\(\int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [625]

Optimal result

Integrand size = 29, antiderivative size = 141 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 x}{128 a}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2918, 2645, 14, 2648, 2715, 8} \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cos ^7(c+d x)}{7 a d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{16 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac {3 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac {3 x}{128 a} \]

[In]

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

[Out]

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

Rule 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 2645

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 2648

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 2715

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] && Integ
erQ[2*n]

Rule 2918

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(375\) vs. \(2(141)=282\).

Time = 6.33 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.66 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1680 (c-d x) \cos \left (\frac {c}{2}\right )-1680 \cos \left (\frac {c}{2}+d x\right )-1680 \cos \left (\frac {3 c}{2}+d x\right )-560 \cos \left (\frac {5 c}{2}+3 d x\right )-560 \cos \left (\frac {7 c}{2}+3 d x\right )+280 \cos \left (\frac {7 c}{2}+4 d x\right )-280 \cos \left (\frac {9 c}{2}+4 d x\right )+112 \cos \left (\frac {9 c}{2}+5 d x\right )+112 \cos \left (\frac {11 c}{2}+5 d x\right )+80 \cos \left (\frac {13 c}{2}+7 d x\right )+80 \cos \left (\frac {15 c}{2}+7 d x\right )-35 \cos \left (\frac {15 c}{2}+8 d x\right )+35 \cos \left (\frac {17 c}{2}+8 d x\right )-3360 \sin \left (\frac {c}{2}\right )+1680 c \sin \left (\frac {c}{2}\right )-1680 d x \sin \left (\frac {c}{2}\right )+1680 \sin \left (\frac {c}{2}+d x\right )-1680 \sin \left (\frac {3 c}{2}+d x\right )+560 \sin \left (\frac {5 c}{2}+3 d x\right )-560 \sin \left (\frac {7 c}{2}+3 d x\right )+280 \sin \left (\frac {7 c}{2}+4 d x\right )+280 \sin \left (\frac {9 c}{2}+4 d x\right )-112 \sin \left (\frac {9 c}{2}+5 d x\right )+112 \sin \left (\frac {11 c}{2}+5 d x\right )-80 \sin \left (\frac {13 c}{2}+7 d x\right )+80 \sin \left (\frac {15 c}{2}+7 d x\right )-35 \sin \left (\frac {15 c}{2}+8 d x\right )-35 \sin \left (\frac {17 c}{2}+8 d x\right )}{71680 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.55

method	result	size
parallelrisch	\(\frac {-840 d x +112 \cos \left (5 d x +5 c \right )-560 \cos \left (3 d x +3 c \right )-1680 \cos \left (d x +c \right )-35 \sin \left (8 d x +8 c \right )+80 \cos \left (7 d x +7 c \right )+280 \sin \left (4 d x +4 c \right )-2048}{35840 d a}\)	\(78\)
risch	\(-\frac {3 x}{128 a}-\frac {3 \cos \left (d x +c \right )}{64 a d}-\frac {\sin \left (8 d x +8 c \right )}{1024 d a}+\frac {\cos \left (7 d x +7 c \right )}{448 a d}+\frac {\cos \left (5 d x +5 c \right )}{320 a d}+\frac {\sin \left (4 d x +4 c \right )}{128 d a}-\frac {\cos \left (3 d x +3 c \right )}{64 a d}\)	\(107\)
derivativedivides	\(\frac {\frac {16 \left (-\frac {1}{140}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}-\frac {333 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {671 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {671 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}+\frac {333 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {23 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {3 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\)	\(207\)
default	\(\frac {\frac {16 \left (-\frac {1}{140}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}-\frac {333 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {671 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {671 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}+\frac {333 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {23 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {3 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\)	\(207\)

[In]

int(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/35840*(-840*d*x+112*cos(5*d*x+5*c)-560*cos(3*d*x+3*c)-1680*cos(d*x+c)-35*sin(8*d*x+8*c)+80*cos(7*d*x+7*c)+28
0*sin(4*d*x+4*c)-2048)/d/a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.57 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {640 \, \cos \left (d x + c\right )^{7} - 896 \, \cos \left (d x + c\right )^{5} - 105 \, d x - 35 \, {\left (16 \, \cos \left (d x + c\right )^{7} - 24 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4480 \, a d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3580 vs. \(2 (116) = 232\).

Time = 49.09 (sec) , antiderivative size = 3580, normalized size of antiderivative = 25.39 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

Piecewise((-105*d*x*tan(c/2 + d*x/2)**16/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125
440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*t
an(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 840*d*x*tan(
c/2 + d*x/2)**14/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)
**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125
440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 2940*d*x*tan(c/2 + d*x/2)**12/(4480*
a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c
/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2
)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 5880*d*x*tan(c/2 + d*x/2)**10/(4480*a*d*tan(c/2 + d*x/2)**1
6 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 31360
0*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/
2 + d*x/2)**2 + 4480*a*d) - 7350*d*x*tan(c/2 + d*x/2)**8/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 +
d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**
8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d
) - 5880*d*x*tan(c/2 + d*x/2)**6/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*
tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 +
d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 2940*d*x*tan(c/2 + d
*x/2)**4/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 2
50880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*
tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 840*d*x*tan(c/2 + d*x/2)**2/(4480*a*d*tan(c/
2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2
)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 358
40*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 105*d*x/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**
14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 2508
80*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 210*
tan(c/2 + d*x/2)**15/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*
x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 +
125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 1610*tan(c/2 + d*x/2)**13/(4480*
a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c
/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2
)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 17920*tan(c/2 + d*x/2)**12/(4480*a*d*tan(c/2 + d*x/2)**16 +
35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a
*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 +
d*x/2)**2 + 4480*a*d) + 23310*tan(c/2 + d*x/2)**11/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2
)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 2
50880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 4
6970*tan(c/2 + d*x/2)**9/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2
+ d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)*
*6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 17920*tan(c/2 + d*x/2)**8/(4
480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*t
an(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d
*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) + 46970*tan(c/2 + d*x/2)**7/(4480*a*d*tan(c/2 + d*x/2)**1
6 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 31360
0*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/
2 + d*x/2)**2 + 4480*a*d) - 28672*tan(c/2 + d*x/2)**6/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x
/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 +
250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) -
23310*tan(c/2 + d*x/2)**5/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/
2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2
)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) + 3584*tan(c/2 + d*x/2)**4/(
4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*
tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 +
d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) + 1610*tan(c/2 + d*x/2)**3/(4480*a*d*tan(c/2 + d*x/2)**1
6 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 31360
0*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/
2 + d*x/2)**2 + 4480*a*d) - 4096*tan(c/2 + d*x/2)**2/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/
2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 +
250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) +
210*tan(c/2 + d*x/2)/(4480*a*d*tan(c/2 + d*x/2)**16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*
x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 313600*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 +
125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c/2 + d*x/2)**2 + 4480*a*d) - 512/(4480*a*d*tan(c/2 + d*x/2)**
16 + 35840*a*d*tan(c/2 + d*x/2)**14 + 125440*a*d*tan(c/2 + d*x/2)**12 + 250880*a*d*tan(c/2 + d*x/2)**10 + 3136
00*a*d*tan(c/2 + d*x/2)**8 + 250880*a*d*tan(c/2 + d*x/2)**6 + 125440*a*d*tan(c/2 + d*x/2)**4 + 35840*a*d*tan(c
/2 + d*x/2)**2 + 4480*a*d), Ne(d, 0)), (x*sin(c)**3*cos(c)**6/(a*sin(c) + a), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (127) = 254\).

Time = 0.32 (sec) , antiderivative size = 461, normalized size of antiderivative = 3.27 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2048 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1792 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {11655 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {14336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {23485 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {8960 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {23485 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {11655 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {8960 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {805 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {105 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - 256}{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}}{2240 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 805 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 8960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 11655 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 23485 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 8960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 23485 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 14336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 11655 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1792 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 805 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2048 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 256\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a}}{4480 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.68 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.41 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3\,x}{128\,a}-\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {333\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {671\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {671\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{64}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {333\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{64}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {4}{35}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \]

[In]

int((cos(c + d*x)^6*sin(c + d*x)^3)/(a + a*sin(c + d*x)),x)

[Out]

- (3*x)/(128*a) - ((32*tan(c/2 + (d*x)/2)^2)/35 - (3*tan(c/2 + (d*x)/2))/64 - (23*tan(c/2 + (d*x)/2)^3)/64 - (
4*tan(c/2 + (d*x)/2)^4)/5 + (333*tan(c/2 + (d*x)/2)^5)/64 + (32*tan(c/2 + (d*x)/2)^6)/5 - (671*tan(c/2 + (d*x)
/2)^7)/64 + 4*tan(c/2 + (d*x)/2)^8 + (671*tan(c/2 + (d*x)/2)^9)/64 - (333*tan(c/2 + (d*x)/2)^11)/64 + 4*tan(c/
2 + (d*x)/2)^12 + (23*tan(c/2 + (d*x)/2)^13)/64 + (3*tan(c/2 + (d*x)/2)^15)/64 + 4/35)/(a*d*(tan(c/2 + (d*x)/2
)^2 + 1)^8)