\(\int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [626]

Optimal result

Integrand size = 29, antiderivative size = 115 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{16 a}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos (c+d x) \sin (c+d x)}{16 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2918, 2648, 2715, 8, 2645, 14} \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{24 a d}+\frac {\sin (c+d x) \cos (c+d x)}{16 a d}+\frac {x}{16 a} \]

[In]

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

[Out]

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

Rule 8

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(351\) vs. \(2(115)=230\).

Time = 8.24 (sec) , antiderivative size = 351, normalized size of antiderivative = 3.05 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {420 x}{a}+\frac {315 \cos (c) \cos (d x)}{a d}+\frac {105 \cos (3 c) \cos (3 d x)}{a d}-\frac {21 \cos (5 c) \cos (5 d x)}{a d}-\frac {15 \cos (7 c) \cos (7 d x)}{a d}+\frac {105 \cos (2 d x) \sin (2 c)}{a d}-\frac {105 \cos (4 d x) \sin (4 c)}{a d}-\frac {35 \cos (6 d x) \sin (6 c)}{a d}-\frac {315 \sin (c) \sin (d x)}{a d}+\frac {105 \cos (2 c) \sin (2 d x)}{a d}-\frac {105 \sin (3 c) \sin (3 d x)}{a d}-\frac {105 \cos (4 c) \sin (4 d x)}{a d}+\frac {21 \sin (5 c) \sin (5 d x)}{a d}-\frac {35 \cos (6 c) \sin (6 d x)}{a d}+\frac {15 \sin (7 c) \sin (7 d x)}{a d}-\frac {525 \sin \left (\frac {d x}{2}\right )}{a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {525 \sin (c+d x)}{2 a d (1+\sin (c+d x))}+\frac {525 \sin ^2\left (\frac {1}{2} (c+d x)\right )}{d (a+a \sin (c+d x))}}{6720} \]

[In]

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

[Out]

((420*x)/a + (315*Cos[c]*Cos[d*x])/(a*d) + (105*Cos[3*c]*Cos[3*d*x])/(a*d) - (21*Cos[5*c]*Cos[5*d*x])/(a*d) -
(15*Cos[7*c]*Cos[7*d*x])/(a*d) + (105*Cos[2*d*x]*Sin[2*c])/(a*d) - (105*Cos[4*d*x]*Sin[4*c])/(a*d) - (35*Cos[6
*d*x]*Sin[6*c])/(a*d) - (315*Sin[c]*Sin[d*x])/(a*d) + (105*Cos[2*c]*Sin[2*d*x])/(a*d) - (105*Sin[3*c]*Sin[3*d*
x])/(a*d) - (105*Cos[4*c]*Sin[4*d*x])/(a*d) + (21*Sin[5*c]*Sin[5*d*x])/(a*d) - (35*Cos[6*c]*Sin[6*d*x])/(a*d)
+ (15*Sin[7*c]*Sin[7*d*x])/(a*d) - (525*Sin[(d*x)/2])/(a*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c +
d*x)/2])) + (525*Sin[c + d*x])/(2*a*d*(1 + Sin[c + d*x])) + (525*Sin[(c + d*x)/2]^2)/(d*(a + a*Sin[c + d*x])))
/6720

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77

method	result	size
parallelrisch	\(\frac {420 d x -21 \cos \left (5 d x +5 c \right )+105 \cos \left (3 d x +3 c \right )+315 \cos \left (d x +c \right )-15 \cos \left (7 d x +7 c \right )-35 \sin \left (6 d x +6 c \right )-105 \sin \left (4 d x +4 c \right )+105 \sin \left (2 d x +2 c \right )+384}{6720 d a}\)	\(89\)
risch	\(\frac {x}{16 a}+\frac {3 \cos \left (d x +c \right )}{64 a d}-\frac {\cos \left (7 d x +7 c \right )}{448 a d}-\frac {\sin \left (6 d x +6 c \right )}{192 d a}-\frac {\cos \left (5 d x +5 c \right )}{320 a d}-\frac {\sin \left (4 d x +4 c \right )}{64 d a}+\frac {\cos \left (3 d x +3 c \right )}{64 a d}+\frac {\sin \left (2 d x +2 c \right )}{64 d a}\)	\(124\)
derivativedivides	\(\frac {\frac {8 \left (\frac {\left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {31 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {31 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}+\frac {1}{70}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\)	\(179\)
default	\(\frac {\frac {8 \left (\frac {\left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {31 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {31 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}+\frac {1}{70}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\)	\(179\)
norman	\(\frac {\frac {7 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {7 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {7 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {7 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {35 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {35 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {7 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {7 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {3}{280 a d}+\frac {7 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {7 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {x}{16 a}-\frac {181 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {59 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{280 d a}+\frac {x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {29 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{140 d a}+\frac {x \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {7 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {227 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a}-\frac {83 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}+\frac {1363 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{840 d a}-\frac {145 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {451 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d a}-\frac {7 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d a}-\frac {311 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d a}-\frac {65 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {137 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\)	\(634\)

[In]

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

[Out]

1/6720*(420*d*x-21*cos(5*d*x+5*c)+105*cos(3*d*x+3*c)+315*cos(d*x+c)-15*cos(7*d*x+7*c)-35*sin(6*d*x+6*c)-105*si
n(4*d*x+4*c)+105*sin(2*d*x+2*c)+384)/d/a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.61 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {240 \, \cos \left (d x + c\right )^{7} - 336 \, \cos \left (d x + c\right )^{5} - 105 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, a d} \]

[In]

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

[Out]

-1/1680*(240*cos(d*x + c)^7 - 336*cos(d*x + c)^5 - 105*d*x + 35*(8*cos(d*x + c)^5 - 2*cos(d*x + c)^3 - 3*cos(d
*x + c))*sin(d*x + c))/(a*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2773 vs. \(2 (90) = 180\).

Time = 31.01 (sec) , antiderivative size = 2773, normalized size of antiderivative = 24.11 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (103) = 206\).

Time = 0.30 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.48 \[ \int \frac {\cos ^6(c+d x) \sin ^2(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 {672 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1344 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1085 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {6720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {3360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {1085 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {3360 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {1540 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 96}{a + \frac {7 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {21 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {35 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {21 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {7 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{840 \, d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.56 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 1540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1085 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1085 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1344 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a}}{1680 \, d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 14.44 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.50 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{16\,a}+\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{6}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{24}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {4}{35}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]

[In]

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

[Out]

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