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]

-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)

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Rubi [A]  time = 0.16757, antiderivative size = 118, normalized size of antiderivative = 1., 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 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 ^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*}

Mathematica [A]  time = 0.0818712, 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]

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

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Maple [A]  time = 0.141, size = 175, normalized size = 1.5 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ( 64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-10}-{\frac{2376}{7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}-30\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-3}+292\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-6}-{\frac{512}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{9}}}- \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-1}-{\frac{932}{5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}+7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}+88\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-4}-{\frac{128}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{11}}}+288\, \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)^4/(a+a*sin(d*x+c))^8,x)

[Out]

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

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Maxima [B]  time = 1.03692, size = 622, normalized size = 5.27 \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [B]  time = 1.62082, size = 760, normalized size = 6.44 \begin{align*} \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 )}} \end{align*}

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

[Out]

Timed out

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Giac [A]  time = 1.19008, size = 204, normalized size = 1.73 \begin{align*} -\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}} \end{align*}

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)