3.635 \(\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=104 \[ \frac{\cos ^5(c+d x)}{10 a^2 d}+\frac{\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{3 x}{16 a^2} \]

[Out]

(3*x)/(16*a^2) + Cos[c + d*x]^5/(10*a^2*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) + (Cos[c + d*x]^3*Sin[c
+ d*x])/(8*a^2*d) + (Cos[c + d*x]^3*(a - a*Sin[c + d*x])^3)/(6*a^5*d)

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Rubi [A]  time = 0.26827, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2875, 2870, 2669, 2635, 8} \[ \frac{\cos ^5(c+d x)}{10 a^2 d}+\frac{\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{3 x}{16 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x)/(16*a^2) + Cos[c + d*x]^5/(10*a^2*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) + (Cos[c + d*x]^3*Sin[c
+ d*x])/(8*a^2*d) + (Cos[c + d*x]^3*(a - a*Sin[c + d*x])^3)/(6*a^5*d)

Rule 2875

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]

Rule 2870

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

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[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]

Rubi steps

\begin{align*} \int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \cos ^2(c+d x) \sin ^2(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac{\int \cos ^4(c+d x) (a-a \sin (c+d x)) \, dx}{2 a^3}\\ &=\frac{\cos ^5(c+d x)}{10 a^2 d}+\frac{\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac{\int \cos ^4(c+d x) \, dx}{2 a^2}\\ &=\frac{\cos ^5(c+d x)}{10 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{8 a^2 d}+\frac{\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac{3 \int \cos ^2(c+d x) \, dx}{8 a^2}\\ &=\frac{\cos ^5(c+d x)}{10 a^2 d}+\frac{3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{8 a^2 d}+\frac{\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac{3 \int 1 \, dx}{16 a^2}\\ &=\frac{3 x}{16 a^2}+\frac{\cos ^5(c+d x)}{10 a^2 d}+\frac{3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{8 a^2 d}+\frac{\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}\\ \end{align*}

Mathematica [B]  time = 1.95445, size = 362, normalized size = 3.48 \[ \frac{360 d x \sin \left (\frac{c}{2}\right )-240 \sin \left (\frac{c}{2}+d x\right )+240 \sin \left (\frac{3 c}{2}+d x\right )-15 \sin \left (\frac{3 c}{2}+2 d x\right )-15 \sin \left (\frac{5 c}{2}+2 d x\right )-40 \sin \left (\frac{5 c}{2}+3 d x\right )+40 \sin \left (\frac{7 c}{2}+3 d x\right )-45 \sin \left (\frac{7 c}{2}+4 d x\right )-45 \sin \left (\frac{9 c}{2}+4 d x\right )+24 \sin \left (\frac{9 c}{2}+5 d x\right )-24 \sin \left (\frac{11 c}{2}+5 d x\right )+5 \sin \left (\frac{11 c}{2}+6 d x\right )+5 \sin \left (\frac{13 c}{2}+6 d x\right )+360 d x \cos \left (\frac{c}{2}\right )+240 \cos \left (\frac{c}{2}+d x\right )+240 \cos \left (\frac{3 c}{2}+d x\right )-15 \cos \left (\frac{3 c}{2}+2 d x\right )+15 \cos \left (\frac{5 c}{2}+2 d x\right )+40 \cos \left (\frac{5 c}{2}+3 d x\right )+40 \cos \left (\frac{7 c}{2}+3 d x\right )-45 \cos \left (\frac{7 c}{2}+4 d x\right )+45 \cos \left (\frac{9 c}{2}+4 d x\right )-24 \cos \left (\frac{9 c}{2}+5 d x\right )-24 \cos \left (\frac{11 c}{2}+5 d x\right )+5 \cos \left (\frac{11 c}{2}+6 d x\right )-5 \cos \left (\frac{13 c}{2}+6 d x\right )+50 \sin \left (\frac{c}{2}\right )}{1920 a^2 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(360*d*x*Cos[c/2] + 240*Cos[c/2 + d*x] + 240*Cos[(3*c)/2 + d*x] - 15*Cos[(3*c)/2 + 2*d*x] + 15*Cos[(5*c)/2 + 2
*d*x] + 40*Cos[(5*c)/2 + 3*d*x] + 40*Cos[(7*c)/2 + 3*d*x] - 45*Cos[(7*c)/2 + 4*d*x] + 45*Cos[(9*c)/2 + 4*d*x]
- 24*Cos[(9*c)/2 + 5*d*x] - 24*Cos[(11*c)/2 + 5*d*x] + 5*Cos[(11*c)/2 + 6*d*x] - 5*Cos[(13*c)/2 + 6*d*x] + 50*
Sin[c/2] + 360*d*x*Sin[c/2] - 240*Sin[c/2 + d*x] + 240*Sin[(3*c)/2 + d*x] - 15*Sin[(3*c)/2 + 2*d*x] - 15*Sin[(
5*c)/2 + 2*d*x] - 40*Sin[(5*c)/2 + 3*d*x] + 40*Sin[(7*c)/2 + 3*d*x] - 45*Sin[(7*c)/2 + 4*d*x] - 45*Sin[(9*c)/2
+ 4*d*x] + 24*Sin[(9*c)/2 + 5*d*x] - 24*Sin[(11*c)/2 + 5*d*x] + 5*Sin[(11*c)/2 + 6*d*x] + 5*Sin[(13*c)/2 + 6*
d*x])/(1920*a^2*d*(Cos[c/2] + Sin[c/2]))

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Maple [B]  time = 0.085, size = 347, normalized size = 3.3 \begin{align*}{\frac{3}{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{13}{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}}+8\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{6}}}-{\frac{25}{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{16}{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{25}{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}}+{\frac{13}{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{16}{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{3}{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{8}{15\,d{a}^{2}} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}+{\frac{3}{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)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^2,x)

[Out]

3/8/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11-13/24/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+
1/2*c)^9+8/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8-25/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2
*d*x+1/2*c)^7+16/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^6+25/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6
*tan(1/2*d*x+1/2*c)^5+13/24/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3+16/5/d/a^2/(1+tan(1/2*d*x+1/
2*c)^2)^6*tan(1/2*d*x+1/2*c)^2-3/8/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)+8/15/d/a^2/(1+tan(1/2*d
*x+1/2*c)^2)^6+3/8/d/a^2*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.5361, size = 477, normalized size = 4.59 \begin{align*} -\frac{\frac{\frac{45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{384 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{65 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{750 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{640 \, \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{960 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{65 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{45 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 64}{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{45 \, \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)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/120*((45*sin(d*x + c)/(cos(d*x + c) + 1) - 384*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 65*sin(d*x + c)^3/(cos
(d*x + c) + 1)^3 - 750*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 640*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 750*sin
(d*x + c)^7/(cos(d*x + c) + 1)^7 - 960*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 65*sin(d*x + c)^9/(cos(d*x + c) +
1)^9 - 45*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 64)/(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/(c
os(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)
- 45*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

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Fricas [A]  time = 1.06272, size = 188, normalized size = 1.81 \begin{align*} -\frac{96 \, \cos \left (d x + c\right )^{5} - 160 \, \cos \left (d x + c\right )^{3} - 45 \, d x - 5 \,{\left (8 \, \cos \left (d x + c\right )^{5} - 26 \, \cos \left (d x + c\right )^{3} + 9 \, \cos \left (d x + c\right )\right )} \sin \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)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/240*(96*cos(d*x + c)^5 - 160*cos(d*x + c)^3 - 45*d*x - 5*(8*cos(d*x + c)^5 - 26*cos(d*x + c)^3 + 9*cos(d*x
+ c))*sin(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)**6*sin(d*x+c)**2/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.24641, size = 207, normalized size = 1.99 \begin{align*} \frac{\frac{45 \,{\left (d x + c\right )}}{a^{2}} + \frac{2 \,{\left (45 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 65 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 960 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 750 \, \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 )^{6} + 750 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 65 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 384 \, \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 )}^{6} a^{2}}}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(45*(d*x + c)/a^2 + 2*(45*tan(1/2*d*x + 1/2*c)^11 - 65*tan(1/2*d*x + 1/2*c)^9 + 960*tan(1/2*d*x + 1/2*c)
^8 - 750*tan(1/2*d*x + 1/2*c)^7 + 640*tan(1/2*d*x + 1/2*c)^6 + 750*tan(1/2*d*x + 1/2*c)^5 + 65*tan(1/2*d*x + 1
/2*c)^3 + 384*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)^6*a^2))/d