\(\int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [634]

Optimal result

Integrand size = 29, antiderivative size = 135 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {x}{8 a^2}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}+\frac {3 \cos ^5(c+d x)}{5 a^2 d}-\frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{3 a^2 d} \]

[Out]

-1/8*x/a^2-2/3*cos(d*x+c)^3/a^2/d+3/5*cos(d*x+c)^5/a^2/d-1/7*cos(d*x+c)^7/a^2/d-1/8*cos(d*x+c)*sin(d*x+c)/a^2/
d+1/4*cos(d*x+c)^3*sin(d*x+c)/a^2/d+1/3*cos(d*x+c)^3*sin(d*x+c)^3/a^2/d

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2954, 2952, 2645, 14, 2648, 2715, 8, 276} \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {3 \cos ^5(c+d x)}{5 a^2 d}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}+\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{3 a^2 d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac {x}{8 a^2} \]

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/8*x/a^2 - (2*Cos[c + d*x]^3)/(3*a^2*d) + (3*Cos[c + d*x]^5)/(5*a^2*d) - Cos[c + d*x]^7/(7*a^2*d) - (Cos[c +
d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(3*a
^2*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 276

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 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 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2954

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e +
f*x])^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(418\) vs. \(2(135)=270\).

Time = 2.12 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.10 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {210 (1+8 d x) \cos \left (\frac {c}{2}\right )+1365 \cos \left (\frac {c}{2}+d x\right )+1365 \cos \left (\frac {3 c}{2}+d x\right )-210 \cos \left (\frac {3 c}{2}+2 d x\right )+210 \cos \left (\frac {5 c}{2}+2 d x\right )+175 \cos \left (\frac {5 c}{2}+3 d x\right )+175 \cos \left (\frac {7 c}{2}+3 d x\right )-210 \cos \left (\frac {7 c}{2}+4 d x\right )+210 \cos \left (\frac {9 c}{2}+4 d x\right )-147 \cos \left (\frac {9 c}{2}+5 d x\right )-147 \cos \left (\frac {11 c}{2}+5 d x\right )+70 \cos \left (\frac {11 c}{2}+6 d x\right )-70 \cos \left (\frac {13 c}{2}+6 d x\right )+15 \cos \left (\frac {13 c}{2}+7 d x\right )+15 \cos \left (\frac {15 c}{2}+7 d x\right )-210 \sin \left (\frac {c}{2}\right )+1680 d x \sin \left (\frac {c}{2}\right )-1365 \sin \left (\frac {c}{2}+d x\right )+1365 \sin \left (\frac {3 c}{2}+d x\right )-210 \sin \left (\frac {3 c}{2}+2 d x\right )-210 \sin \left (\frac {5 c}{2}+2 d x\right )-175 \sin \left (\frac {5 c}{2}+3 d x\right )+175 \sin \left (\frac {7 c}{2}+3 d x\right )-210 \sin \left (\frac {7 c}{2}+4 d x\right )-210 \sin \left (\frac {9 c}{2}+4 d x\right )+147 \sin \left (\frac {9 c}{2}+5 d x\right )-147 \sin \left (\frac {11 c}{2}+5 d x\right )+70 \sin \left (\frac {11 c}{2}+6 d x\right )+70 \sin \left (\frac {13 c}{2}+6 d x\right )-15 \sin \left (\frac {13 c}{2}+7 d x\right )+15 \sin \left (\frac {15 c}{2}+7 d x\right )}{13440 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/13440*(210*(1 + 8*d*x)*Cos[c/2] + 1365*Cos[c/2 + d*x] + 1365*Cos[(3*c)/2 + d*x] - 210*Cos[(3*c)/2 + 2*d*x]
+ 210*Cos[(5*c)/2 + 2*d*x] + 175*Cos[(5*c)/2 + 3*d*x] + 175*Cos[(7*c)/2 + 3*d*x] - 210*Cos[(7*c)/2 + 4*d*x] +
210*Cos[(9*c)/2 + 4*d*x] - 147*Cos[(9*c)/2 + 5*d*x] - 147*Cos[(11*c)/2 + 5*d*x] + 70*Cos[(11*c)/2 + 6*d*x] - 7
0*Cos[(13*c)/2 + 6*d*x] + 15*Cos[(13*c)/2 + 7*d*x] + 15*Cos[(15*c)/2 + 7*d*x] - 210*Sin[c/2] + 1680*d*x*Sin[c/
2] - 1365*Sin[c/2 + d*x] + 1365*Sin[(3*c)/2 + d*x] - 210*Sin[(3*c)/2 + 2*d*x] - 210*Sin[(5*c)/2 + 2*d*x] - 175
*Sin[(5*c)/2 + 3*d*x] + 175*Sin[(7*c)/2 + 3*d*x] - 210*Sin[(7*c)/2 + 4*d*x] - 210*Sin[(9*c)/2 + 4*d*x] + 147*S
in[(9*c)/2 + 5*d*x] - 147*Sin[(11*c)/2 + 5*d*x] + 70*Sin[(11*c)/2 + 6*d*x] + 70*Sin[(13*c)/2 + 6*d*x] - 15*Sin
[(13*c)/2 + 7*d*x] + 15*Sin[(15*c)/2 + 7*d*x])/(a^2*d*(Cos[c/2] + Sin[c/2]))

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.66

method	result	size
parallelrisch	\(\frac {-840 d x +147 \cos \left (5 d x +5 c \right )-175 \cos \left (3 d x +3 c \right )-1365 \cos \left (d x +c \right )-15 \cos \left (7 d x +7 c \right )-70 \sin \left (6 d x +6 c \right )+210 \sin \left (4 d x +4 c \right )+210 \sin \left (2 d x +2 c \right )-1408}{6720 d \,a^{2}}\)	\(89\)
risch	\(-\frac {x}{8 a^{2}}-\frac {13 \cos \left (d x +c \right )}{64 a^{2} d}-\frac {\cos \left (7 d x +7 c \right )}{448 d \,a^{2}}-\frac {\sin \left (6 d x +6 c \right )}{96 d \,a^{2}}+\frac {7 \cos \left (5 d x +5 c \right )}{320 d \,a^{2}}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{2}}-\frac {5 \cos \left (3 d x +3 c \right )}{192 d \,a^{2}}+\frac {\sin \left (2 d x +2 c \right )}{32 d \,a^{2}}\)	\(124\)
derivativedivides	\(\frac {\frac {16 \left (-\frac {\left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {5 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {97 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {13 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}+\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {97 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}+\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}-\frac {11 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}-\frac {11}{420}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\)	\(181\)
default	\(\frac {\frac {16 \left (-\frac {\left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {5 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {97 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {13 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}+\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {97 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}+\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}-\frac {11 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}-\frac {11}{420}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\)	\(181\)

[In]

int(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/6720*(-840*d*x+147*cos(5*d*x+5*c)-175*cos(3*d*x+3*c)-1365*cos(d*x+c)-15*cos(7*d*x+7*c)-70*sin(6*d*x+6*c)+210
*sin(4*d*x+4*c)+210*sin(2*d*x+2*c)-1408)/d/a^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {120 \, \cos \left (d x + c\right )^{7} - 504 \, \cos \left (d x + c\right )^{5} + 560 \, \cos \left (d x + c\right )^{3} + 105 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 14 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, a^{2} d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/840*(120*cos(d*x + c)^7 - 504*cos(d*x + c)^5 + 560*cos(d*x + c)^3 + 105*d*x + 35*(8*cos(d*x + c)^5 - 14*cos
(d*x + c)^3 + 3*cos(d*x + c))*sin(d*x + c))/(a^2*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3046 vs. \(2 (121) = 242\).

Time = 83.58 (sec) , antiderivative size = 3046, normalized size of antiderivative = 22.56 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**3/(a+a*sin(d*x+c))**2,x)

[Out]

Piecewise((-105*d*x*tan(c/2 + d*x/2)**14/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 +
17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 176
40*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 735*d*x*tan(c/2 + d*x/2)**12/(
840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400
*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*
d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 2205*d*x*tan(c/2 + d*x/2)**10/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a*
*2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*
d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 367
5*d*x*tan(c/2 + d*x/2)**8/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*t
an(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c
/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 3675*d*x*tan(c/2 + d*x/2)**6/(840*a**2*d*tan(
c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2
+ d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x
/2)**2 + 840*a**2*d) - 2205*d*x*tan(c/2 + d*x/2)**4/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d
*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/
2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 735*d*x*tan(c/2 + d
*x/2)**2/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**
10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 +
5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 105*d*x/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2
+ d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 +
d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 210*tan(c/2 + d
*x/2)**13/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)*
*10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 +
5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 1400*tan(c/2 + d*x/2)**11/(840*a**2*d*tan(c/2 + d*x/2)**14 +
5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 2940
0*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d
) - 3360*tan(c/2 + d*x/2)**10/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2
*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*t
an(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) + 6790*tan(c/2 + d*x/2)**9/(840*a**2*d*tan(
c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2
+ d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x
/2)**2 + 840*a**2*d) - 14560*tan(c/2 + d*x/2)**8/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/
2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)*
*6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) + 2240*tan(c/2 + d*x/2)*
*6/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 2
9400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a
**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 6790*tan(c/2 + d*x/2)**5/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**
2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d
*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 4032
*tan(c/2 + d*x/2)**4/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/
2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 +
d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) + 1400*tan(c/2 + d*x/2)**3/(840*a**2*d*tan(c/2 + d*x
/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)
**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 +
840*a**2*d) - 2464*tan(c/2 + d*x/2)**2/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 1
7640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640
*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) + 210*tan(c/2 + d*x/2)/(840*a**2*d
*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*ta
n(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2
+ d*x/2)**2 + 840*a**2*d) - 352/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a*
*2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d
*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d), Ne(d, 0)), (x*sin(c)**3*cos(c)**6/(a*sin
(c) + a)**2, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (121) = 242\).

Time = 0.31 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.08 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1232 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {700 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2016 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3395 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1120 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {7280 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {3395 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {1680 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {700 \, \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}} - 176}{a^{2} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a^{2} \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^{2}}}{420 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/420*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 1232*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 700*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 - 2016*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3395*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 112
0*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 7280*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 3395*sin(d*x + c)^9/(cos(d*
x + c) + 1)^9 - 1680*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 700*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 105*s
in(d*x + c)^13/(cos(d*x + c) + 1)^13 - 176)/(a^2 + 7*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 21*a^2*sin(d*x
+ c)^4/(cos(d*x + c) + 1)^4 + 35*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 35*a^2*sin(d*x + c)^8/(cos(d*x + c)
+ 1)^8 + 21*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 7*a^2*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + a^2*sin
(d*x + c)^14/(cos(d*x + c) + 1)^14) - 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 700 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 3395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 7280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 700 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1232 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 176\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/840*(105*(d*x + c)/a^2 + 2*(105*tan(1/2*d*x + 1/2*c)^13 + 700*tan(1/2*d*x + 1/2*c)^11 + 1680*tan(1/2*d*x +
1/2*c)^10 - 3395*tan(1/2*d*x + 1/2*c)^9 + 7280*tan(1/2*d*x + 1/2*c)^8 - 1120*tan(1/2*d*x + 1/2*c)^6 + 3395*tan
(1/2*d*x + 1/2*c)^5 + 2016*tan(1/2*d*x + 1/2*c)^4 - 700*tan(1/2*d*x + 1/2*c)^3 + 1232*tan(1/2*d*x + 1/2*c)^2 -
105*tan(1/2*d*x + 1/2*c) + 176)/((tan(1/2*d*x + 1/2*c)^2 + 1)^7*a^2))/d

Mupad [B] (verification not implemented)

Time = 14.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.28 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {x}{8\,a^2}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {97\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+\frac {52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {97\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {44}{105}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]

[In]

int((cos(c + d*x)^6*sin(c + d*x)^3)/(a + a*sin(c + d*x))^2,x)

[Out]

- x/(8*a^2) - ((44*tan(c/2 + (d*x)/2)^2)/15 - tan(c/2 + (d*x)/2)/4 - (5*tan(c/2 + (d*x)/2)^3)/3 + (24*tan(c/2
+ (d*x)/2)^4)/5 + (97*tan(c/2 + (d*x)/2)^5)/12 - (8*tan(c/2 + (d*x)/2)^6)/3 + (52*tan(c/2 + (d*x)/2)^8)/3 - (9
7*tan(c/2 + (d*x)/2)^9)/12 + 4*tan(c/2 + (d*x)/2)^10 + (5*tan(c/2 + (d*x)/2)^11)/3 + tan(c/2 + (d*x)/2)^13/4 +
44/105)/(a^2*d*(tan(c/2 + (d*x)/2)^2 + 1)^7)