3.624 \(\int \frac {\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=159 \[ \frac {\cos ^9(c+d x)}{9 a d}-\frac {2 \cos ^7(c+d x)}{7 a d}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{16 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{64 a d}+\frac {3 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac {3 x}{128 a} \]

[Out]

3/128*x/a+1/5*cos(d*x+c)^5/a/d-2/7*cos(d*x+c)^7/a/d+1/9*cos(d*x+c)^9/a/d+3/128*cos(d*x+c)*sin(d*x+c)/a/d+1/64*
cos(d*x+c)^3*sin(d*x+c)/a/d-1/16*cos(d*x+c)^5*sin(d*x+c)/a/d-1/8*cos(d*x+c)^5*sin(d*x+c)^3/a/d

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Rubi [A]  time = 0.22, antiderivative size = 159, normalized size of antiderivative = 1.00, 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, 2568, 2635, 8, 2565, 270} \[ \frac {\cos ^9(c+d x)}{9 a d}-\frac {2 \cos ^7(c+d x)}{7 a d}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{16 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{64 a d}+\frac {3 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac {3 x}{128 a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(3*x)/(128*a) + Cos[c + d*x]^5/(5*a*d) - (2*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])/(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 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]

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 ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{a}-\frac {\int \cos ^4(c+d x) \sin ^5(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 )^2 \, 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^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\cos ^5(c+d x)}{5 a d}-\frac {2 \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)}{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 {2 \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)}{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 {2 \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)}{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*}

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Mathematica [B]  time = 8.41, size = 429, normalized size = 2.70 \[ \frac {15120 d x \sin \left (\frac {c}{2}\right )-7560 \sin \left (\frac {c}{2}+d x\right )+7560 \sin \left (\frac {3 c}{2}+d x\right )-1680 \sin \left (\frac {5 c}{2}+3 d x\right )+1680 \sin \left (\frac {7 c}{2}+3 d x\right )-2520 \sin \left (\frac {7 c}{2}+4 d x\right )-2520 \sin \left (\frac {9 c}{2}+4 d x\right )+1008 \sin \left (\frac {9 c}{2}+5 d x\right )-1008 \sin \left (\frac {11 c}{2}+5 d x\right )+180 \sin \left (\frac {13 c}{2}+7 d x\right )-180 \sin \left (\frac {15 c}{2}+7 d x\right )+315 \sin \left (\frac {15 c}{2}+8 d x\right )+315 \sin \left (\frac {17 c}{2}+8 d x\right )-140 \sin \left (\frac {17 c}{2}+9 d x\right )+140 \sin \left (\frac {19 c}{2}+9 d x\right )+2520 \cos \left (\frac {c}{2}\right ) (5 c+6 d x)+7560 \cos \left (\frac {c}{2}+d x\right )+7560 \cos \left (\frac {3 c}{2}+d x\right )+1680 \cos \left (\frac {5 c}{2}+3 d x\right )+1680 \cos \left (\frac {7 c}{2}+3 d x\right )-2520 \cos \left (\frac {7 c}{2}+4 d x\right )+2520 \cos \left (\frac {9 c}{2}+4 d x\right )-1008 \cos \left (\frac {9 c}{2}+5 d x\right )-1008 \cos \left (\frac {11 c}{2}+5 d x\right )-180 \cos \left (\frac {13 c}{2}+7 d x\right )-180 \cos \left (\frac {15 c}{2}+7 d x\right )+315 \cos \left (\frac {15 c}{2}+8 d x\right )-315 \cos \left (\frac {17 c}{2}+8 d x\right )+140 \cos \left (\frac {17 c}{2}+9 d x\right )+140 \cos \left (\frac {19 c}{2}+9 d x\right )+12600 c \sin \left (\frac {c}{2}\right )+12600 \sin \left (\frac {c}{2}\right )}{645120 a d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(2520*(5*c + 6*d*x)*Cos[c/2] + 7560*Cos[c/2 + d*x] + 7560*Cos[(3*c)/2 + d*x] + 1680*Cos[(5*c)/2 + 3*d*x] + 168
0*Cos[(7*c)/2 + 3*d*x] - 2520*Cos[(7*c)/2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 1008*Cos[(9*c)/2 + 5*d*x] - 1
008*Cos[(11*c)/2 + 5*d*x] - 180*Cos[(13*c)/2 + 7*d*x] - 180*Cos[(15*c)/2 + 7*d*x] + 315*Cos[(15*c)/2 + 8*d*x]
- 315*Cos[(17*c)/2 + 8*d*x] + 140*Cos[(17*c)/2 + 9*d*x] + 140*Cos[(19*c)/2 + 9*d*x] + 12600*Sin[c/2] + 12600*c
*Sin[c/2] + 15120*d*x*Sin[c/2] - 7560*Sin[c/2 + d*x] + 7560*Sin[(3*c)/2 + d*x] - 1680*Sin[(5*c)/2 + 3*d*x] + 1
680*Sin[(7*c)/2 + 3*d*x] - 2520*Sin[(7*c)/2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4*d*x] + 1008*Sin[(9*c)/2 + 5*d*x] -
1008*Sin[(11*c)/2 + 5*d*x] + 180*Sin[(13*c)/2 + 7*d*x] - 180*Sin[(15*c)/2 + 7*d*x] + 315*Sin[(15*c)/2 + 8*d*x
] + 315*Sin[(17*c)/2 + 8*d*x] - 140*Sin[(17*c)/2 + 9*d*x] + 140*Sin[(19*c)/2 + 9*d*x])/(645120*a*d*(Cos[c/2] +
Sin[c/2]))

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fricas [A]  time = 0.91, size = 90, normalized size = 0.57 \[ \frac {4480 \, \cos \left (d x + c\right )^{9} - 11520 \, \cos \left (d x + c\right )^{7} + 8064 \, \cos \left (d x + c\right )^{5} + 945 \, d x + 315 \, {\left (16 \, \cos \left (d x + c\right )^{7} - 24 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/40320*(4480*cos(d*x + c)^9 - 11520*cos(d*x + c)^7 + 8064*cos(d*x + c)^5 + 945*d*x + 315*(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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giac [A]  time = 0.21, size = 218, normalized size = 1.37 \[ \frac {\frac {945 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 8190 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 97650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 215040 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 106470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 322560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 451584 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 106470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 129024 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 97650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36864 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8190 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9216 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1024\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{9} a}}{40320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/40320*(945*(d*x + c)/a + 2*(945*tan(1/2*d*x + 1/2*c)^17 + 8190*tan(1/2*d*x + 1/2*c)^15 - 97650*tan(1/2*d*x +
1/2*c)^13 + 215040*tan(1/2*d*x + 1/2*c)^12 + 106470*tan(1/2*d*x + 1/2*c)^11 - 322560*tan(1/2*d*x + 1/2*c)^10
+ 451584*tan(1/2*d*x + 1/2*c)^8 - 106470*tan(1/2*d*x + 1/2*c)^7 - 129024*tan(1/2*d*x + 1/2*c)^6 + 97650*tan(1/
2*d*x + 1/2*c)^5 + 36864*tan(1/2*d*x + 1/2*c)^4 - 8190*tan(1/2*d*x + 1/2*c)^3 + 9216*tan(1/2*d*x + 1/2*c)^2 -
945*tan(1/2*d*x + 1/2*c) + 1024)/((tan(1/2*d*x + 1/2*c)^2 + 1)^9*a))/d

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maple [B]  time = 0.30, size = 517, normalized size = 3.25 \[ \frac {16}{315 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {3 \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 )^{9}}+\frac {16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {13 \left (\tan ^{3}\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 )^{9}}+\frac {64 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {155 \left (\tan ^{5}\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 )^{9}}-\frac {32 \left (\tan ^{6}\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 )^{9}}-\frac {169 \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 )^{9}}+\frac {112 \left (\tan ^{8}\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 )^{9}}-\frac {16 \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 )^{9}}+\frac {169 \left (\tan ^{11}\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 )^{9}}+\frac {32 \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 )^{9}}-\frac {155 \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 )^{9}}+\frac {13 \left (\tan ^{15}\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 )^{9}}+\frac {3 \left (\tan ^{17}\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 )^{9}}+\frac {3 \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)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x)

[Out]

16/315/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9-3/64/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)+16/35/a/d/(1+tan(
1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^2-13/32/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^3+64/35/a/d/(
1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^4+155/32/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^5-32/5
/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^6-169/32/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^
7+112/5/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^8-16/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*
c)^10+169/32/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^11+32/3/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*
d*x+1/2*c)^12-155/32/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^13+13/32/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9
*tan(1/2*d*x+1/2*c)^15+3/64/a/d/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^17+3/64/a/d*arctan(tan(1/2*d*x+1
/2*c))

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maxima [B]  time = 0.48, size = 502, normalized size = 3.16 \[ -\frac {\frac {\frac {945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9216 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {8190 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {36864 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {97650 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {129024 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {106470 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {451584 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {322560 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {106470 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {215040 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {97650 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {8190 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {945 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - 1024}{a + \frac {9 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {36 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {126 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {126 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {84 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {36 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {9 \, a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {a \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}}} - \frac {945 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{20160 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/20160*((945*sin(d*x + c)/(cos(d*x + c) + 1) - 9216*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 8190*sin(d*x + c)^
3/(cos(d*x + c) + 1)^3 - 36864*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 97650*sin(d*x + c)^5/(cos(d*x + c) + 1)^5
+ 129024*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 106470*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 451584*sin(d*x +
c)^8/(cos(d*x + c) + 1)^8 + 322560*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 106470*sin(d*x + c)^11/(cos(d*x + c
) + 1)^11 - 215040*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 97650*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 8190*
sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 945*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 - 1024)/(a + 9*a*sin(d*x + c
)^2/(cos(d*x + c) + 1)^2 + 36*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 84*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6
+ 126*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 126*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 84*a*sin(d*x + c)
^12/(cos(d*x + c) + 1)^12 + 36*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 9*a*sin(d*x + c)^16/(cos(d*x + c) + 1
)^16 + a*sin(d*x + c)^18/(cos(d*x + c) + 1)^18) - 945*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

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mupad [B]  time = 11.41, size = 211, normalized size = 1.33 \[ \frac {3\,x}{128\,a}+\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{32}-\frac {155\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+\frac {169\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}-16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {112\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}-\frac {169\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {155\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {16}{315}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^6*sin(c + d*x)^4)/(a + a*sin(c + d*x)),x)

[Out]

(3*x)/(128*a) + ((16*tan(c/2 + (d*x)/2)^2)/35 - (3*tan(c/2 + (d*x)/2))/64 - (13*tan(c/2 + (d*x)/2)^3)/32 + (64
*tan(c/2 + (d*x)/2)^4)/35 + (155*tan(c/2 + (d*x)/2)^5)/32 - (32*tan(c/2 + (d*x)/2)^6)/5 - (169*tan(c/2 + (d*x)
/2)^7)/32 + (112*tan(c/2 + (d*x)/2)^8)/5 - 16*tan(c/2 + (d*x)/2)^10 + (169*tan(c/2 + (d*x)/2)^11)/32 + (32*tan
(c/2 + (d*x)/2)^12)/3 - (155*tan(c/2 + (d*x)/2)^13)/32 + (13*tan(c/2 + (d*x)/2)^15)/32 + (3*tan(c/2 + (d*x)/2)
^17)/64 + 16/315)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^9)

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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)**6*sin(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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