\(\int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx\) [1]

Optimal result

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

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

Mathematica [A] (verified)

Time = 2.59 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13 \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {c^3 \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (405 \cos (2 (e+f x))+90 \cos (4 (e+f x))-5 \cos (6 (e+f x))+1080 \sin (e+f x)+20 \sin (3 (e+f x))-36 \sin (5 (e+f x)))}{960 f} \] Input:

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

Output:

(c^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)
 

Rubi [A] (verified)

Time = 0.71 (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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3320

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3217

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

Input:

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

Output:

(-1/15*(a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(f*Sqrt[a + a*Sin[e + 
 f*x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9 
/2))/(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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

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

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

Output:

-1/30*(5*c^3*cos(f*x + e)^6 - 30*c^3*cos(f*x + e)^4 + 25*c^3 + 2*(9*c^3*co 
s(f*x + e)^4 - 8*c^3*cos(f*x + e)^2 - 16*c^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. \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.11 \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {32 \, {\left (5 \, c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 6 \, c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10}\right )} \sqrt {a} \sqrt {c}}{15 \, f} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 19.63 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.32 \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {c^3\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (405\,\cos \left (e+f\,x\right )+495\,\cos \left (3\,e+3\,f\,x\right )+85\,\cos \left (5\,e+5\,f\,x\right )-5\,\cos \left (7\,e+7\,f\,x\right )+1100\,\sin \left (2\,e+2\,f\,x\right )-16\,\sin \left (4\,e+4\,f\,x\right )-36\,\sin \left (6\,e+6\,f\,x\right )\right )}{960\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \] Input:

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

Output:

(c^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)))/(960 
*f*(cos(2*e + 2*f*x) + 1))
 

Reduce [F]

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

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

Output:

sqrt(c)*sqrt(a)*c**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))