\(\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx\) [34]

Optimal result

Integrand size = 38, antiderivative size = 92 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {c \cos (e+f x) (a+a \sin (e+f x))^{9/2}}{15 a f \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)}}{6 a f} \] Output:

1/15*c*cos(f*x+e)*(a+a*sin(f*x+e))^(9/2)/a/f/(c-c*sin(f*x+e))^(1/2)+1/6*co 
s(f*x+e)*(a+a*sin(f*x+e))^(9/2)*(c-c*sin(f*x+e))^(1/2)/a/f
 

Mathematica [A] (verified)

Time = 2.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {a^3 \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (-405 \cos (2 (e+f x))-90 \cos (4 (e+f x))+5 \cos (6 (e+f x))+1080 \sin (e+f x)+20 \sin (3 (e+f x))-36 \sin (5 (e+f x)))}{960 f} \] Input:

Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x 
]],x]
 

Output:

(a^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-40 
5*Cos[2*(e + f*x)] - 90*Cos[4*(e + f*x)] + 5*Cos[6*(e + f*x)] + 1080*Sin[e 
 + f*x] + 20*Sin[3*(e + f*x)] - 36*Sin[5*(e + f*x)]))/(960*f)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3042, 3320, 3042, 3219, 3042, 3217}

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.

\(\displaystyle \int \cos ^2(e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (e+f x)^2 (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)}dx\)

\(\Big \downarrow \) 3320

\(\displaystyle \frac {\int (\sin (e+f x) a+a)^{9/2} (c-c \sin (e+f x))^{3/2}dx}{a c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (\sin (e+f x) a+a)^{9/2} (c-c \sin (e+f x))^{3/2}dx}{a c}\)

\(\Big \downarrow \) 3219

\(\displaystyle \frac {\frac {1}{3} c \int (\sin (e+f x) a+a)^{9/2} \sqrt {c-c \sin (e+f x)}dx+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt {c-c \sin (e+f x)}}{6 f}}{a c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} c \int (\sin (e+f x) a+a)^{9/2} \sqrt {c-c \sin (e+f x)}dx+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt {c-c \sin (e+f x)}}{6 f}}{a c}\)

\(\Big \downarrow \) 3217

\(\displaystyle \frac {\frac {c^2 \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{15 f \sqrt {c-c \sin (e+f x)}}+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt {c-c \sin (e+f x)}}{6 f}}{a c}\)

Input:

Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]],x]
 

Output:

((c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(15*f*Sqrt[c - c*Sin[e + f* 
x]]) + (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*Sqrt[c - c*Sin[e + f*x]] 
)/(6*f))/(a*c)
 

Defintions of rubi rules used

rule 3042

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

rule 3217

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f 
_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^ 
n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f, n 
}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]
 

rule 3219

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^ 
(m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[a*((2*m - 1)/(m + n 
))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; Fre 
eQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I 
GtQ[m - 1/2, 0] &&  !LtQ[n, -1] &&  !(IGtQ[n - 1/2, 0] && LtQ[n, m]) &&  !( 
ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
 

rule 3320

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ 
.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(a^(p/ 
2)*c^(p/2))   Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(n + 
p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && 
EqQ[a^2 - b^2, 0] && IntegerQ[p/2]
 

Maple [F]

\[\int \cos \left (f x +e \right )^{2} \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {c -c \sin \left (f x +e \right )}d x\]

Input:

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x)
 

Output:

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {{\left (5 \, a^{3} \cos \left (f x + e\right )^{6} - 30 \, a^{3} \cos \left (f x + e\right )^{4} + 25 \, a^{3} - 2 \, {\left (9 \, a^{3} \cos \left (f x + e\right )^{4} - 8 \, a^{3} \cos \left (f x + e\right )^{2} - 16 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{30 \, f \cos \left (f x + e\right )} \] Input:

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, al 
gorithm="fricas")
 

Output:

1/30*(5*a^3*cos(f*x + e)^6 - 30*a^3*cos(f*x + e)^4 + 25*a^3 - 2*(9*a^3*cos 
(f*x + e)^4 - 8*a^3*cos(f*x + e)^2 - 16*a^3)*sin(f*x + e))*sqrt(a*sin(f*x 
+ e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**(7/2)*(c-c*sin(f*x+e))**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \sqrt {-c \sin \left (f x + e\right ) + c} \cos \left (f x + e\right )^{2} \,d x } \] Input:

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, al 
gorithm="maxima")
 

Output:

integrate((a*sin(f*x + e) + a)^(7/2)*sqrt(-c*sin(f*x + e) + c)*cos(f*x + e 
)^2, x)
 

Giac [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.11 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {32 \, {\left (5 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 6 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a} \sqrt {c}}{15 \, f} \] Input:

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, al 
gorithm="giac")
 

Output:

32/15*(5*a^3*cos(-1/4*pi + 1/2*f*x + 1/2*e)^12*sgn(cos(-1/4*pi + 1/2*f*x + 
 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 6*a^3*cos(-1/4*pi + 1/2*f*x 
 + 1/2*e)^10*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x 
 + 1/2*e)))*sqrt(a)*sqrt(c)/f
 

Mupad [B] (verification not implemented)

Time = 19.70 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.32 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {a^3\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (405\,\cos \left (e+f\,x\right )+495\,\cos \left (3\,e+3\,f\,x\right )+85\,\cos \left (5\,e+5\,f\,x\right )-5\,\cos \left (7\,e+7\,f\,x\right )-1100\,\sin \left (2\,e+2\,f\,x\right )+16\,\sin \left (4\,e+4\,f\,x\right )+36\,\sin \left (6\,e+6\,f\,x\right )\right )}{960\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \] Input:

int(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(1/2),x 
)
 

Output:

-(a^3*(a*(sin(e + f*x) + 1))^(1/2)*(-c*(sin(e + f*x) - 1))^(1/2)*(405*cos( 
e + f*x) + 495*cos(3*e + 3*f*x) + 85*cos(5*e + 5*f*x) - 5*cos(7*e + 7*f*x) 
 - 1100*sin(2*e + 2*f*x) + 16*sin(4*e + 4*f*x) + 36*sin(6*e + 6*f*x)))/(96 
0*f*(cos(2*e + 2*f*x) + 1))
 

Reduce [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\sqrt {c}\, \sqrt {a}\, a^{3} \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{3}d x +3 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{2}d x \right )+3 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )d x \right )+\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2}d x \right ) \] Input:

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x)
 

Output:

sqrt(c)*sqrt(a)*a**3*(int(sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1) 
*cos(e + f*x)**2*sin(e + f*x)**3,x) + 3*int(sqrt(sin(e + f*x) + 1)*sqrt( - 
 sin(e + f*x) + 1)*cos(e + f*x)**2*sin(e + f*x)**2,x) + 3*int(sqrt(sin(e + 
 f*x) + 1)*sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2*sin(e + f*x),x) + int 
(sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2,x))