3.297 \(\int \frac{\cos ^2(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=104 \[ \frac{\cos ^5(c+d x)}{5 a d}-\frac{2 \cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x}{8 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.135656, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2839, 2635, 8, 2633} \[ \frac{\cos ^5(c+d x)}{5 a d}-\frac{2 \cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^2*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(3*x)/(8*a) + Cos[c + d*x]/(a*d) - (2*Cos[c + d*x]^3)/(3*a*d) + Cos[c + d*x]^5/(5*a*d) - (3*Cos[c + d*x]*Sin[c
+ d*x])/(8*a*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*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 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 2633

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]

Rubi steps

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

Mathematica [B]  time = 5.10352, size = 281, normalized size = 2.7 \[ \frac{1}{480} \left (\frac{60 \sin ^2\left (\frac{1}{2} (c+d x)\right )}{d (a \sin (c+d x)+a)}-\frac{300 \sin (c) \sin (d x)}{a d}+\frac{50 \sin (3 c) \sin (3 d x)}{a d}-\frac{6 \sin (5 c) \sin (5 d x)}{a d}+\frac{30 \sin (c+d x)}{a d (\sin (c+d x)+1)}+\frac{300 \cos (c) \cos (d x)}{a d}-\frac{50 \cos (3 c) \cos (3 d x)}{a d}+\frac{6 \cos (5 c) \cos (5 d x)}{a d}-\frac{120 \sin (2 c) \cos (2 d x)}{a d}+\frac{15 \sin (4 c) \cos (4 d x)}{a d}-\frac{120 \cos (2 c) \sin (2 d x)}{a d}+\frac{15 \cos (4 c) \sin (4 d x)}{a d}-\frac{60 \sin \left (\frac{d x}{2}\right )}{a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{180 x}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

((180*x)/a + (300*Cos[c]*Cos[d*x])/(a*d) - (50*Cos[3*c]*Cos[3*d*x])/(a*d) + (6*Cos[5*c]*Cos[5*d*x])/(a*d) - (1
20*Cos[2*d*x]*Sin[2*c])/(a*d) + (15*Cos[4*d*x]*Sin[4*c])/(a*d) - (300*Sin[c]*Sin[d*x])/(a*d) - (120*Cos[2*c]*S
in[2*d*x])/(a*d) + (50*Sin[3*c]*Sin[3*d*x])/(a*d) + (15*Cos[4*c]*Sin[4*d*x])/(a*d) - (6*Sin[5*c]*Sin[5*d*x])/(
a*d) - (60*Sin[(d*x)/2])/(a*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (30*Sin[c + d*x])
/(a*d*(1 + Sin[c + d*x])) + (60*Sin[(c + d*x)/2]^2)/(d*(a + a*Sin[c + d*x])))/480

________________________________________________________________________________________

Maple [B]  time = 0.091, size = 245, normalized size = 2.4 \begin{align*}{\frac{3}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{7}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{32}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-{\frac{7}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{16}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-{\frac{3}{4\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{16}{15\,da} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{3}{4\,da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*sin(d*x+c)^4/(a+a*sin(d*x+c)),x)

[Out]

3/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9+7/2/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^
7+32/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4-7/2/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*
c)^3+16/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^2-3/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1
/2*c)+16/15/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5+3/4/a/d*arctan(tan(1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [B]  time = 1.55767, size = 348, normalized size = 3.35 \begin{align*} -\frac{\frac{\frac{45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{320 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{210 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{640 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{210 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{45 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 64}{a + \frac{5 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{10 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{10 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{5 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac{45 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*((45*sin(d*x + c)/(cos(d*x + c) + 1) - 320*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 210*sin(d*x + c)^3/(cos
(d*x + c) + 1)^3 - 640*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 210*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 45*sin(
d*x + c)^9/(cos(d*x + c) + 1)^9 - 64)/(a + 5*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*a*sin(d*x + c)^4/(cos(
d*x + c) + 1)^4 + 10*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a*sin(d
*x + c)^10/(cos(d*x + c) + 1)^10) - 45*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

________________________________________________________________________________________

Fricas [A]  time = 1.72003, size = 182, normalized size = 1.75 \begin{align*} \frac{24 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} + 45 \, d x + 15 \,{\left (2 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 120 \, \cos \left (d x + c\right )}{120 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/120*(24*cos(d*x + c)^5 - 80*cos(d*x + c)^3 + 45*d*x + 15*(2*cos(d*x + c)^3 - 5*cos(d*x + c))*sin(d*x + c) +
120*cos(d*x + c))/(a*d)

________________________________________________________________________________________

Sympy [A]  time = 74.3806, size = 1360, normalized size = 13.08 \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)**2*sin(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Piecewise((45*d*x*tan(c/2 + d*x/2)**10/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*
tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 225*d*x*tan(c/2
+ d*x/2)**8/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*
a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 450*d*x*tan(c/2 + d*x/2)**6/(120*a*d*tan(c/
2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 6
00*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 450*d*x*tan(c/2 + d*x/2)**4/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*ta
n(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2
+ 120*a*d) + 225*d*x*tan(c/2 + d*x/2)**2/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*
d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 45*d*x/(120*a*
d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2
)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 90*tan(c/2 + d*x/2)**9/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d
*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)*
*2 + 120*a*d) + 420*tan(c/2 + d*x/2)**7/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d
*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 1280*tan(c/2 +
d*x/2)**4/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*
d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) - 420*tan(c/2 + d*x/2)**3/(120*a*d*tan(c/2 + d*
x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d
*tan(c/2 + d*x/2)**2 + 120*a*d) + 640*tan(c/2 + d*x/2)**2/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*
x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d)
- 90*tan(c/2 + d*x/2)/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)
**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 128/(120*a*d*tan(c/2 + d*x/2)**1
0 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/
2 + d*x/2)**2 + 120*a*d), Ne(d, 0)), (x*sin(c)**4*cos(c)**2/(a*sin(c) + a), True))

________________________________________________________________________________________

Giac [A]  time = 1.34131, size = 154, normalized size = 1.48 \begin{align*} \frac{\frac{45 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (45 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 640 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 320 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 45 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 64\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{5} a}}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/120*(45*(d*x + c)/a + 2*(45*tan(1/2*d*x + 1/2*c)^9 + 210*tan(1/2*d*x + 1/2*c)^7 + 640*tan(1/2*d*x + 1/2*c)^4
- 210*tan(1/2*d*x + 1/2*c)^3 + 320*tan(1/2*d*x + 1/2*c)^2 - 45*tan(1/2*d*x + 1/2*c) + 64)/((tan(1/2*d*x + 1/2
*c)^2 + 1)^5*a))/d