3.184 \(\int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ -\frac{8 a^2 \cos ^9(c+d x)}{99 d (a \sin (c+d x)+a)^{9/2}}-\frac{2 a \cos ^9(c+d x)}{11 d (a \sin (c+d x)+a)^{7/2}} \]

[Out]

(-8*a^2*Cos[c + d*x]^9)/(99*d*(a + a*Sin[c + d*x])^(9/2)) - (2*a*Cos[c + d*x]^9)/(11*d*(a + a*Sin[c + d*x])^(7
/2))

________________________________________________________________________________________

Rubi [A]  time = 0.11884, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ -\frac{8 a^2 \cos ^9(c+d x)}{99 d (a \sin (c+d x)+a)^{9/2}}-\frac{2 a \cos ^9(c+d x)}{11 d (a \sin (c+d x)+a)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-8*a^2*Cos[c + d*x]^9)/(99*d*(a + a*Sin[c + d*x])^(9/2)) - (2*a*Cos[c + d*x]^9)/(11*d*(a + a*Sin[c + d*x])^(7
/2))

Rule 2674

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 - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), 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]
&& IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]

Rule 2673

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 - 1))/(f*g*(m - 1)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rubi steps

\begin{align*} \int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac{2 a \cos ^9(c+d x)}{11 d (a+a \sin (c+d x))^{7/2}}+\frac{1}{11} (4 a) \int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^{7/2}} \, dx\\ &=-\frac{8 a^2 \cos ^9(c+d x)}{99 d (a+a \sin (c+d x))^{9/2}}-\frac{2 a \cos ^9(c+d x)}{11 d (a+a \sin (c+d x))^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.335438, size = 49, normalized size = 0.78 \[ -\frac{2 (9 \sin (c+d x)+13) \cos ^9(c+d x)}{99 d (\sin (c+d x)+1)^2 (a (\sin (c+d x)+1))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-2*Cos[c + d*x]^9*(13 + 9*Sin[c + d*x]))/(99*d*(1 + Sin[c + d*x])^2*(a*(1 + Sin[c + d*x]))^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.12, size = 57, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ) \left ( \sin \left ( dx+c \right ) -1 \right ) ^{5} \left ( 9\,\sin \left ( dx+c \right ) +13 \right ) }{99\,{a}^{2}\cos \left ( dx+c \right ) d}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8/(a+a*sin(d*x+c))^(5/2),x)

[Out]

2/99/a^2*(1+sin(d*x+c))*(sin(d*x+c)-1)^5*(9*sin(d*x+c)+13)/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{8}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^8/(a*sin(d*x + c) + a)^(5/2), x)

________________________________________________________________________________________

Fricas [B]  time = 2.23003, size = 431, normalized size = 6.84 \begin{align*} -\frac{2 \,{\left (9 \, \cos \left (d x + c\right )^{6} - 23 \, \cos \left (d x + c\right )^{5} - 52 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} +{\left (9 \, \cos \left (d x + c\right )^{5} + 32 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} - 24 \, \cos \left (d x + c\right )^{2} - 32 \, \cos \left (d x + c\right ) - 64\right )} \sin \left (d x + c\right ) + 32 \, \cos \left (d x + c\right ) + 64\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{99 \,{\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-2/99*(9*cos(d*x + c)^6 - 23*cos(d*x + c)^5 - 52*cos(d*x + c)^4 + 4*cos(d*x + c)^3 - 8*cos(d*x + c)^2 + (9*cos
(d*x + c)^5 + 32*cos(d*x + c)^4 - 20*cos(d*x + c)^3 - 24*cos(d*x + c)^2 - 32*cos(d*x + c) - 64)*sin(d*x + c) +
32*cos(d*x + c) + 64)*sqrt(a*sin(d*x + c) + a)/(a^3*d*cos(d*x + c) + a^3*d*sin(d*x + c) + a^3*d)

________________________________________________________________________________________

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)**8/(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.32693, size = 497, normalized size = 7.89 \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)^8/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

1/1584*((((((((((((13*sgn(tan(1/2*d*x + 1/2*c) + 1)*tan(1/2*d*x + 1/2*c)/a^15 - 99*sgn(tan(1/2*d*x + 1/2*c) +
1)/a^15)*tan(1/2*d*x + 1/2*c) + 319*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) - 561*sgn(tan(1/2
*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) + 594*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) -
462*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) + 462*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/
2*d*x + 1/2*c) - 594*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) + 561*sgn(tan(1/2*d*x + 1/2*c) +
1)/a^15)*tan(1/2*d*x + 1/2*c) - 319*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) + 99*sgn(tan(1/2
*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) - 13*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)/(a*tan(1/2*d*x + 1/2*c)
^2 + a)^(11/2) + 64*sqrt(2)*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^(41/2))/d