3.91 \(\int \frac {\cos ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=65 \[ -\frac {1}{3 a^5 d (a \sin (c+d x)+a)^3}-\frac {4}{5 a^3 d (a \sin (c+d x)+a)^5}+\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )^4} \]

[Out]

-4/5/a^3/d/(a+a*sin(d*x+c))^5-1/3/a^5/d/(a+a*sin(d*x+c))^3+1/d/(a^2+a^2*sin(d*x+c))^4

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Rubi [A]  time = 0.06, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ -\frac {1}{3 a^5 d (a \sin (c+d x)+a)^3}+\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {4}{5 a^3 d (a \sin (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]

[Out]

-4/(5*a^3*d*(a + a*Sin[c + d*x])^5) - 1/(3*a^5*d*(a + a*Sin[c + d*x])^3) + 1/(d*(a^2 + a^2*Sin[c + d*x])^4)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2}{(a+x)^6} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {4 a^2}{(a+x)^6}-\frac {4 a}{(a+x)^5}+\frac {1}{(a+x)^4}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac {4}{5 a^3 d (a+a \sin (c+d x))^5}-\frac {1}{3 a^5 d (a+a \sin (c+d x))^3}+\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )^4}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 58, normalized size = 0.89 \[ \frac {\left (5 \sin ^2(c+d x)-5 \sin (c+d x)+2\right ) \cos ^6(c+d x)}{15 a^8 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]

[Out]

(Cos[c + d*x]^6*(2 - 5*Sin[c + d*x] + 5*Sin[c + d*x]^2))/(15*a^8*d*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^8)

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fricas [A]  time = 0.65, size = 100, normalized size = 1.54 \[ \frac {5 \, \cos \left (d x + c\right )^{2} + 5 \, \sin \left (d x + c\right ) - 7}{15 \, {\left (5 \, a^{8} d \cos \left (d x + c\right )^{4} - 20 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, a^{8} d + {\left (a^{8} d \cos \left (d x + c\right )^{4} - 12 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/15*(5*cos(d*x + c)^2 + 5*sin(d*x + c) - 7)/(5*a^8*d*cos(d*x + c)^4 - 20*a^8*d*cos(d*x + c)^2 + 16*a^8*d + (a
^8*d*cos(d*x + c)^4 - 12*a^8*d*cos(d*x + c)^2 + 16*a^8*d)*sin(d*x + c))

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giac [B]  time = 1.44, size = 137, normalized size = 2.11 \[ \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 30 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 170 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 282 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 170 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{15 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

2/15*(15*tan(1/2*d*x + 1/2*c)^9 + 30*tan(1/2*d*x + 1/2*c)^8 + 140*tan(1/2*d*x + 1/2*c)^7 + 170*tan(1/2*d*x + 1
/2*c)^6 + 282*tan(1/2*d*x + 1/2*c)^5 + 170*tan(1/2*d*x + 1/2*c)^4 + 140*tan(1/2*d*x + 1/2*c)^3 + 30*tan(1/2*d*
x + 1/2*c)^2 + 15*tan(1/2*d*x + 1/2*c))/(a^8*d*(tan(1/2*d*x + 1/2*c) + 1)^10)

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maple [A]  time = 0.25, size = 43, normalized size = 0.66 \[ \frac {-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {1}{\left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {4}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}}{d \,a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+a*sin(d*x+c))^8,x)

[Out]

1/d/a^8*(-1/3/(1+sin(d*x+c))^3+1/(1+sin(d*x+c))^4-4/5/(1+sin(d*x+c))^5)

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maxima [A]  time = 0.32, size = 93, normalized size = 1.43 \[ -\frac {5 \, \sin \left (d x + c\right )^{2} - 5 \, \sin \left (d x + c\right ) + 2}{15 \, {\left (a^{8} \sin \left (d x + c\right )^{5} + 5 \, a^{8} \sin \left (d x + c\right )^{4} + 10 \, a^{8} \sin \left (d x + c\right )^{3} + 10 \, a^{8} \sin \left (d x + c\right )^{2} + 5 \, a^{8} \sin \left (d x + c\right ) + a^{8}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/15*(5*sin(d*x + c)^2 - 5*sin(d*x + c) + 2)/((a^8*sin(d*x + c)^5 + 5*a^8*sin(d*x + c)^4 + 10*a^8*sin(d*x + c
)^3 + 10*a^8*sin(d*x + c)^2 + 5*a^8*sin(d*x + c) + a^8)*d)

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mupad [B]  time = 4.71, size = 54, normalized size = 0.83 \[ \frac {1}{a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^4}-\frac {1}{3\,a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^3}-\frac {4}{5\,a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5/(a + a*sin(c + d*x))^8,x)

[Out]

1/(a^8*d*(sin(c + d*x) + 1)^4) - 1/(3*a^8*d*(sin(c + d*x) + 1)^3) - 4/(5*a^8*d*(sin(c + d*x) + 1)^5)

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sympy [A]  time = 42.11, size = 1120, normalized size = 17.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5/(a+a*sin(d*x+c))**8,x)

[Out]

Piecewise((-8*sin(c + d*x)**4/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 2205*a**8*d*sin(c + d
*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c + d*x)**2 + 735*a**8*d*
sin(c + d*x) + 105*a**8*d) - 21*sin(c + d*x)**3/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 220
5*a**8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c + d*x
)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d) + 12*sin(c + d*x)**2*cos(c + d*x)**2/(105*a**8*d*sin(c + d*x)**7
+ 735*a**8*d*sin(c + d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c +
d*x)**3 + 2205*a**8*d*sin(c + d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d) - 19*sin(c + d*x)**2/(105*a**8*
d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3
675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c + d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d) + 14*sin(c +
d*x)*cos(c + d*x)**2/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 +
3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c + d*x)**2 + 735*a**8*d*sin(c + d
*x) + 105*a**8*d) - 7*sin(c + d*x)/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 2205*a**8*d*sin(
c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c + d*x)**2 + 735*a*
*8*d*sin(c + d*x) + 105*a**8*d) - 15*cos(c + d*x)**4/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6
+ 2205*a**8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c
+ d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d) + 2*cos(c + d*x)**2/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d
*sin(c + d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2
205*a**8*d*sin(c + d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d) - 1/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*
d*sin(c + d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 +
2205*a**8*d*sin(c + d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d), Ne(d, 0)), (x*cos(c)**5/(a*sin(c) + a)**8
, True))

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