2)**5/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 +
75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*
d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 16128*tan(
c/2 + d*x/2)**4/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**
12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264
*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 5558*tan(c/2 + d*x/2)**3/(2688*a*d*tan(
c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/
2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 215
04*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 768*tan(c/2 + d*x/2)**2/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*ta
n(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d
*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 26
88*a*d) + 210*tan(c/2 + d*x/2)/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan
(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*
x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 768/(2688*a*d*tan(c/2 +
d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10
+ 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d
*tan(c/2 + d*x/2)**2 + 2688*a*d), Ne(d, 0)), (x*sin(c)*cos(c)**8/(a*sin(c) + a), True))

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