\(\int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [645]

Optimal result

Integrand size = 29, antiderivative size = 105 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {13 x}{8 a^3}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d} \]

[Out]

13/8*x/a^3+4*cos(d*x+c)/a^3/d-5/3*cos(d*x+c)^3/a^3/d+1/5*cos(d*x+c)^5/a^3/d-13/8*cos(d*x+c)*sin(d*x+c)/a^3/d-3
/4*cos(d*x+c)*sin(d*x+c)^3/a^3/d

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2948, 2836, 2715, 8, 2713} \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {3 \sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac {13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {13 x}{8 a^3} \]

[In]

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

[Out]

(13*x)/(8*a^3) + (4*Cos[c + d*x])/(a^3*d) - (5*Cos[c + d*x]^3)/(3*a^3*d) + Cos[c + d*x]^5/(5*a^3*d) - (13*Cos[
c + d*x]*Sin[c + d*x])/(8*a^3*d) - (3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^3*d)

Rule 8

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

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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 2836

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

Rule 2948

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin ^2(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (a^3 \sin ^2(c+d x)-3 a^3 \sin ^3(c+d x)+3 a^3 \sin ^4(c+d x)-a^3 \sin ^5(c+d x)\right ) \, dx}{a^6} \\ & = \frac {\int \sin ^2(c+d x) \, dx}{a^3}-\frac {\int \sin ^5(c+d x) \, dx}{a^3}-\frac {3 \int \sin ^3(c+d x) \, dx}{a^3}+\frac {3 \int \sin ^4(c+d x) \, dx}{a^3} \\ & = -\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac {\int 1 \, dx}{2 a^3}+\frac {9 \int \sin ^2(c+d x) \, dx}{4 a^3}+\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}+\frac {3 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d} \\ & = \frac {x}{2 a^3}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac {9 \int 1 \, dx}{8 a^3} \\ & = \frac {13 x}{8 a^3}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(310\) vs. \(2(105)=210\).

Time = 1.33 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.95 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {1560 d x \cos \left (\frac {c}{2}\right )+1380 \cos \left (\frac {c}{2}+d x\right )+1380 \cos \left (\frac {3 c}{2}+d x\right )-480 \cos \left (\frac {3 c}{2}+2 d x\right )+480 \cos \left (\frac {5 c}{2}+2 d x\right )-170 \cos \left (\frac {5 c}{2}+3 d x\right )-170 \cos \left (\frac {7 c}{2}+3 d x\right )+45 \cos \left (\frac {7 c}{2}+4 d x\right )-45 \cos \left (\frac {9 c}{2}+4 d x\right )+6 \cos \left (\frac {9 c}{2}+5 d x\right )+6 \cos \left (\frac {11 c}{2}+5 d x\right )+10 \sin \left (\frac {c}{2}\right )+1560 d x \sin \left (\frac {c}{2}\right )-1380 \sin \left (\frac {c}{2}+d x\right )+1380 \sin \left (\frac {3 c}{2}+d x\right )-480 \sin \left (\frac {3 c}{2}+2 d x\right )-480 \sin \left (\frac {5 c}{2}+2 d x\right )+170 \sin \left (\frac {5 c}{2}+3 d x\right )-170 \sin \left (\frac {7 c}{2}+3 d x\right )+45 \sin \left (\frac {7 c}{2}+4 d x\right )+45 \sin \left (\frac {9 c}{2}+4 d x\right )-6 \sin \left (\frac {9 c}{2}+5 d x\right )+6 \sin \left (\frac {11 c}{2}+5 d x\right )}{960 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]

[In]

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

[Out]

(1560*d*x*Cos[c/2] + 1380*Cos[c/2 + d*x] + 1380*Cos[(3*c)/2 + d*x] - 480*Cos[(3*c)/2 + 2*d*x] + 480*Cos[(5*c)/
2 + 2*d*x] - 170*Cos[(5*c)/2 + 3*d*x] - 170*Cos[(7*c)/2 + 3*d*x] + 45*Cos[(7*c)/2 + 4*d*x] - 45*Cos[(9*c)/2 +
4*d*x] + 6*Cos[(9*c)/2 + 5*d*x] + 6*Cos[(11*c)/2 + 5*d*x] + 10*Sin[c/2] + 1560*d*x*Sin[c/2] - 1380*Sin[c/2 + d
*x] + 1380*Sin[(3*c)/2 + d*x] - 480*Sin[(3*c)/2 + 2*d*x] - 480*Sin[(5*c)/2 + 2*d*x] + 170*Sin[(5*c)/2 + 3*d*x]
- 170*Sin[(7*c)/2 + 3*d*x] + 45*Sin[(7*c)/2 + 4*d*x] + 45*Sin[(9*c)/2 + 4*d*x] - 6*Sin[(9*c)/2 + 5*d*x] + 6*S
in[(11*c)/2 + 5*d*x])/(960*a^3*d*(Cos[c/2] + Sin[c/2]))

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64

method	result	size
parallelrisch	\(\frac {780 d x +6 \cos \left (5 d x +5 c \right )-170 \cos \left (3 d x +3 c \right )+1380 \cos \left (d x +c \right )+45 \sin \left (4 d x +4 c \right )-480 \sin \left (2 d x +2 c \right )+1216}{480 a^{3} d}\)	\(67\)
risch	\(\frac {13 x}{8 a^{3}}+\frac {23 \cos \left (d x +c \right )}{8 a^{3} d}+\frac {\cos \left (5 d x +5 c \right )}{80 d \,a^{3}}+\frac {3 \sin \left (4 d x +4 c \right )}{32 d \,a^{3}}-\frac {17 \cos \left (3 d x +3 c \right )}{48 d \,a^{3}}-\frac {\sin \left (2 d x +2 c \right )}{d \,a^{3}}\)	\(90\)
derivativedivides	\(\frac {\frac {8 \left (\frac {13 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {25 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {29 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {19 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {19}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{3} d}\)	\(129\)
default	\(\frac {\frac {8 \left (\frac {13 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {25 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {29 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {19 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {19}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{3} d}\)	\(129\)

[In]

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

[Out]

1/480*(780*d*x+6*cos(5*d*x+5*c)-170*cos(3*d*x+3*c)+1380*cos(d*x+c)+45*sin(4*d*x+4*c)-480*sin(2*d*x+2*c)+1216)/
a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {24 \, \cos \left (d x + c\right )^{5} - 200 \, \cos \left (d x + c\right )^{3} + 195 \, d x + 15 \, {\left (6 \, \cos \left (d x + c\right )^{3} - 19 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 480 \, \cos \left (d x + c\right )}{120 \, a^{3} d} \]

[In]

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

[Out]

1/120*(24*cos(d*x + c)^5 - 200*cos(d*x + c)^3 + 195*d*x + 15*(6*cos(d*x + c)^3 - 19*cos(d*x + c))*sin(d*x + c)
+ 480*cos(d*x + c))/(a^3*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1608 vs. \(2 (99) = 198\).

Time = 89.95 (sec) , antiderivative size = 1608, normalized size of antiderivative = 15.31 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((195*d*x*tan(c/2 + d*x/2)**10/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 12
00*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d)
+ 975*d*x*tan(c/2 + d*x/2)**8/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d
*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 1950*d
*x*tan(c/2 + d*x/2)**6/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2
+ d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 1950*d*x*tan(c
/2 + d*x/2)**4/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2
)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 975*d*x*tan(c/2 + d*x/
2)**2/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 12
00*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 195*d*x/(120*a**3*d*tan(c/2 + d
*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4
+ 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 390*tan(c/2 + d*x/2)**9/(120*a**3*d*tan(c/2 + d*x/2)**10 + 6
00*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d
*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 1500*tan(c/2 + d*x/2)**7/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*ta
n(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d
*x/2)**2 + 120*a**3*d) + 1440*tan(c/2 + d*x/2)**6/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/
2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 1
20*a**3*d) + 4640*tan(c/2 + d*x/2)**4/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200
*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) -
1500*tan(c/2 + d*x/2)**3/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(
c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 3040*tan(c/
2 + d*x/2)**2/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)
**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) - 390*tan(c/2 + d*x/2)/(1
20*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*
d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 608/(120*a**3*d*tan(c/2 + d*x/2)**10 +
600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*
d*tan(c/2 + d*x/2)**2 + 120*a**3*d), Ne(d, 0)), (x*sin(c)**2*cos(c)**6/(a*sin(c) + a)**3, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (95) = 190\).

Time = 0.36 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.76 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {195 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1520 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2320 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {750 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {195 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 304}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {195 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, d} \]

[In]

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

[Out]

-1/60*((195*sin(d*x + c)/(cos(d*x + c) + 1) - 1520*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 750*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 - 2320*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 720*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 750*
sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 195*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 304)/(a^3 + 5*a^3*sin(d*x + c)
^2/(cos(d*x + c) + 1)^2 + 10*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1
)^6 + 5*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10) - 195*arctan(sin(
d*x + c)/(cos(d*x + c) + 1))/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.21 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {195 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (195 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 195 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 304\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a^{3}}}{120 \, d} \]

[In]

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

[Out]

1/120*(195*(d*x + c)/a^3 + 2*(195*tan(1/2*d*x + 1/2*c)^9 + 750*tan(1/2*d*x + 1/2*c)^7 + 720*tan(1/2*d*x + 1/2*
c)^6 + 2320*tan(1/2*d*x + 1/2*c)^4 - 750*tan(1/2*d*x + 1/2*c)^3 + 1520*tan(1/2*d*x + 1/2*c)^2 - 195*tan(1/2*d*
x + 1/2*c) + 304)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*a^3))/d

Mupad [B] (verification not implemented)

Time = 10.39 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {13\,x}{8\,a^3}+\frac {4\,\cos \left (c+d\,x\right )}{a^3\,d}-\frac {5\,{\cos \left (c+d\,x\right )}^3}{3\,a^3\,d}+\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3\,d}+\frac {3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^3\,d}-\frac {19\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^3\,d} \]

[In]

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

[Out]

(13*x)/(8*a^3) + (4*cos(c + d*x))/(a^3*d) - (5*cos(c + d*x)^3)/(3*a^3*d) + cos(c + d*x)^5/(5*a^3*d) + (3*cos(c
+ d*x)^3*sin(c + d*x))/(4*a^3*d) - (19*cos(c + d*x)*sin(c + d*x))/(8*a^3*d)