3.157 \(\int \frac{\cos ^6(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{16 a^2 \cos ^7(c+d x)}{99 d (a \sin (c+d x)+a)^{5/2}}-\frac{64 a^3 \cos ^7(c+d x)}{693 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}} \]

[Out]

(-64*a^3*Cos[c + d*x]^7)/(693*d*(a + a*Sin[c + d*x])^(7/2)) - (16*a^2*Cos[c + d*x]^7)/(99*d*(a + a*Sin[c + d*x
])^(5/2)) - (2*a*Cos[c + d*x]^7)/(11*d*(a + a*Sin[c + d*x])^(3/2))

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Rubi [A]  time = 0.170434, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ -\frac{16 a^2 \cos ^7(c+d x)}{99 d (a \sin (c+d x)+a)^{5/2}}-\frac{64 a^3 \cos ^7(c+d x)}{693 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-64*a^3*Cos[c + d*x]^7)/(693*d*(a + a*Sin[c + d*x])^(7/2)) - (16*a^2*Cos[c + d*x]^7)/(99*d*(a + a*Sin[c + d*x
])^(5/2)) - (2*a*Cos[c + d*x]^7)/(11*d*(a + a*Sin[c + d*x])^(3/2))

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 ^6(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=-\frac{2 a \cos ^7(c+d x)}{11 d (a+a \sin (c+d x))^{3/2}}+\frac{1}{11} (8 a) \int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{16 a^2 \cos ^7(c+d x)}{99 d (a+a \sin (c+d x))^{5/2}}-\frac{2 a \cos ^7(c+d x)}{11 d (a+a \sin (c+d x))^{3/2}}+\frac{1}{99} \left (32 a^2\right ) \int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\\ &=-\frac{64 a^3 \cos ^7(c+d x)}{693 d (a+a \sin (c+d x))^{7/2}}-\frac{16 a^2 \cos ^7(c+d x)}{99 d (a+a \sin (c+d x))^{5/2}}-\frac{2 a \cos ^7(c+d x)}{11 d (a+a \sin (c+d x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.245876, size = 59, normalized size = 0.62 \[ -\frac{2 \left (63 \sin ^2(c+d x)+182 \sin (c+d x)+151\right ) \cos ^7(c+d x)}{693 d (\sin (c+d x)+1)^3 \sqrt{a (\sin (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-2*Cos[c + d*x]^7*(151 + 182*Sin[c + d*x] + 63*Sin[c + d*x]^2))/(693*d*(1 + Sin[c + d*x])^3*Sqrt[a*(1 + Sin[c
+ d*x])])

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Maple [A]  time = 0.125, size = 64, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ) \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4} \left ( 63\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+182\,\sin \left ( dx+c \right ) +151 \right ) }{693\,d\cos \left ( dx+c \right ) }{\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)^6/(a+a*sin(d*x+c))^(1/2),x)

[Out]

-2/693*(1+sin(d*x+c))*(sin(d*x+c)-1)^4*(63*sin(d*x+c)^2+182*sin(d*x+c)+151)/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/
d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{6}}{\sqrt{a \sin \left (d x + c\right ) + a}}\,{d x} \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))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^6/sqrt(a*sin(d*x + c) + a), x)

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Fricas [A]  time = 1.89483, size = 432, normalized size = 4.55 \begin{align*} \frac{2 \,{\left (63 \, \cos \left (d x + c\right )^{6} - 7 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{3} + 32 \, \cos \left (d x + c\right )^{2} +{\left (63 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} + 96 \, \cos \left (d x + c\right )^{2} + 128 \, \cos \left (d x + c\right ) + 256\right )} \sin \left (d x + c\right ) - 128 \, \cos \left (d x + c\right ) - 256\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{693 \,{\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\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))^(1/2),x, algorithm="fricas")

[Out]

2/693*(63*cos(d*x + c)^6 - 7*cos(d*x + c)^5 + 10*cos(d*x + c)^4 - 16*cos(d*x + c)^3 + 32*cos(d*x + c)^2 + (63*
cos(d*x + c)^5 + 70*cos(d*x + c)^4 + 80*cos(d*x + c)^3 + 96*cos(d*x + c)^2 + 128*cos(d*x + c) + 256)*sin(d*x +
c) - 128*cos(d*x + c) - 256)*sqrt(a*sin(d*x + c) + a)/(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)

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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))**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.1936, size = 497, normalized size = 5.23 \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)^6/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/11088*((((((((((((151*sgn(tan(1/2*d*x + 1/2*c) + 1)*tan(1/2*d*x + 1/2*c)/a^13 - 693*sgn(tan(1/2*d*x + 1/2*c)
+ 1)/a^13)*tan(1/2*d*x + 1/2*c) + 1177*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^13)*tan(1/2*d*x + 1/2*c) - 1155*sgn(ta
n(1/2*d*x + 1/2*c) + 1)/a^13)*tan(1/2*d*x + 1/2*c) + 1782*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^13)*tan(1/2*d*x + 1/
2*c) - 3234*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^13)*tan(1/2*d*x + 1/2*c) + 3234*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^13
)*tan(1/2*d*x + 1/2*c) - 1782*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^13)*tan(1/2*d*x + 1/2*c) + 1155*sgn(tan(1/2*d*x
+ 1/2*c) + 1)/a^13)*tan(1/2*d*x + 1/2*c) - 1177*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^13)*tan(1/2*d*x + 1/2*c) + 693
*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^13)*tan(1/2*d*x + 1/2*c) - 151*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^13)/(a*tan(1/2
*d*x + 1/2*c)^2 + a)^(11/2) + 256*sqrt(2)*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^(37/2))/d