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

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

[Out]

-64/693*a^3*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(7/2)-16/99*a^2*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(5/2)-2/11*a*cos(d
*x+c)^7/d/(a+a*sin(d*x+c))^(3/2)

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Rubi [A]  time = 0.17, antiderivative size = 95, normalized size of antiderivative = 1.00, 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 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 ^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*}

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Mathematica [A]  time = 0.26, 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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fricas [A]  time = 0.62, size = 155, normalized size = 1.63 \[ \frac {2 \, {\left (63 \, \cos \left (d x + c\right )^{6} - 7 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{3} + 32 \, \cos \left (d x + c\right )^{2} + {\left (63 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} + 96 \, \cos \left (d x + c\right )^{2} + 128 \, \cos \left (d x + c\right ) + 256\right )} \sin \left (d x + c\right ) - 128 \, \cos \left (d x + c\right ) - 256\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{693 \, {\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \]

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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giac [B]  time = 4.62, size = 402, normalized size = 4.23 \[ \frac {2 \, {\left (\frac {256 \, \sqrt {2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a}} - \frac {\frac {151 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {693 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {1177 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {1155 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {1782 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {3234 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {3234 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {1782 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {1155 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {1177 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {151 \, a^{5} \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 {693 \, a^{5}}{\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 )}}{693 \, d} \]

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]

2/693*(256*sqrt(2)*sgn(tan(1/2*d*x + 1/2*c) + 1)/sqrt(a) - (151*a^5/sgn(tan(1/2*d*x + 1/2*c) + 1) - (693*a^5/s
gn(tan(1/2*d*x + 1/2*c) + 1) - (1177*a^5/sgn(tan(1/2*d*x + 1/2*c) + 1) - (1155*a^5/sgn(tan(1/2*d*x + 1/2*c) +
1) - (1782*a^5/sgn(tan(1/2*d*x + 1/2*c) + 1) - (3234*a^5/sgn(tan(1/2*d*x + 1/2*c) + 1) - (3234*a^5/sgn(tan(1/2
*d*x + 1/2*c) + 1) - (1782*a^5/sgn(tan(1/2*d*x + 1/2*c) + 1) - (1155*a^5/sgn(tan(1/2*d*x + 1/2*c) + 1) - (1177
*a^5/sgn(tan(1/2*d*x + 1/2*c) + 1) + (151*a^5*tan(1/2*d*x + 1/2*c)/sgn(tan(1/2*d*x + 1/2*c) + 1) - 693*a^5/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 = 0.20, size = 64, normalized size = 0.67 \[ -\frac {2 \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right )^{4} \left (63 \left (\sin ^{2}\left (d x +c \right )\right )+182 \sin \left (d x +c \right )+151\right )}{693 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{6}}{\sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^6}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(a + a*sin(c + d*x))^(1/2),x)

[Out]

int(cos(c + d*x)^6/(a + a*sin(c + d*x))^(1/2), 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)**6/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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