3.90 \(\int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=58 \[ -\frac {\cos ^7(c+d x)}{63 a d (a \sin (c+d x)+a)^7}-\frac {\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8} \]

[Out]

-1/9*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^8-1/63*cos(d*x+c)^7/a/d/(a+a*sin(d*x+c))^7

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Rubi [A]  time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2672, 2671} \[ -\frac {\cos ^7(c+d x)}{63 a d (a \sin (c+d x)+a)^7}-\frac {\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^8,x]

[Out]

-Cos[c + d*x]^7/(9*d*(a + a*Sin[c + d*x])^8) - Cos[c + d*x]^7/(63*a*d*(a + a*Sin[c + d*x])^7)

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 ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac {\cos ^7(c+d x)}{9 d (a+a \sin (c+d x))^8}+\frac {\int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{9 a}\\ &=-\frac {\cos ^7(c+d x)}{9 d (a+a \sin (c+d x))^8}-\frac {\cos ^7(c+d x)}{63 a d (a+a \sin (c+d x))^7}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 36, normalized size = 0.62 \[ -\frac {(\sin (c+d x)+8) \cos ^7(c+d x)}{63 a^8 d (\sin (c+d x)+1)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^8,x]

[Out]

-1/63*(Cos[c + d*x]^7*(8 + Sin[c + d*x]))/(a^8*d*(1 + Sin[c + d*x])^8)

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fricas [B]  time = 0.59, size = 239, normalized size = 4.12 \[ \frac {\cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{4} + 19 \, \cos \left (d x + c\right )^{3} + 52 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + 24 \, \cos \left (d x + c\right )^{2} - 28 \, \cos \left (d x + c\right ) - 56\right )} \sin \left (d x + c\right ) - 28 \, \cos \left (d x + c\right ) - 56}{63 \, {\left (a^{8} d \cos \left (d x + c\right )^{5} + 5 \, a^{8} d \cos \left (d x + c\right )^{4} - 8 \, a^{8} d \cos \left (d x + c\right )^{3} - 20 \, a^{8} d \cos \left (d x + c\right )^{2} + 8 \, a^{8} d \cos \left (d x + c\right ) + 16 \, a^{8} d + {\left (a^{8} d \cos \left (d x + c\right )^{4} - 4 \, a^{8} d \cos \left (d x + c\right )^{3} - 12 \, a^{8} d \cos \left (d x + c\right )^{2} + 8 \, a^{8} d \cos \left (d x + c\right ) + 16 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/63*(cos(d*x + c)^5 - 4*cos(d*x + c)^4 + 19*cos(d*x + c)^3 + 52*cos(d*x + c)^2 - (cos(d*x + c)^4 + 5*cos(d*x
+ c)^3 + 24*cos(d*x + c)^2 - 28*cos(d*x + c) - 56)*sin(d*x + c) - 28*cos(d*x + c) - 56)/(a^8*d*cos(d*x + c)^5
+ 5*a^8*d*cos(d*x + c)^4 - 8*a^8*d*cos(d*x + c)^3 - 20*a^8*d*cos(d*x + c)^2 + 8*a^8*d*cos(d*x + c) + 16*a^8*d
+ (a^8*d*cos(d*x + c)^4 - 4*a^8*d*cos(d*x + c)^3 - 12*a^8*d*cos(d*x + c)^2 + 8*a^8*d*cos(d*x + c) + 16*a^8*d)*
sin(d*x + c))

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giac [B]  time = 0.61, size = 125, normalized size = 2.16 \[ -\frac {2 \, {\left (63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 483 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 693 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{63 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-2/63*(63*tan(1/2*d*x + 1/2*c)^8 + 63*tan(1/2*d*x + 1/2*c)^7 + 483*tan(1/2*d*x + 1/2*c)^6 + 315*tan(1/2*d*x +
1/2*c)^5 + 693*tan(1/2*d*x + 1/2*c)^4 + 189*tan(1/2*d*x + 1/2*c)^3 + 225*tan(1/2*d*x + 1/2*c)^2 + 9*tan(1/2*d*
x + 1/2*c) + 8)/(a^8*d*(tan(1/2*d*x + 1/2*c) + 1)^9)

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maple [B]  time = 0.28, size = 145, normalized size = 2.50 \[ \frac {-\frac {172}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {256}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {1856}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {152}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {272}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {992}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}}{d \,a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6/(a+a*sin(d*x+c))^8,x)

[Out]

2/d/a^8*(-86/3/(tan(1/2*d*x+1/2*c)+1)^3-128/9/(tan(1/2*d*x+1/2*c)+1)^9-928/7/(tan(1/2*d*x+1/2*c)+1)^7+7/(tan(1
/2*d*x+1/2*c)+1)^2-1/(tan(1/2*d*x+1/2*c)+1)+76/(tan(1/2*d*x+1/2*c)+1)^4-136/(tan(1/2*d*x+1/2*c)+1)^5+64/(tan(1
/2*d*x+1/2*c)+1)^8+496/3/(tan(1/2*d*x+1/2*c)+1)^6)

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maxima [B]  time = 0.39, size = 375, normalized size = 6.47 \[ -\frac {2 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {225 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {189 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {693 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {315 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {483 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {63 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {63 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 8\right )}}{63 \, {\left (a^{8} + \frac {9 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {36 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {126 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {126 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {84 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {36 \, a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {9 \, a^{8} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{8} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-2/63*(9*sin(d*x + c)/(cos(d*x + c) + 1) + 225*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 189*sin(d*x + c)^3/(cos(d
*x + c) + 1)^3 + 693*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 315*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 483*sin(d
*x + c)^6/(cos(d*x + c) + 1)^6 + 63*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 63*sin(d*x + c)^8/(cos(d*x + c) + 1)
^8 + 8)/((a^8 + 9*a^8*sin(d*x + c)/(cos(d*x + c) + 1) + 36*a^8*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 84*a^8*si
n(d*x + c)^3/(cos(d*x + c) + 1)^3 + 126*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 126*a^8*sin(d*x + c)^5/(cos(
d*x + c) + 1)^5 + 84*a^8*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 36*a^8*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 9*
a^8*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a^8*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)*d)

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mupad [B]  time = 6.63, size = 118, normalized size = 2.03 \[ -\frac {\sqrt {2}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {63\,\sin \left (c+d\,x\right )}{2}-\frac {257\,\cos \left (c+d\,x\right )}{8}-\frac {113\,\cos \left (2\,c+2\,d\,x\right )}{4}+\frac {37\,\cos \left (3\,c+3\,d\,x\right )}{8}+\frac {7\,\cos \left (4\,c+4\,d\,x\right )}{16}-\frac {63\,\sin \left (2\,c+2\,d\,x\right )}{8}-\frac {9\,\sin \left (3\,c+3\,d\,x\right )}{2}+\frac {9\,\sin \left (4\,c+4\,d\,x\right )}{16}+\frac {1013}{16}\right )}{1008\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(a + a*sin(c + d*x))^8,x)

[Out]

-(2^(1/2)*cos(c/2 + (d*x)/2)*((63*sin(c + d*x))/2 - (257*cos(c + d*x))/8 - (113*cos(2*c + 2*d*x))/4 + (37*cos(
3*c + 3*d*x))/8 + (7*cos(4*c + 4*d*x))/16 - (63*sin(2*c + 2*d*x))/8 - (9*sin(3*c + 3*d*x))/2 + (9*sin(4*c + 4*
d*x))/16 + 1013/16))/(1008*a^8*d*cos(c/2 - pi/4 + (d*x)/2)^9)

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

[Out]

Timed out

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