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]

-8/99*a^2*cos(d*x+c)^9/d/(a+a*sin(d*x+c))^(9/2)-2/11*a*cos(d*x+c)^9/d/(a+a*sin(d*x+c))^(7/2)

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Rubi [A]  time = 0.12, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.35, 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))

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fricas [B]  time = 0.76, size = 161, normalized size = 2.56 \[ -\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{6} - 23 \, \cos \left (d x + c\right )^{5} - 52 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} + {\left (9 \, \cos \left (d x + c\right )^{5} + 32 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} - 24 \, \cos \left (d x + c\right )^{2} - 32 \, \cos \left (d x + c\right ) - 64\right )} \sin \left (d x + c\right ) + 32 \, \cos \left (d x + c\right ) + 64\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{99 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]

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)

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giac [B]  time = 3.16, size = 402, normalized size = 6.38 \[ \frac {2 \, {\left (\frac {64 \, \sqrt {2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}}} - \frac {\frac {13 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {99 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {319 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {561 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {594 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {462 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {462 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {594 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {561 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {319 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {13 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - \frac {99 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {11}{2}}}\right )}}{99 \, d} \]

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]

2/99*(64*sqrt(2)*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^(5/2) - (13*a^3/sgn(tan(1/2*d*x + 1/2*c) + 1) - (99*a^3/sgn(t
an(1/2*d*x + 1/2*c) + 1) - (319*a^3/sgn(tan(1/2*d*x + 1/2*c) + 1) - (561*a^3/sgn(tan(1/2*d*x + 1/2*c) + 1) - (
594*a^3/sgn(tan(1/2*d*x + 1/2*c) + 1) - (462*a^3/sgn(tan(1/2*d*x + 1/2*c) + 1) - (462*a^3/sgn(tan(1/2*d*x + 1/
2*c) + 1) - (594*a^3/sgn(tan(1/2*d*x + 1/2*c) + 1) - (561*a^3/sgn(tan(1/2*d*x + 1/2*c) + 1) - (319*a^3/sgn(tan
(1/2*d*x + 1/2*c) + 1) + (13*a^3*tan(1/2*d*x + 1/2*c)/sgn(tan(1/2*d*x + 1/2*c) + 1) - 99*a^3/sgn(tan(1/2*d*x +
1/2*c) + 1))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*
d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*
d*x + 1/2*c))/(a*tan(1/2*d*x + 1/2*c)^2 + a)^(11/2))/d

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maple [A]  time = 0.21, size = 57, normalized size = 0.90 \[ \frac {2 \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right )^{5} \left (9 \sin \left (d x +c \right )+13\right )}{99 a^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{8}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\cos \left (c+d\,x\right )}^8}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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