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

Optimal. Leaf size=129 \[ \frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{4 \cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x)}{6 a^2 d}-\frac{11 \sin ^3(c+d x) \cos (c+d x)}{24 a^2 d}-\frac{11 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{11 x}{16 a^2} \]

[Out]

(11*x)/(16*a^2) + (2*Cos[c + d*x])/(a^2*d) - (4*Cos[c + d*x]^3)/(3*a^2*d) + (2*Cos[c + d*x]^5)/(5*a^2*d) - (11
*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) - (11*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^2*d) - (Cos[c + d*x]*Sin[c + d
*x]^5)/(6*a^2*d)

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Rubi [A]  time = 0.225079, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2869, 2757, 2635, 8, 2633} \[ \frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{4 \cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x)}{6 a^2 d}-\frac{11 \sin ^3(c+d x) \cos (c+d x)}{24 a^2 d}-\frac{11 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{11 x}{16 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*x)/(16*a^2) + (2*Cos[c + d*x])/(a^2*d) - (4*Cos[c + d*x]^3)/(3*a^2*d) + (2*Cos[c + d*x]^5)/(5*a^2*d) - (11
*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) - (11*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^2*d) - (Cos[c + d*x]*Sin[c + d
*x]^5)/(6*a^2*d)

Rule 2869

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]

Rule 2757

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 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 ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \sin ^4(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \sin ^4(c+d x)-2 a^2 \sin ^5(c+d x)+a^2 \sin ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \sin ^4(c+d x) \, dx}{a^2}+\frac{\int \sin ^6(c+d x) \, dx}{a^2}-\frac{2 \int \sin ^5(c+d x) \, dx}{a^2}\\ &=-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 a^2 d}+\frac{3 \int \sin ^2(c+d x) \, dx}{4 a^2}+\frac{5 \int \sin ^4(c+d x) \, dx}{6 a^2}+\frac{2 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac{2 \cos (c+d x)}{a^2 d}-\frac{4 \cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{3 \cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac{11 \cos (c+d x) \sin ^3(c+d x)}{24 a^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 a^2 d}+\frac{3 \int 1 \, dx}{8 a^2}+\frac{5 \int \sin ^2(c+d x) \, dx}{8 a^2}\\ &=\frac{3 x}{8 a^2}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{4 \cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{11 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac{11 \cos (c+d x) \sin ^3(c+d x)}{24 a^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 a^2 d}+\frac{5 \int 1 \, dx}{16 a^2}\\ &=\frac{11 x}{16 a^2}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{4 \cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{11 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac{11 \cos (c+d x) \sin ^3(c+d x)}{24 a^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.247646, size = 76, normalized size = 0.59 \[ \frac{-465 \sin (2 (c+d x))+75 \sin (4 (c+d x))-5 \sin (6 (c+d x))+1200 \cos (c+d x)-200 \cos (3 (c+d x))+24 \cos (5 (c+d x))+660 c+660 d x}{960 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(660*c + 660*d*x + 1200*Cos[c + d*x] - 200*Cos[3*(c + d*x)] + 24*Cos[5*(c + d*x)] - 465*Sin[2*(c + d*x)] + 75*
Sin[4*(c + d*x)] - 5*Sin[6*(c + d*x)])/(960*a^2*d)

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Maple [B]  time = 0.109, size = 347, normalized size = 2.7 \begin{align*}{\frac{11}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}+{\frac{187}{24\,d{a}^{2}} \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 ) ^{-6}}+{\frac{47}{4\,d{a}^{2}} \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 ) ^{-6}}+{\frac{64}{3\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{47}{4\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}+32\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{6}}}-{\frac{187}{24\,d{a}^{2}} \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 ) ^{-6}}+{\frac{64}{5\,d{a}^{2}} \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 ) ^{-6}}-{\frac{11}{8\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}+{\frac{32}{15\,d{a}^{2}} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}+{\frac{11}{8\,d{a}^{2}}\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)^4*sin(d*x+c)^4/(a+a*sin(d*x+c))^2,x)

[Out]

11/8/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11+187/24/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*
x+1/2*c)^9+47/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7+64/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*ta
n(1/2*d*x+1/2*c)^6-47/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5+32/d/a^2/(1+tan(1/2*d*x+1/2*c)^2
)^6*tan(1/2*d*x+1/2*c)^4-187/24/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3+64/5/d/a^2/(1+tan(1/2*d*
x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2-11/8/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)+32/15/d/a^2/(1+tan
(1/2*d*x+1/2*c)^2)^6+11/8/d/a^2*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.6535, size = 477, normalized size = 3.7 \begin{align*} -\frac{\frac{\frac{165 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1536 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{935 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{3840 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1410 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{2560 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{1410 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{935 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{165 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 256}{a^{2} + \frac{6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac{165 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{120 \, 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))^2,x, algorithm="maxima")

[Out]

-1/120*((165*sin(d*x + c)/(cos(d*x + c) + 1) - 1536*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 935*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 - 3840*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1410*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 25
60*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 1410*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 935*sin(d*x + c)^9/(cos(d*
x + c) + 1)^9 - 165*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 256)/(a^2 + 6*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1
)^2 + 15*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a^2*sin(d*x
+ c)^8/(cos(d*x + c) + 1)^8 + 6*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a^2*sin(d*x + c)^12/(cos(d*x + c)
+ 1)^12) - 165*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

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Fricas [A]  time = 1.15177, size = 215, normalized size = 1.67 \begin{align*} \frac{96 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} + 165 \, d x - 5 \,{\left (8 \, \cos \left (d x + c\right )^{5} - 38 \, \cos \left (d x + c\right )^{3} + 63 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 480 \, \cos \left (d x + c\right )}{240 \, a^{2} 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))^2,x, algorithm="fricas")

[Out]

1/240*(96*cos(d*x + c)^5 - 320*cos(d*x + c)^3 + 165*d*x - 5*(8*cos(d*x + c)^5 - 38*cos(d*x + c)^3 + 63*cos(d*x
+ c))*sin(d*x + c) + 480*cos(d*x + c))/(a^2*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.31003, size = 207, normalized size = 1.6 \begin{align*} \frac{\frac{165 \,{\left (d x + c\right )}}{a^{2}} + \frac{2 \,{\left (165 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 935 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 1410 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 2560 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1410 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3840 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 935 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1536 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 165 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 256\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{6} a^{2}}}{240 \, 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))^2,x, algorithm="giac")

[Out]

1/240*(165*(d*x + c)/a^2 + 2*(165*tan(1/2*d*x + 1/2*c)^11 + 935*tan(1/2*d*x + 1/2*c)^9 + 1410*tan(1/2*d*x + 1/
2*c)^7 + 2560*tan(1/2*d*x + 1/2*c)^6 - 1410*tan(1/2*d*x + 1/2*c)^5 + 3840*tan(1/2*d*x + 1/2*c)^4 - 935*tan(1/2
*d*x + 1/2*c)^3 + 1536*tan(1/2*d*x + 1/2*c)^2 - 165*tan(1/2*d*x + 1/2*c) + 256)/((tan(1/2*d*x + 1/2*c)^2 + 1)^
6*a^2))/d