\(\int \frac {\cos ^2(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\) [303]

Optimal result

Integrand size = 25, antiderivative size = 219 \[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {\cos (e+f x) \sin (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(a+2 b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(a+2 b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {\frac {a+b \sin ^2(e+f x)}{a}}}{3 a b (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {\frac {a+b \sin ^2(e+f x)}{a}}}{3 a b f \sqrt {a+b \sin ^2(e+f x)}} \] Output:

1/3*cos(f*x+e)*sin(f*x+e)/a/f/(a+b*sin(f*x+e)^2)^(3/2)+1/3*(a+2*b)*cos(f*x 
+e)*sin(f*x+e)/a^2/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)+1/3*(a+2*b)*EllipticE( 
sin(f*x+e),(-b/a)^(1/2))*((a+b*sin(f*x+e)^2)/a)^(1/2)/a/b/(a+b)/f/(a+b*sin 
(f*x+e)^2)^(1/2)-1/3*InverseJacobiAM(f*x+e,(-b/a)^(1/2))*((a+b*sin(f*x+e)^ 
2)/a)^(1/2)/a/b/f/(a+b*sin(f*x+e)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 2.04 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 a^2 (a+2 b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )-2 a^2 (a+b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )-\sqrt {2} b \left (-4 a^2-7 a b-2 b^2+b (a+2 b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{6 a^2 b (a+b) f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \] Input:

Integrate[Cos[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]
 

Output:

(2*a^2*(a + 2*b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f* 
x, -(b/a)] - 2*a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*Ellipt 
icF[e + f*x, -(b/a)] - Sqrt[2]*b*(-4*a^2 - 7*a*b - 2*b^2 + b*(a + 2*b)*Cos 
[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*a^2*b*(a + b)*f*(2*a + b - b*Cos[2*(e 
+ f*x)])^(3/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3671, 314, 25, 402, 25, 399, 323, 321, 330, 327}

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.

\(\displaystyle \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (e+f x)^2}{\left (a+b \sin (e+f x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 3671

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\sqrt {1-\sin ^2(e+f x)}}{\left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 314

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {2-\sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\int \frac {2-\sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {(a+2 b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int -\frac {(a+2 b) \sin ^2(e+f x)+a}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\int \frac {(a+2 b) \sin ^2(e+f x)+a}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}+\frac {(a+2 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\frac {(a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}-\frac {a (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{b}}{a (a+b)}+\frac {(a+2 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\frac {(a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{b \sqrt {a+b \sin ^2(e+f x)}}}{a (a+b)}+\frac {(a+2 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\frac {(a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}}{a (a+b)}+\frac {(a+2 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\frac {(a+2 b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}}{a (a+b)}+\frac {(a+2 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\frac {(a+2 b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}}{a (a+b)}+\frac {(a+2 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

Input:

Int[Cos[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]
 

Output:

(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*((Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2] 
)/(3*a*(a + b*Sin[e + f*x]^2)^(3/2)) + (((a + 2*b)*Sin[e + f*x]*Sqrt[1 - S 
in[e + f*x]^2])/(a*(a + b)*Sqrt[a + b*Sin[e + f*x]^2]) + (((a + 2*b)*Ellip 
ticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(b*Sqrt[1 + 
 (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*EllipticF[ArcSin[Sin[e + f*x]], -(b/a 
)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*Sqrt[a + b*Sin[e + f*x]^2]))/(a*(a + 
 b)))/(3*a)))/f
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, 
 x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 3042

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

rule 3671

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( 
p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff*(Sqrt[ 
Cos[e + f*x]^2]/(f*Cos[e + f*x]))   Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a 
 + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] 
 && IntegerQ[m/2] &&  !IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(548\) vs. \(2(204)=408\).

Time = 4.17 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.51

method	result	size
default	\(-\frac {\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b \sin \left (f x +e \right )^{2}+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2} \sin \left (f x +e \right )^{2}-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b \sin \left (f x +e \right )^{2}-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2} \sin \left (f x +e \right )^{2}+a \,b^{2} \sin \left (f x +e \right )^{5}+2 b^{3} \sin \left (f x +e \right )^{5}+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+a^{2} \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +2 a^{2} b \sin \left (f x +e \right )^{3}+2 a \,b^{2} \sin \left (f x +e \right )^{3}-2 b^{3} \sin \left (f x +e \right )^{3}-2 a^{2} b \sin \left (f x +e \right )-3 a \,b^{2} \sin \left (f x +e \right )}{3 \left (a +b \right ) a^{2} \left (a +b \sin \left (f x +e \right )^{2}\right )^{\frac {3}{2}} b \cos \left (f x +e \right ) f}\)	\(549\)

Input:

int(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

-1/3*((cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticF(sin(f*x+ 
e),(-b/a)^(1/2))*a^2*b*sin(f*x+e)^2+(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^ 
2)/a)^(1/2)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*a*b^2*sin(f*x+e)^2-(cos(f*x 
+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticE(sin(f*x+e),(-b/a)^(1/2 
))*a^2*b*sin(f*x+e)^2-2*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)* 
EllipticE(sin(f*x+e),(-b/a)^(1/2))*a*b^2*sin(f*x+e)^2+a*b^2*sin(f*x+e)^5+2 
*b^3*sin(f*x+e)^5+(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*Ellipt 
icF(sin(f*x+e),(-b/a)^(1/2))*a^3+a^2*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e) 
^2)/a)^(1/2)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*b-(cos(f*x+e)^2)^(1/2)*((a 
+b*sin(f*x+e)^2)/a)^(1/2)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*a^3-2*(cos(f* 
x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticE(sin(f*x+e),(-b/a)^(1/ 
2))*a^2*b+2*a^2*b*sin(f*x+e)^3+2*a*b^2*sin(f*x+e)^3-2*b^3*sin(f*x+e)^3-2*a 
^2*b*sin(f*x+e)-3*a*b^2*sin(f*x+e))/(a+b)/a^2/(a+b*sin(f*x+e)^2)^(3/2)/b/c 
os(f*x+e)/f
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 1394, normalized size of antiderivative = 6.37 \[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="fricas")
 

Output:

1/6*((2*((I*a*b^3 + 2*I*b^4)*cos(f*x + e)^4 + I*a^3*b + 4*I*a^2*b^2 + 5*I* 
a*b^3 + 2*I*b^4 - 2*(I*a^2*b^2 + 3*I*a*b^3 + 2*I*b^4)*cos(f*x + e)^2)*sqrt 
(-b)*sqrt((a^2 + a*b)/b^2) - ((-2*I*a^2*b^2 - 5*I*a*b^3 - 2*I*b^4)*cos(f*x 
 + e)^4 - 2*I*a^4 - 9*I*a^3*b - 14*I*a^2*b^2 - 9*I*a*b^3 - 2*I*b^4 + 2*(2* 
I*a^3*b + 7*I*a^2*b^2 + 7*I*a*b^3 + 2*I*b^4)*cos(f*x + e)^2)*sqrt(-b))*sqr 
t((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqr 
t((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 
+ 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*((-I*a*b^ 
3 - 2*I*b^4)*cos(f*x + e)^4 - I*a^3*b - 4*I*a^2*b^2 - 5*I*a*b^3 - 2*I*b^4 
- 2*(-I*a^2*b^2 - 3*I*a*b^3 - 2*I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 
+ a*b)/b^2) - ((2*I*a^2*b^2 + 5*I*a*b^3 + 2*I*b^4)*cos(f*x + e)^4 + 2*I*a^ 
4 + 9*I*a^3*b + 14*I*a^2*b^2 + 9*I*a*b^3 + 2*I*b^4 + 2*(-2*I*a^3*b - 7*I*a 
^2*b^2 - 7*I*a*b^3 - 2*I*b^4)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2*b*sqrt((a^ 
2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b 
^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 
 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - 2*(4*((I*a*b^3 + I*b^4)*cos 
(f*x + e)^4 + I*a^3*b + 3*I*a^2*b^2 + 3*I*a*b^3 + I*b^4 + 2*(-I*a^2*b^2 - 
2*I*a*b^3 - I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) + ((-2*I 
*a^2*b^2 - I*a*b^3)*cos(f*x + e)^4 - 2*I*a^4 - 5*I*a^3*b - 4*I*a^2*b^2 - I 
*a*b^3 + 2*(2*I*a^3*b + 3*I*a^2*b^2 + I*a*b^3)*cos(f*x + e)^2)*sqrt(-b)...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**2/(a+b*sin(f*x+e)**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(cos(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="giac")
 

Output:

integrate(cos(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \] Input:

int(cos(e + f*x)^2/(a + b*sin(e + f*x)^2)^(5/2),x)
 

Output:

int(cos(e + f*x)^2/(a + b*sin(e + f*x)^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right )^{2} b +a}\, \cos \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{6} b^{3}+3 \sin \left (f x +e \right )^{4} a \,b^{2}+3 \sin \left (f x +e \right )^{2} a^{2} b +a^{3}}d x \] Input:

int(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x)
 

Output:

int((sqrt(sin(e + f*x)**2*b + a)*cos(e + f*x)**2)/(sin(e + f*x)**6*b**3 + 
3*sin(e + f*x)**4*a*b**2 + 3*sin(e + f*x)**2*a**2*b + a**3),x)