\(\int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx\) [1190]

Optimal result

Integrand size = 35, antiderivative size = 510 \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{3/2}}+\frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 \left (a^2-b^2\right )^2 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {32 b \left (2 a^2-b^2\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{35 a^5 (a-b) (a+b)^{3/2} \sqrt {d} f}-\frac {8 \left (5 a^2-3 a b-4 b^2\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{35 a^4 (a-b) (a+b)^{3/2} \sqrt {d} f} \]

[Out]

2/7*cos(f*x+e)^3*(d*sin(f*x+e))^(1/2)/a/d/f/(a+b*sin(f*x+e))^(7/2)+12/35*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^2/d
/f/(a+b*sin(f*x+e))^(5/2)+8/35*(a^2-2*b^2)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^3/(a^2-b^2)/d/f/(a+b*sin(f*x+e))^
(3/2)+32/35*b*(2*a^2-b^2)*cos(f*x+e)/a^3/(a^2-b^2)^2/f/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)-32/35*b*(2*
a^2-b^2)*EllipticE(d^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2))*(a*(1
-csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/a^5/(a-b)/(a+b)^(3/2)/f/d^(1/2)-8/35*(5*a^
2-3*a*b-4*b^2)*EllipticF(d^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2))
*(a*(1-csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/a^4/(a-b)/(a+b)^(3/2)/f/d^(1/2)

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2966, 2968, 3135, 3072, 3077, 2895, 3073} \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}-\frac {32 b \left (2 a^2-b^2\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{35 a^5 \sqrt {d} f (a-b) (a+b)^{3/2}}-\frac {8 \left (5 a^2-3 a b-4 b^2\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{35 a^4 \sqrt {d} f (a-b) (a+b)^{3/2}}+\frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 f \left (a^2-b^2\right )^2 \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}} \]

[In]

Int[Cos[e + f*x]^4/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2)),x]

[Out]

(2*Cos[e + f*x]^3*Sqrt[d*Sin[e + f*x]])/(7*a*d*f*(a + b*Sin[e + f*x])^(7/2)) + (12*Cos[e + f*x]*Sqrt[d*Sin[e +
f*x]])/(35*a^2*d*f*(a + b*Sin[e + f*x])^(5/2)) + (8*(a^2 - 2*b^2)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(35*a^3*
(a^2 - b^2)*d*f*(a + b*Sin[e + f*x])^(3/2)) + (32*b*(2*a^2 - b^2)*Cos[e + f*x])/(35*a^3*(a^2 - b^2)^2*f*Sqrt[d
*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (32*b*(2*a^2 - b^2)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1
+ Csc[e + f*x]))/(a - b)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x
]])], -((a + b)/(a - b))]*Tan[e + f*x])/(35*a^5*(a - b)*(a + b)^(3/2)*Sqrt[d]*f) - (8*(5*a^2 - 3*a*b - 4*b^2)*
Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b
*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(35*a^4*(a - b)*(a + b)
^(3/2)*Sqrt[d]*f)

Rule 2895

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(
Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqrt[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]
*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]

Rule 2966

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_))/Sqrt[(d_.)*sin[(e_.) +
(f_.)*(x_)]], x_Symbol] :> Simp[(-g)*(g*Cos[e + f*x])^(p - 1)*Sqrt[d*Sin[e + f*x]]*((a + b*Sin[e + f*x])^(m +
1)/(a*d*f*(m + 1))), x] + Dist[g^2*((2*m + 3)/(2*a*(m + 1))), Int[(g*Cos[e + f*x])^(p - 2)*((a + b*Sin[e + f*x
])^(m + 1)/Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&
EqQ[m + p + 1/2, 0]

Rule 2968

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f,
m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])

Rule 3072

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)])^(3/2)), x_Symbol] :> Simp[2*(A*b - a*B)*(Cos[e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[d
*Sin[e + f*x]])), x] + Dist[d/(a^2 - b^2), Int[(A*b - a*B + (a*A - b*B)*Sin[e + f*x])/(Sqrt[a + b*Sin[e + f*x]
]*(d*Sin[e + f*x])^(3/2)), x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[a^2 - b^2, 0]

Rule 3073

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A*(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e +
f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e +
f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ
[A, B] && PosQ[(c + d)/b]

Rule 3077

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]

Rule 3135

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c
+ d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C
)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n +
3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && L
tQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {6 \int \frac {\cos ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}} \, dx}{7 a} \\ & = \frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {6 \int \frac {1-\sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}} \, dx}{7 a} \\ & = \frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {12 \int \frac {2 \left (a^2-b^2\right ) d-\left (a^2-b^2\right ) d \sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}} \, dx}{35 a^2 \left (a^2-b^2\right ) d} \\ & = \frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{3/2}}+\frac {8 \int \frac {\frac {1}{2} \left (5 a^4-9 a^2 b^2+4 b^4\right ) d^2-\frac {3}{2} a b \left (a^2-b^2\right ) d^2 \sin (e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}} \, dx}{35 a^3 \left (a^2-b^2\right )^2 d^2} \\ & = \frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{3/2}}+\frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 \left (a^2-b^2\right )^2 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {8 \int \frac {\frac {3}{2} a^2 b \left (a^2-b^2\right ) d^2+\frac {1}{2} b \left (5 a^4-9 a^2 b^2+4 b^4\right ) d^2+\left (\frac {3}{2} a b^2 \left (a^2-b^2\right ) d^2+\frac {1}{2} a \left (5 a^4-9 a^2 b^2+4 b^4\right ) d^2\right ) \sin (e+f x)}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}} \, dx}{35 a^3 \left (a^2-b^2\right )^3 d} \\ & = \frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{3/2}}+\frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 \left (a^2-b^2\right )^2 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {\left (4 \left (5 a^2-3 a b-4 b^2\right )\right ) \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx}{35 a^3 (a-b) (a+b)^2}+\frac {\left (16 b \left (2 a^2-b^2\right ) d\right ) \int \frac {1+\sin (e+f x)}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}} \, dx}{35 a^3 \left (a^2-b^2\right )^2} \\ & = \frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{3/2}}+\frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 \left (a^2-b^2\right )^2 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {32 b \left (2 a^2-b^2\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{35 a^5 (a-b) (a+b)^{3/2} \sqrt {d} f}-\frac {8 \left (5 a^2-3 a b-4 b^2\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{35 a^4 (a-b) (a+b)^{3/2} \sqrt {d} f} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.41 (sec) , antiderivative size = 1670, normalized size of antiderivative = 3.27 \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\frac {\sin (e+f x) \sqrt {a+b \sin (e+f x)} \left (-\frac {2 \left (a^2 \cos (e+f x)-b^2 \cos (e+f x)\right )}{7 a b^2 (a+b \sin (e+f x))^4}+\frac {4 \left (5 a^2 \cos (e+f x)+3 b^2 \cos (e+f x)\right )}{35 a^2 b^2 (a+b \sin (e+f x))^3}-\frac {2 \left (5 a^4 \cos (e+f x)-9 a^2 b^2 \cos (e+f x)+8 b^4 \cos (e+f x)\right )}{35 a^3 b^2 \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {32 \left (2 a^2 b^2 \cos (e+f x)-b^4 \cos (e+f x)\right )}{35 a^4 \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}\right )}{f \sqrt {d \sin (e+f x)}}+\frac {4 \sqrt {\sin (e+f x)} \left (\frac {4 a \left (5 a^4-9 a^2 b^2+4 b^4\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+4 a \left (-8 a^3 b+4 a b^3\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{b \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )+2 \left (8 a^2 b^2-4 b^4\right ) \left (\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{b \sqrt {\sin (e+f x)}}+\frac {i \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \csc (e+f x) E\left (i \text {arcsinh}\left (\frac {\sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\sin (e+f x)}}\right )|-\frac {2 a}{-a-b}\right ) \sqrt {a+b \sin (e+f x)}}{b \sqrt {\cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \csc (e+f x)} \sqrt {\frac {\csc (e+f x) (a+b \sin (e+f x))}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{b \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{b}\right )\right )}{35 a^4 (a-b)^2 (a+b)^2 f \sqrt {d \sin (e+f x)}} \]

[In]

Integrate[Cos[e + f*x]^4/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2)),x]

[Out]

(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((-2*(a^2*Cos[e + f*x] - b^2*Cos[e + f*x]))/(7*a*b^2*(a + b*Sin[e + f*x
])^4) + (4*(5*a^2*Cos[e + f*x] + 3*b^2*Cos[e + f*x]))/(35*a^2*b^2*(a + b*Sin[e + f*x])^3) - (2*(5*a^4*Cos[e +
f*x] - 9*a^2*b^2*Cos[e + f*x] + 8*b^4*Cos[e + f*x]))/(35*a^3*b^2*(a^2 - b^2)*(a + b*Sin[e + f*x])^2) - (32*(2*
a^2*b^2*Cos[e + f*x] - b^4*Cos[e + f*x]))/(35*a^4*(a^2 - b^2)^2*(a + b*Sin[e + f*x]))))/(f*Sqrt[d*Sin[e + f*x]
]) + (4*Sqrt[Sin[e + f*x]]*((4*a*(5*a^4 - 9*a^2*b^2 + 4*b^4)*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b
)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e
+ f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e
+ Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + 4*a*(-8*a
^3*b + 4*a*b^3)*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 -
f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-((
(a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]
)/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]
*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)
]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(C
sc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])) + 2*(8*a^
2*b^2 - 4*b^4)*((Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Sin[e + f*x]]) + (I*Cos[(-e + Pi/2 - f*x)/2]*C
sc[e + f*x]*EllipticE[I*ArcSinh[Sin[(-e + Pi/2 - f*x)/2]/Sqrt[Sin[e + f*x]]], (-2*a)/(-a - b)]*Sqrt[a + b*Sin[
e + f*x]])/(b*Sqrt[Cos[(-e + Pi/2 - f*x)/2]^2*Csc[e + f*x]]*Sqrt[(Csc[e + f*x]*(a + b*Sin[e + f*x]))/(a + b)])
+ (2*a*((a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2
]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b
)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a
+ b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*Ell
ipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Se
c[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(
-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])))/b)))/(35*a^4
*(a - b)^2*(a + b)^2*f*Sqrt[d*Sin[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(17279\) vs. \(2(458)=916\).

Time = 9.72 (sec) , antiderivative size = 17280, normalized size of antiderivative = 33.88

method	result	size
default	\(\text {Expression too large to display}\)	\(17280\)

[In]

int(cos(f*x+e)^4/(a+b*sin(f*x+e))^(9/2)/(d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate(cos(f*x+e)^4/(a+b*sin(f*x+e))^(9/2)/(d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))*cos(f*x + e)^4/(b^5*d*cos(f*x + e)^6 - (10*a^2*b^3 + 3
*b^5)*d*cos(f*x + e)^4 + (5*a^4*b + 20*a^2*b^3 + 3*b^5)*d*cos(f*x + e)^2 - (5*a^4*b + 10*a^2*b^3 + b^5)*d - (5
*a*b^4*d*cos(f*x + e)^4 - 10*(a^3*b^2 + a*b^4)*d*cos(f*x + e)^2 + (a^5 + 10*a^3*b^2 + 5*a*b^4)*d)*sin(f*x + e)
), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**4/(a+b*sin(f*x+e))**(9/2)/(d*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate(cos(f*x+e)^4/(a+b*sin(f*x+e))^(9/2)/(d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(f*x + e)^4/((b*sin(f*x + e) + a)^(9/2)*sqrt(d*sin(f*x + e))), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)^4/(a+b*sin(f*x+e))^(9/2)/(d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^4}{\sqrt {d\,\sin \left (e+f\,x\right )}\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]

[In]

int(cos(e + f*x)^4/((d*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(9/2)),x)

[Out]

int(cos(e + f*x)^4/((d*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(9/2)), x)