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]

-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)

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Rubi [A]  time = 0.0803006, antiderivative size = 58, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0830468, 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]

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

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Maple [B]  time = 0.131, size = 145, normalized size = 2.5 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ( 7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}+{\frac{496}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{928}{7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}-{\frac{86}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}-{\frac{128}{9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{9}}}- \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-1}-136\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-5}+76\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-4}+64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-8} \right ) } \end{align*}

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

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Maxima [B]  time = 1.02298, size = 506, normalized size = 8.72 \begin{align*} -\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} \end{align*}

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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Fricas [B]  time = 1.54647, size = 614, normalized size = 10.59 \begin{align*} \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 )}} \end{align*}

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

[Out]

Timed out

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Giac [B]  time = 1.20341, size = 169, normalized size = 2.91 \begin{align*} -\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}} \end{align*}

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)