3.92 \(\int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=118 \[ -\frac {2 \cos ^5(c+d x)}{1155 a^3 d (a \sin (c+d x)+a)^5}-\frac {2 \cos ^5(c+d x)}{231 a^2 d (a \sin (c+d x)+a)^6}-\frac {\cos ^5(c+d x)}{33 a d (a \sin (c+d x)+a)^7}-\frac {\cos ^5(c+d x)}{11 d (a \sin (c+d x)+a)^8} \]

[Out]

-1/11*cos(d*x+c)^5/d/(a+a*sin(d*x+c))^8-1/33*cos(d*x+c)^5/a/d/(a+a*sin(d*x+c))^7-2/231*cos(d*x+c)^5/a^2/d/(a+a
*sin(d*x+c))^6-2/1155*cos(d*x+c)^5/a^3/d/(a+a*sin(d*x+c))^5

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Rubi [A]  time = 0.17, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2672, 2671} \[ -\frac {2 \cos ^5(c+d x)}{1155 a^3 d (a \sin (c+d x)+a)^5}-\frac {2 \cos ^5(c+d x)}{231 a^2 d (a \sin (c+d x)+a)^6}-\frac {\cos ^5(c+d x)}{33 a d (a \sin (c+d x)+a)^7}-\frac {\cos ^5(c+d x)}{11 d (a \sin (c+d x)+a)^8} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^8,x]

[Out]

-Cos[c + d*x]^5/(11*d*(a + a*Sin[c + d*x])^8) - Cos[c + d*x]^5/(33*a*d*(a + a*Sin[c + d*x])^7) - (2*Cos[c + d*
x]^5)/(231*a^2*d*(a + a*Sin[c + d*x])^6) - (2*Cos[c + d*x]^5)/(1155*a^3*d*(a + a*Sin[c + d*x])^5)

Rule 2671

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

Rule 2672

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(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] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac {\cos ^5(c+d x)}{11 d (a+a \sin (c+d x))^8}+\frac {3 \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{11 a}\\ &=-\frac {\cos ^5(c+d x)}{11 d (a+a \sin (c+d x))^8}-\frac {\cos ^5(c+d x)}{33 a d (a+a \sin (c+d x))^7}+\frac {2 \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{33 a^2}\\ &=-\frac {\cos ^5(c+d x)}{11 d (a+a \sin (c+d x))^8}-\frac {\cos ^5(c+d x)}{33 a d (a+a \sin (c+d x))^7}-\frac {2 \cos ^5(c+d x)}{231 a^2 d (a+a \sin (c+d x))^6}+\frac {2 \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{231 a^3}\\ &=-\frac {\cos ^5(c+d x)}{11 d (a+a \sin (c+d x))^8}-\frac {\cos ^5(c+d x)}{33 a d (a+a \sin (c+d x))^7}-\frac {2 \cos ^5(c+d x)}{231 a^2 d (a+a \sin (c+d x))^6}-\frac {2 \cos ^5(c+d x)}{1155 a^3 d (a+a \sin (c+d x))^5}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 58, normalized size = 0.49 \[ -\frac {\left (2 \sin ^3(c+d x)+16 \sin ^2(c+d x)+61 \sin (c+d x)+152\right ) \cos ^5(c+d x)}{1155 a^8 d (\sin (c+d x)+1)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^8,x]

[Out]

-1/1155*(Cos[c + d*x]^5*(152 + 61*Sin[c + d*x] + 16*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3))/(a^8*d*(1 + Sin[c + d*
x])^8)

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fricas [B]  time = 0.69, size = 291, normalized size = 2.47 \[ \frac {2 \, \cos \left (d x + c\right )^{6} + 12 \, \cos \left (d x + c\right )^{5} - 25 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{3} - 245 \, \cos \left (d x + c\right )^{2} + {\left (2 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{4} - 35 \, \cos \left (d x + c\right )^{3} + 35 \, \cos \left (d x + c\right )^{2} - 210 \, \cos \left (d x + c\right ) - 420\right )} \sin \left (d x + c\right ) + 210 \, \cos \left (d x + c\right ) + 420}{1155 \, {\left (a^{8} d \cos \left (d x + c\right )^{6} - 5 \, a^{8} d \cos \left (d x + c\right )^{5} - 18 \, a^{8} d \cos \left (d x + c\right )^{4} + 20 \, a^{8} d \cos \left (d x + c\right )^{3} + 48 \, a^{8} d \cos \left (d x + c\right )^{2} - 16 \, a^{8} d \cos \left (d x + c\right ) - 32 \, a^{8} d - {\left (a^{8} d \cos \left (d x + c\right )^{5} + 6 \, a^{8} d \cos \left (d x + c\right )^{4} - 12 \, a^{8} d \cos \left (d x + c\right )^{3} - 32 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, a^{8} d \cos \left (d x + c\right ) + 32 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/1155*(2*cos(d*x + c)^6 + 12*cos(d*x + c)^5 - 25*cos(d*x + c)^4 - 70*cos(d*x + c)^3 - 245*cos(d*x + c)^2 + (2
*cos(d*x + c)^5 - 10*cos(d*x + c)^4 - 35*cos(d*x + c)^3 + 35*cos(d*x + c)^2 - 210*cos(d*x + c) - 420)*sin(d*x
+ c) + 210*cos(d*x + c) + 420)/(a^8*d*cos(d*x + c)^6 - 5*a^8*d*cos(d*x + c)^5 - 18*a^8*d*cos(d*x + c)^4 + 20*a
^8*d*cos(d*x + c)^3 + 48*a^8*d*cos(d*x + c)^2 - 16*a^8*d*cos(d*x + c) - 32*a^8*d - (a^8*d*cos(d*x + c)^5 + 6*a
^8*d*cos(d*x + c)^4 - 12*a^8*d*cos(d*x + c)^3 - 32*a^8*d*cos(d*x + c)^2 + 16*a^8*d*cos(d*x + c) + 32*a^8*d)*si
n(d*x + c))

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giac [A]  time = 0.60, size = 151, normalized size = 1.28 \[ -\frac {2 \, {\left (1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13860 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 23100 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 37422 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 32802 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 27060 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 11220 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4895 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 517 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 152\right )}}{1155 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-2/1155*(1155*tan(1/2*d*x + 1/2*c)^10 + 3465*tan(1/2*d*x + 1/2*c)^9 + 13860*tan(1/2*d*x + 1/2*c)^8 + 23100*tan
(1/2*d*x + 1/2*c)^7 + 37422*tan(1/2*d*x + 1/2*c)^6 + 32802*tan(1/2*d*x + 1/2*c)^5 + 27060*tan(1/2*d*x + 1/2*c)
^4 + 11220*tan(1/2*d*x + 1/2*c)^3 + 4895*tan(1/2*d*x + 1/2*c)^2 + 517*tan(1/2*d*x + 1/2*c) + 152)/(a^8*d*(tan(
1/2*d*x + 1/2*c) + 1)^11)

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maple [A]  time = 0.30, size = 175, normalized size = 1.48 \[ \frac {\frac {584}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {576}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {60}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {256}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1024}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4752}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {176}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1864}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}}{d \,a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4/(a+a*sin(d*x+c))^8,x)

[Out]

2/d/a^8*(292/(tan(1/2*d*x+1/2*c)+1)^6+288/(tan(1/2*d*x+1/2*c)+1)^8-30/(tan(1/2*d*x+1/2*c)+1)^3-128/11/(tan(1/2
*d*x+1/2*c)+1)^11+7/(tan(1/2*d*x+1/2*c)+1)^2-512/3/(tan(1/2*d*x+1/2*c)+1)^9-2376/7/(tan(1/2*d*x+1/2*c)+1)^7+64
/(tan(1/2*d*x+1/2*c)+1)^10-1/(tan(1/2*d*x+1/2*c)+1)+88/(tan(1/2*d*x+1/2*c)+1)^4-932/5/(tan(1/2*d*x+1/2*c)+1)^5
)

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maxima [B]  time = 0.42, size = 461, normalized size = 3.91 \[ -\frac {2 \, {\left (\frac {517 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4895 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {11220 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {27060 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {32802 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {37422 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {23100 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {13860 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {3465 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {1155 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 152\right )}}{1155 \, {\left (a^{8} + \frac {11 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {55 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {165 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {330 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {462 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {462 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {330 \, a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {165 \, a^{8} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {55 \, a^{8} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {11 \, a^{8} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{8} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-2/1155*(517*sin(d*x + c)/(cos(d*x + c) + 1) + 4895*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 11220*sin(d*x + c)^3
/(cos(d*x + c) + 1)^3 + 27060*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 32802*sin(d*x + c)^5/(cos(d*x + c) + 1)^5
+ 37422*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 23100*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 13860*sin(d*x + c)^8
/(cos(d*x + c) + 1)^8 + 3465*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 1155*sin(d*x + c)^10/(cos(d*x + c) + 1)^10
+ 152)/((a^8 + 11*a^8*sin(d*x + c)/(cos(d*x + c) + 1) + 55*a^8*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 165*a^8*s
in(d*x + c)^3/(cos(d*x + c) + 1)^3 + 330*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 462*a^8*sin(d*x + c)^5/(cos
(d*x + c) + 1)^5 + 462*a^8*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 330*a^8*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 +
165*a^8*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 55*a^8*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 11*a^8*sin(d*x + c
)^10/(cos(d*x + c) + 1)^10 + a^8*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)*d)

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mupad [B]  time = 7.15, size = 140, normalized size = 1.19 \[ -\frac {\sqrt {2}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {7623\,\sin \left (c+d\,x\right )}{4}-697\,\cos \left (c+d\,x\right )-\frac {3977\,\cos \left (2\,c+2\,d\,x\right )}{4}+\frac {3203\,\cos \left (3\,c+3\,d\,x\right )}{16}+\frac {461\,\cos \left (4\,c+4\,d\,x\right )}{8}-\frac {75\,\cos \left (5\,c+5\,d\,x\right )}{16}-462\,\sin \left (2\,c+2\,d\,x\right )-\frac {4983\,\sin \left (3\,c+3\,d\,x\right )}{16}+\frac {187\,\sin \left (4\,c+4\,d\,x\right )}{4}+\frac {77\,\sin \left (5\,c+5\,d\,x\right )}{16}+\frac {12721}{8}\right )}{36960\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^4/(a + a*sin(c + d*x))^8,x)

[Out]

-(2^(1/2)*cos(c/2 + (d*x)/2)*((7623*sin(c + d*x))/4 - 697*cos(c + d*x) - (3977*cos(2*c + 2*d*x))/4 + (3203*cos
(3*c + 3*d*x))/16 + (461*cos(4*c + 4*d*x))/8 - (75*cos(5*c + 5*d*x))/16 - 462*sin(2*c + 2*d*x) - (4983*sin(3*c
+ 3*d*x))/16 + (187*sin(4*c + 4*d*x))/4 + (77*sin(5*c + 5*d*x))/16 + 12721/8))/(36960*a^8*d*cos(c/2 - pi/4 +
(d*x)/2)^11)

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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)**4/(a+a*sin(d*x+c))**8,x)

[Out]

Timed out

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