3.707 \(\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=165 \[ \frac {\cos ^9(c+d x)}{9 a d}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}+\frac {3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{160 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{128 a d}-\frac {3 \sin (c+d x) \cos (c+d x)}{256 a d}-\frac {3 x}{256 a} \]

[Out]

-3/256*x/a-1/7*cos(d*x+c)^7/a/d+1/9*cos(d*x+c)^9/a/d-3/256*cos(d*x+c)*sin(d*x+c)/a/d-1/128*cos(d*x+c)^3*sin(d*
x+c)/a/d-1/160*cos(d*x+c)^5*sin(d*x+c)/a/d+3/80*cos(d*x+c)^7*sin(d*x+c)/a/d+1/10*cos(d*x+c)^7*sin(d*x+c)^3/a/d

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Rubi [A]  time = 0.24, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {\cos ^9(c+d x)}{9 a d}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}+\frac {3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{160 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{128 a d}-\frac {3 \sin (c+d x) \cos (c+d x)}{256 a d}-\frac {3 x}{256 a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

(-3*x)/(256*a) - Cos[c + d*x]^7/(7*a*d) + Cos[c + d*x]^9/(9*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(256*a*d) - (
Cos[c + d*x]^3*Sin[c + d*x])/(128*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(160*a*d) + (3*Cos[c + d*x]^7*Sin[c + d
*x])/(80*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*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 ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos ^6(c+d x) \sin ^3(c+d x) \, dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^4(c+d x) \, dx}{a}\\ &=\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac {3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{10 a}-\frac {\operatorname {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac {3 \int \cos ^6(c+d x) \, dx}{80 a}-\frac {\operatorname {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^9(c+d x)}{9 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac {\int \cos ^4(c+d x) \, dx}{32 a}\\ &=-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^9(c+d x)}{9 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac {3 \int \cos ^2(c+d x) \, dx}{128 a}\\ &=-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^9(c+d x)}{9 a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{256 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac {3 \int 1 \, dx}{256 a}\\ &=-\frac {3 x}{256 a}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^9(c+d x)}{9 a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{256 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}\\ \end {align*}

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Mathematica [B]  time = 14.16, size = 533, normalized size = 3.23 \[ -\frac {15120 d x \sin \left (\frac {c}{2}\right )-15120 \sin \left (\frac {c}{2}+d x\right )+15120 \sin \left (\frac {3 c}{2}+d x\right )+1260 \sin \left (\frac {3 c}{2}+2 d x\right )+1260 \sin \left (\frac {5 c}{2}+2 d x\right )-6720 \sin \left (\frac {5 c}{2}+3 d x\right )+6720 \sin \left (\frac {7 c}{2}+3 d x\right )-2520 \sin \left (\frac {7 c}{2}+4 d x\right )-2520 \sin \left (\frac {9 c}{2}+4 d x\right )-630 \sin \left (\frac {11 c}{2}+6 d x\right )-630 \sin \left (\frac {13 c}{2}+6 d x\right )+1080 \sin \left (\frac {13 c}{2}+7 d x\right )-1080 \sin \left (\frac {15 c}{2}+7 d x\right )+315 \sin \left (\frac {15 c}{2}+8 d x\right )+315 \sin \left (\frac {17 c}{2}+8 d x\right )+280 \sin \left (\frac {17 c}{2}+9 d x\right )-280 \sin \left (\frac {19 c}{2}+9 d x\right )+126 \sin \left (\frac {19 c}{2}+10 d x\right )+126 \sin \left (\frac {21 c}{2}+10 d x\right )-1260 \cos \left (\frac {c}{2}\right ) (25 c-12 d x)+15120 \cos \left (\frac {c}{2}+d x\right )+15120 \cos \left (\frac {3 c}{2}+d x\right )+1260 \cos \left (\frac {3 c}{2}+2 d x\right )-1260 \cos \left (\frac {5 c}{2}+2 d x\right )+6720 \cos \left (\frac {5 c}{2}+3 d x\right )+6720 \cos \left (\frac {7 c}{2}+3 d x\right )-2520 \cos \left (\frac {7 c}{2}+4 d x\right )+2520 \cos \left (\frac {9 c}{2}+4 d x\right )-630 \cos \left (\frac {11 c}{2}+6 d x\right )+630 \cos \left (\frac {13 c}{2}+6 d x\right )-1080 \cos \left (\frac {13 c}{2}+7 d x\right )-1080 \cos \left (\frac {15 c}{2}+7 d x\right )+315 \cos \left (\frac {15 c}{2}+8 d x\right )-315 \cos \left (\frac {17 c}{2}+8 d x\right )-280 \cos \left (\frac {17 c}{2}+9 d x\right )-280 \cos \left (\frac {19 c}{2}+9 d x\right )+126 \cos \left (\frac {19 c}{2}+10 d x\right )-126 \cos \left (\frac {21 c}{2}+10 d x\right )-31500 c \sin \left (\frac {c}{2}\right )+37800 \sin \left (\frac {c}{2}\right )}{1290240 a d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

-1/1290240*(-1260*(25*c - 12*d*x)*Cos[c/2] + 15120*Cos[c/2 + d*x] + 15120*Cos[(3*c)/2 + d*x] + 1260*Cos[(3*c)/
2 + 2*d*x] - 1260*Cos[(5*c)/2 + 2*d*x] + 6720*Cos[(5*c)/2 + 3*d*x] + 6720*Cos[(7*c)/2 + 3*d*x] - 2520*Cos[(7*c
)/2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 630*Cos[(11*c)/2 + 6*d*x] + 630*Cos[(13*c)/2 + 6*d*x] - 1080*Cos[(1
3*c)/2 + 7*d*x] - 1080*Cos[(15*c)/2 + 7*d*x] + 315*Cos[(15*c)/2 + 8*d*x] - 315*Cos[(17*c)/2 + 8*d*x] - 280*Cos
[(17*c)/2 + 9*d*x] - 280*Cos[(19*c)/2 + 9*d*x] + 126*Cos[(19*c)/2 + 10*d*x] - 126*Cos[(21*c)/2 + 10*d*x] + 378
00*Sin[c/2] - 31500*c*Sin[c/2] + 15120*d*x*Sin[c/2] - 15120*Sin[c/2 + d*x] + 15120*Sin[(3*c)/2 + d*x] + 1260*S
in[(3*c)/2 + 2*d*x] + 1260*Sin[(5*c)/2 + 2*d*x] - 6720*Sin[(5*c)/2 + 3*d*x] + 6720*Sin[(7*c)/2 + 3*d*x] - 2520
*Sin[(7*c)/2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4*d*x] - 630*Sin[(11*c)/2 + 6*d*x] - 630*Sin[(13*c)/2 + 6*d*x] + 10
80*Sin[(13*c)/2 + 7*d*x] - 1080*Sin[(15*c)/2 + 7*d*x] + 315*Sin[(15*c)/2 + 8*d*x] + 315*Sin[(17*c)/2 + 8*d*x]
+ 280*Sin[(17*c)/2 + 9*d*x] - 280*Sin[(19*c)/2 + 9*d*x] + 126*Sin[(19*c)/2 + 10*d*x] + 126*Sin[(21*c)/2 + 10*d
*x])/(a*d*(Cos[c/2] + Sin[c/2]))

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fricas [A]  time = 0.47, size = 90, normalized size = 0.55 \[ \frac {8960 \, \cos \left (d x + c\right )^{9} - 11520 \, \cos \left (d x + c\right )^{7} - 945 \, d x - 63 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 176 \, \cos \left (d x + c\right )^{7} + 8 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/80640*(8960*cos(d*x + c)^9 - 11520*cos(d*x + c)^7 - 945*d*x - 63*(128*cos(d*x + c)^9 - 176*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 [A]  time = 0.21, size = 257, normalized size = 1.56 \[ -\frac {\frac {945 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 9135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 161280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} - 218484 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 107520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 653940 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 537600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 1183770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 322560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1183770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 653940 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 414720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 218484 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 46080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 9135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 25600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2560\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{10} a}}{80640 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/80640*(945*(d*x + c)/a + 2*(945*tan(1/2*d*x + 1/2*c)^19 + 9135*tan(1/2*d*x + 1/2*c)^17 + 161280*tan(1/2*d*x
+ 1/2*c)^16 - 218484*tan(1/2*d*x + 1/2*c)^15 - 107520*tan(1/2*d*x + 1/2*c)^14 + 653940*tan(1/2*d*x + 1/2*c)^1
3 + 537600*tan(1/2*d*x + 1/2*c)^12 - 1183770*tan(1/2*d*x + 1/2*c)^11 + 322560*tan(1/2*d*x + 1/2*c)^10 + 118377
0*tan(1/2*d*x + 1/2*c)^9 - 653940*tan(1/2*d*x + 1/2*c)^7 + 414720*tan(1/2*d*x + 1/2*c)^6 + 218484*tan(1/2*d*x
+ 1/2*c)^5 - 46080*tan(1/2*d*x + 1/2*c)^4 - 9135*tan(1/2*d*x + 1/2*c)^3 + 25600*tan(1/2*d*x + 1/2*c)^2 - 945*t
an(1/2*d*x + 1/2*c) + 2560)/((tan(1/2*d*x + 1/2*c)^2 + 1)^10*a))/d

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maple [B]  time = 0.31, size = 619, normalized size = 3.75 \[ -\frac {4}{63 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {40 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {29 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {8 \left (\tan ^{4}\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 )^{10}}-\frac {867 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {72 \left (\tan ^{6}\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 )^{10}}+\frac {519 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {1879 \left (\tan ^{9}\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 )^{10}}-\frac {8 \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 )^{10}}+\frac {1879 \left (\tan ^{11}\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 )^{10}}-\frac {40 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {519 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {8 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {867 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {4 \left (\tan ^{16}\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 )^{10}}-\frac {29 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {3 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c)),x)

[Out]

-4/63/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10+3/128/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)-40/63/a/d/(1+ta
n(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^2+29/128/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^3+8/7/a/
d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^4-867/160/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^
5-72/7/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^6+519/32/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x
+1/2*c)^7-1879/64/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^9-8/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1
/2*d*x+1/2*c)^10+1879/64/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^11-40/3/a/d/(1+tan(1/2*d*x+1/2*c)^
2)^10*tan(1/2*d*x+1/2*c)^12-519/32/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^13+8/3/a/d/(1+tan(1/2*d*
x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^14+867/160/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^15-4/a/d/(1+ta
n(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^16-29/128/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^17-3/12
8/a/d/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^19-3/128/a/d*arctan(tan(1/2*d*x+1/2*c))

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maxima [B]  time = 0.44, size = 583, normalized size = 3.53 \[ \frac {\frac {\frac {945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25600 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {9135 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {46080 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {218484 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {414720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {653940 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {1183770 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {322560 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {1183770 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {537600 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {653940 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} + \frac {107520 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {218484 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {161280 \, \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} - \frac {9135 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - \frac {945 \, \sin \left (d x + c\right )^{19}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{19}} - 2560}{a + \frac {10 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {120 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {210 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {252 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {210 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {120 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {45 \, a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {10 \, a \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}} + \frac {a \sin \left (d x + c\right )^{20}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{20}}} - \frac {945 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{40320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/40320*((945*sin(d*x + c)/(cos(d*x + c) + 1) - 25600*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 9135*sin(d*x + c)^
3/(cos(d*x + c) + 1)^3 + 46080*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 218484*sin(d*x + c)^5/(cos(d*x + c) + 1)^
5 - 414720*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 653940*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 1183770*sin(d*x
+ c)^9/(cos(d*x + c) + 1)^9 - 322560*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 1183770*sin(d*x + c)^11/(cos(d*x
+ c) + 1)^11 - 537600*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 653940*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 + 1
07520*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 218484*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 161280*sin(d*x +
c)^16/(cos(d*x + c) + 1)^16 - 9135*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 - 945*sin(d*x + c)^19/(cos(d*x + c) +
1)^19 - 2560)/(a + 10*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 45*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 120*
a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 210*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 252*a*sin(d*x + c)^10/(cos
(d*x + c) + 1)^10 + 210*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 120*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14
+ 45*a*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 + 10*a*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 + a*sin(d*x + c)^20/
(cos(d*x + c) + 1)^20) - 945*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

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mupad [B]  time = 11.52, size = 251, normalized size = 1.52 \[ -\frac {3\,x}{256\,a}-\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-\frac {867\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}+\frac {519\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-\frac {1879\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {1879\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {519\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {72\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {867\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}-\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}+\frac {40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {4}{63}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^8*sin(c + d*x)^3)/(a + a*sin(c + d*x)),x)

[Out]

- (3*x)/(256*a) - ((40*tan(c/2 + (d*x)/2)^2)/63 - (3*tan(c/2 + (d*x)/2))/128 - (29*tan(c/2 + (d*x)/2)^3)/128 -
(8*tan(c/2 + (d*x)/2)^4)/7 + (867*tan(c/2 + (d*x)/2)^5)/160 + (72*tan(c/2 + (d*x)/2)^6)/7 - (519*tan(c/2 + (d
*x)/2)^7)/32 + (1879*tan(c/2 + (d*x)/2)^9)/64 + 8*tan(c/2 + (d*x)/2)^10 - (1879*tan(c/2 + (d*x)/2)^11)/64 + (4
0*tan(c/2 + (d*x)/2)^12)/3 + (519*tan(c/2 + (d*x)/2)^13)/32 - (8*tan(c/2 + (d*x)/2)^14)/3 - (867*tan(c/2 + (d*
x)/2)^15)/160 + 4*tan(c/2 + (d*x)/2)^16 + (29*tan(c/2 + (d*x)/2)^17)/128 + (3*tan(c/2 + (d*x)/2)^19)/128 + 4/6
3)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^10)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**8*sin(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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