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

Optimal. Leaf size=135 \[ \frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}-\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos (c+d x)}{16 a d}+\frac{x}{16 a} \]

[Out]

x/(16*a) + Cos[c + d*x]^3/(3*a*d) - (2*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])/(8*a*d) - (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.197698, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 2568, 2635, 8, 2565, 270} \[ \frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}-\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos (c+d x)}{16 a d}+\frac{x}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(16*a) + Cos[c + d*x]^3/(3*a*d) - (2*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])/(8*a*d) - (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a*d)

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

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

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.256363, size = 86, normalized size = 0.64 \[ \frac{-105 \sin (2 (c+d x))-105 \sin (4 (c+d x))+35 \sin (6 (c+d x))+525 \cos (c+d x)+35 \cos (3 (c+d x))-63 \cos (5 (c+d x))+15 \cos (7 (c+d x))+420 c+420 d x}{6720 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(420*c + 420*d*x + 525*Cos[c + d*x] + 35*Cos[3*(c + d*x)] - 63*Cos[5*(c + d*x)] + 15*Cos[7*(c + d*x)] - 105*Si
n[2*(c + d*x)] - 105*Sin[4*(c + d*x)] + 35*Sin[6*(c + d*x)])/(6720*a*d)

________________________________________________________________________________________

Maple [B]  time = 0.085, size = 381, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13+5/6/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)
^11-97/24/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^9+32/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+
1/2*c)^8-16/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^6+97/24/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2
*d*x+1/2*c)^5+16/5/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4-5/6/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(
1/2*d*x+1/2*c)^3+16/15/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2-1/8/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7*
tan(1/2*d*x+1/2*c)+16/105/d/a/(1+tan(1/2*d*x+1/2*c)^2)^7+1/8/a/d*arctan(tan(1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [B]  time = 1.68898, size = 513, normalized size = 3.8 \begin{align*} -\frac{\frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{896 \, \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{2688 \, \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{4480 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{8960 \, \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{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}} - 128}{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/840*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 896*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 700*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 - 2688*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3395*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 448
0*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 8960*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 3395*sin(d*x + c)^9/(cos(d*
x + c) + 1)^9 - 700*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 105*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 128)/(
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/(cos(d*x + c) + 1)^14) - 105*arctan(sin(d*x + c)
/(cos(d*x + c) + 1))/a)/d

________________________________________________________________________________________

Fricas [A]  time = 1.13487, size = 217, normalized size = 1.61 \begin{align*} \frac{240 \, \cos \left (d x + c\right )^{7} - 672 \, \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 )}{1680 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(240*cos(d*x + c)^7 - 672*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*d)

________________________________________________________________________________________

Sympy [A]  time = 125.481, size = 3048, normalized size = 22.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((210*d*x*tan(c/2 + d*x/2)**14/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 7056
0*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c
/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) + 1470*d*x*tan(c/2 + d*x/2)**12/(3360*a*d*tan(c/2 +
d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8
+ 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d)
+ 4410*d*x*tan(c/2 + d*x/2)**10/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*ta
n(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x
/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) + 7350*d*x*tan(c/2 + d*x/2)**8/(3360*a*d*tan(c/2 + d*x/2)**
14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600
*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) + 7350*d*
x*tan(c/2 + d*x/2)**6/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*
x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 2
3520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) + 4410*d*x*tan(c/2 + d*x/2)**4/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520
*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c
/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) + 1470*d*x*tan(c/2
+ d*x/2)**2/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 +
117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*t
an(c/2 + d*x/2)**2 + 3360*a*d) + 210*d*x/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 705
60*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(
c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) - 115*tan(c/2 + d*x/2)**14/(3360*a*d*tan(c/2 + d*x
/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 1
17600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) + 42
0*tan(c/2 + d*x/2)**13/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d
*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 +
23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) - 805*tan(c/2 + d*x/2)**12/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*
d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2
+ d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) + 2800*tan(c/2 + d*x/2
)**11/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 11760
0*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2
+ d*x/2)**2 + 3360*a*d) - 2415*tan(c/2 + d*x/2)**10/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/
2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70
560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) - 13580*tan(c/2 + d*x/2)**9/(3360*a*d*
tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 +
d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3
360*a*d) + 31815*tan(c/2 + d*x/2)**8/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a
*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2
+ d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) - 21945*tan(c/2 + d*x/2)**6/(3360*a*d*tan(c/2 + d*x/2)
**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 1176
00*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) + 13580
*tan(c/2 + d*x/2)**5/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x
/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23
520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) + 8337*tan(c/2 + d*x/2)**4/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*
tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 +
d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) - 2800*tan(c/2 + d*x/2)*
*3/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a
*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 +
d*x/2)**2 + 3360*a*d) + 2779*tan(c/2 + d*x/2)**2/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**
12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*
a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d) - 420*tan(c/2 + d*x/2)/(3360*a*d*tan(c/2 +
d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 117600*a*d*tan(c/2 + d*x/2)**8
+ 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2 + d*x/2)**2 + 3360*a*d)
+ 397/(3360*a*d*tan(c/2 + d*x/2)**14 + 23520*a*d*tan(c/2 + d*x/2)**12 + 70560*a*d*tan(c/2 + d*x/2)**10 + 11760
0*a*d*tan(c/2 + d*x/2)**8 + 117600*a*d*tan(c/2 + d*x/2)**6 + 70560*a*d*tan(c/2 + d*x/2)**4 + 23520*a*d*tan(c/2
+ d*x/2)**2 + 3360*a*d), Ne(d, 0)), (x*sin(c)**4*cos(c)**4/(a*sin(c) + a), True))

________________________________________________________________________________________

Giac [A]  time = 1.44052, size = 224, normalized size = 1.66 \begin{align*} \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} + 700 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 3395 \, \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} - 4480 \, \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} + 2688 \, \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} + 896 \, \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 ) + 128\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{7} a}}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4/(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 + 700*tan(1/2*d*x + 1/2*c)^11 - 3395*tan(1/2*d*x + 1/
2*c)^9 + 8960*tan(1/2*d*x + 1/2*c)^8 - 4480*tan(1/2*d*x + 1/2*c)^6 + 3395*tan(1/2*d*x + 1/2*c)^5 + 2688*tan(1/
2*d*x + 1/2*c)^4 - 700*tan(1/2*d*x + 1/2*c)^3 + 896*tan(1/2*d*x + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 128)/(
(tan(1/2*d*x + 1/2*c)^2 + 1)^7*a))/d