3.2.2 \(\int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\) [102]

3.2.2.1 Optimal result

Integrand size = 23, antiderivative size = 127 \[ \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {256 a^4 \cos ^7(c+d x)}{3003 d (a+a \sin (c+d x))^{7/2}}-\frac {64 a^3 \cos ^7(c+d x)}{429 d (a+a \sin (c+d x))^{5/2}}-\frac {24 a^2 \cos ^7(c+d x)}{143 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a+a \sin (c+d x)}} \]

output

-256/3003*a^4*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(7/2)-64/429*a^3*cos(d*x+c)^ 
7/d/(a+a*sin(d*x+c))^(5/2)-24/143*a^2*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(3/2 
)-2/13*a*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(1/2)
 

3.2.2.2 Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.54 \[ \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \cos ^7(c+d x) \sqrt {a (1+\sin (c+d x))} \left (835+1421 \sin (c+d x)+945 \sin ^2(c+d x)+231 \sin ^3(c+d x)\right )}{3003 d (1+\sin (c+d x))^4} \]

input

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

output

(-2*Cos[c + d*x]^7*Sqrt[a*(1 + Sin[c + d*x])]*(835 + 1421*Sin[c + d*x] + 9 
45*Sin[c + d*x]^2 + 231*Sin[c + d*x]^3))/(3003*d*(1 + Sin[c + d*x])^4)
 

3.2.2.3 Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 3153, 3042, 3153, 3042, 3153, 3042, 3152}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3153

\(\displaystyle \frac {12}{13} a \int \frac {\cos ^6(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {12}{13} a \int \frac {\cos (c+d x)^6}{\sqrt {\sin (c+d x) a+a}}dx-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 3153

\(\displaystyle \frac {12}{13} a \left (\frac {8}{11} a \int \frac {\cos ^6(c+d x)}{(\sin (c+d x) a+a)^{3/2}}dx-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {12}{13} a \left (\frac {8}{11} a \int \frac {\cos (c+d x)^6}{(\sin (c+d x) a+a)^{3/2}}dx-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 3153

\(\displaystyle \frac {12}{13} a \left (\frac {8}{11} a \left (\frac {4}{9} a \int \frac {\cos ^6(c+d x)}{(\sin (c+d x) a+a)^{5/2}}dx-\frac {2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}\right )-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {12}{13} a \left (\frac {8}{11} a \left (\frac {4}{9} a \int \frac {\cos (c+d x)^6}{(\sin (c+d x) a+a)^{5/2}}dx-\frac {2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}\right )-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 3152

\(\displaystyle \frac {12}{13} a \left (\frac {8}{11} a \left (-\frac {8 a^2 \cos ^7(c+d x)}{63 d (a \sin (c+d x)+a)^{7/2}}-\frac {2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}\right )-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\)

input

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

output

(-2*a*Cos[c + d*x]^7)/(13*d*Sqrt[a + a*Sin[c + d*x]]) + (12*a*((-2*a*Cos[c 
 + d*x]^7)/(11*d*(a + a*Sin[c + d*x])^(3/2)) + (8*a*((-8*a^2*Cos[c + d*x]^ 
7)/(63*d*(a + a*Sin[c + d*x])^(7/2)) - (2*a*Cos[c + d*x]^7)/(9*d*(a + a*Si 
n[c + d*x])^(5/2))))/11))/13
 

3.2.2.3.1 Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3152

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] && EqQ[a^2 
 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
 

rule 3153

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] + Simp[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]
 

3.2.2.4 Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.59

method	result	size
default	\(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right )^{4} \left (231 \left (\sin ^{3}\left (d x +c \right )\right )+945 \left (\sin ^{2}\left (d x +c \right )\right )+1421 \sin \left (d x +c \right )+835\right )}{3003 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(75\)

input

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

output

-2/3003*(1+sin(d*x+c))*a*(sin(d*x+c)-1)^4*(231*sin(d*x+c)^3+945*sin(d*x+c) 
^2+1421*sin(d*x+c)+835)/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
 

3.2.2.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.35 \[ \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \, {\left (231 \, \cos \left (d x + c\right )^{7} - 21 \, \cos \left (d x + c\right )^{6} + 28 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{4} + 64 \, \cos \left (d x + c\right )^{3} - 128 \, \cos \left (d x + c\right )^{2} - {\left (231 \, \cos \left (d x + c\right )^{6} + 252 \, \cos \left (d x + c\right )^{5} + 280 \, \cos \left (d x + c\right )^{4} + 320 \, \cos \left (d x + c\right )^{3} + 384 \, \cos \left (d x + c\right )^{2} + 512 \, \cos \left (d x + c\right ) + 1024\right )} \sin \left (d x + c\right ) + 512 \, \cos \left (d x + c\right ) + 1024\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3003 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]

input

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")
 

output

-2/3003*(231*cos(d*x + c)^7 - 21*cos(d*x + c)^6 + 28*cos(d*x + c)^5 - 40*c 
os(d*x + c)^4 + 64*cos(d*x + c)^3 - 128*cos(d*x + c)^2 - (231*cos(d*x + c) 
^6 + 252*cos(d*x + c)^5 + 280*cos(d*x + c)^4 + 320*cos(d*x + c)^3 + 384*co 
s(d*x + c)^2 + 512*cos(d*x + c) + 1024)*sin(d*x + c) + 512*cos(d*x + c) + 
1024)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)
 

3.2.2.6 Sympy [F]

\[ \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \cos ^{6}{\left (c + d x \right )}\, dx \]

input

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

output

Integral(sqrt(a*(sin(c + d*x) + 1))*cos(c + d*x)**6, x)
 

3.2.2.7 Maxima [F]

\[ \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{6} \,d x } \]

input

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")
 

output

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

3.2.2.8 Giac [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.58 \[ \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} {\left (60060 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15015 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - 9009 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2574 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 2002 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right ) - 273 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right ) - 231 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {13}{4} \, \pi + \frac {13}{2} \, d x + \frac {13}{2} \, c\right )\right )} \sqrt {a}}{96096 \, d} \]

input

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")
 

output

1/96096*sqrt(2)*(60060*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1 
/2*d*x + 1/2*c) - 15015*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-3/4*pi + 
3/2*d*x + 3/2*c) - 9009*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-5/4*pi + 
5/2*d*x + 5/2*c) + 2574*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-7/4*pi + 
7/2*d*x + 7/2*c) + 2002*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-9/4*pi + 
9/2*d*x + 9/2*c) - 273*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-11/4*pi + 
11/2*d*x + 11/2*c) - 231*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-13/4*pi 
+ 13/2*d*x + 13/2*c))*sqrt(a)/d
 

3.2.2.9 Mupad [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^6\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]

input

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

output

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