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

Optimal. Leaf size=105 \[ \frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {3 \sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac {13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {13 x}{8 a^3} \]

[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

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Rubi [A]  time = 0.22, antiderivative size = 105, normalized size of antiderivative = 1.00, 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} \[ \frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {3 \sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac {13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {13 x}{8 a^3} \]

Antiderivative was successfully verified.

[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 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]

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 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 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]

Rubi steps

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

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Mathematica [B]  time = 1.77, size = 310, normalized size = 2.95 \[ \frac {1560 d x \sin \left (\frac {c}{2}\right )-1380 \sin \left (\frac {c}{2}+d x\right )+1380 \sin \left (\frac {3 c}{2}+d x\right )-480 \sin \left (\frac {3 c}{2}+2 d x\right )-480 \sin \left (\frac {5 c}{2}+2 d x\right )+170 \sin \left (\frac {5 c}{2}+3 d x\right )-170 \sin \left (\frac {7 c}{2}+3 d x\right )+45 \sin \left (\frac {7 c}{2}+4 d x\right )+45 \sin \left (\frac {9 c}{2}+4 d x\right )-6 \sin \left (\frac {9 c}{2}+5 d x\right )+6 \sin \left (\frac {11 c}{2}+5 d x\right )+1560 d x \cos \left (\frac {c}{2}\right )+1380 \cos \left (\frac {c}{2}+d x\right )+1380 \cos \left (\frac {3 c}{2}+d x\right )-480 \cos \left (\frac {3 c}{2}+2 d x\right )+480 \cos \left (\frac {5 c}{2}+2 d x\right )-170 \cos \left (\frac {5 c}{2}+3 d x\right )-170 \cos \left (\frac {7 c}{2}+3 d x\right )+45 \cos \left (\frac {7 c}{2}+4 d x\right )-45 \cos \left (\frac {9 c}{2}+4 d x\right )+6 \cos \left (\frac {9 c}{2}+5 d x\right )+6 \cos \left (\frac {11 c}{2}+5 d x\right )+10 \sin \left (\frac {c}{2}\right )}{960 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

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fricas [A]  time = 0.69, size = 68, normalized size = 0.65 \[ \frac {24 \, \cos \left (d x + c\right )^{5} - 200 \, \cos \left (d x + c\right )^{3} + 195 \, d x + 15 \, {\left (6 \, \cos \left (d x + c\right )^{3} - 19 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 480 \, \cos \left (d x + c\right )}{120 \, a^{3} d} \]

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))^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)

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giac [A]  time = 0.21, size = 127, normalized size = 1.21 \[ \frac {\frac {195 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (195 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 195 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 304\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a^{3}}}{120 \, d} \]

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))^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

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maple [B]  time = 0.43, size = 279, normalized size = 2.66 \[ \frac {13 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {25 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {12 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {116 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {76 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {76}{15 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d} \]

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))^3,x)

[Out]

13/4/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9+25/2/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1
/2*c)^7+12/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^6+116/3/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/
2*d*x+1/2*c)^4-25/2/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^3+76/3/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^
5*tan(1/2*d*x+1/2*c)^2-13/4/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)+76/15/a^3/d/(1+tan(1/2*d*x+1/2
*c)^2)^5+13/4/a^3/d*arctan(tan(1/2*d*x+1/2*c))

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maxima [B]  time = 1.47, size = 290, normalized size = 2.76 \[ -\frac {\frac {\frac {195 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1520 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2320 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {750 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {195 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 304}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {195 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, d} \]

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))^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

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mupad [B]  time = 9.03, size = 95, normalized size = 0.90 \[ \frac {13\,x}{8\,a^3}+\frac {4\,\cos \left (c+d\,x\right )}{a^3\,d}-\frac {5\,{\cos \left (c+d\,x\right )}^3}{3\,a^3\,d}+\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3\,d}+\frac {3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^3\,d}-\frac {19\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))**3,x)

[Out]

Timed out

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