\(\int \frac {\cos ^6(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\) [301]

Optimal result

Integrand size = 25, antiderivative size = 245 \[ \int \frac {\cos ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {(a+b) \cos ^3(e+f x) \sin (e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) (a+b) \cos (e+f x) \sin (e+f x)}{3 a^2 b^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (8 a^2+3 a b-2 b^2\right ) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b^3 f \sqrt {\frac {a+b \sin ^2(e+f x)}{a}}}+\frac {(8 a-b) (a+b) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {\frac {a+b \sin ^2(e+f x)}{a}}}{3 a b^3 f \sqrt {a+b \sin ^2(e+f x)}} \] Output:

1/3*(a+b)*cos(f*x+e)^3*sin(f*x+e)/a/b/f/(a+b*sin(f*x+e)^2)^(3/2)-2/3*(2*a- 
b)*(a+b)*cos(f*x+e)*sin(f*x+e)/a^2/b^2/f/(a+b*sin(f*x+e)^2)^(1/2)-1/3*(8*a 
^2+3*a*b-2*b^2)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*(a+b*sin(f*x+e)^2)^(1/2 
)/a^2/b^3/f/((a+b*sin(f*x+e)^2)/a)^(1/2)+1/3*(8*a-b)*(a+b)*InverseJacobiAM 
(f*x+e,(-b/a)^(1/2))*((a+b*sin(f*x+e)^2)/a)^(1/2)/a/b^3/f/(a+b*sin(f*x+e)^ 
2)^(1/2)
 

Mathematica [A] (verified)

Time = 3.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {-2 a^2 \left (8 a^2+3 a b-2 b^2\right ) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )+\frac {1}{2} (a+b) \left (4 a^2 (8 a-b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )-2 \sqrt {2} b \left (8 a^2-a b-2 b^2+b (-5 a+2 b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))\right )}{6 a^2 b^3 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \] Input:

Integrate[Cos[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(5/2),x]
 

Output:

(-2*a^2*(8*a^2 + 3*a*b - 2*b^2)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*E 
llipticE[e + f*x, -(b/a)] + ((a + b)*(4*a^2*(8*a - b)*((2*a + b - b*Cos[2* 
(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - 2*Sqrt[2]*b*(8*a^2 - a*b 
 - 2*b^2 + b*(-5*a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))/2)/(6*a^2*b 
^3*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3671, 315, 25, 401, 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.

\(\displaystyle \int \frac {\cos ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (e+f x)^6}{\left (a+b \sin (e+f x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 3671

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

\(\Big \downarrow \) 315

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\int -\frac {\sqrt {1-\sin ^2(e+f x)} \left (-\left ((4 a+b) \sin ^2(e+f x)\right )+a-2 b\right )}{\left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a b}+\frac {(a+b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2}}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {(a+b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2}}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {\sqrt {1-\sin ^2(e+f x)} \left (-\left ((4 a+b) \sin ^2(e+f x)\right )+a-2 b\right )}{\left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a b}\right )}{f}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {(a+b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2}}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 \left (\frac {2 a}{b}-\frac {b}{a}+1\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{\sqrt {a+b \sin ^2(e+f x)}}-\frac {\int \frac {a (4 a+b)-\left (8 a^2+3 b a-2 b^2\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a b}}{3 a b}\right )}{f}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {(a+b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2}}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 \left (\frac {2 a}{b}-\frac {b}{a}+1\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{\sqrt {a+b \sin ^2(e+f x)}}-\frac {\frac {a (8 a-b) (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{b}-\frac {\left (8 a^2+3 a b-2 b^2\right ) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}}{a b}}{3 a b}\right )}{f}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {(a+b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2}}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 \left (\frac {2 a}{b}-\frac {b}{a}+1\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{\sqrt {a+b \sin ^2(e+f x)}}-\frac {\frac {a (8 a-b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (8 a^2+3 a b-2 b^2\right ) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}}{a b}}{3 a b}\right )}{f}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {(a+b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2}}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 \left (\frac {2 a}{b}-\frac {b}{a}+1\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{\sqrt {a+b \sin ^2(e+f x)}}-\frac {\frac {a (8 a-b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (8 a^2+3 a b-2 b^2\right ) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}}{a b}}{3 a b}\right )}{f}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {(a+b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2}}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 \left (\frac {2 a}{b}-\frac {b}{a}+1\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{\sqrt {a+b \sin ^2(e+f x)}}-\frac {\frac {a (8 a-b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (8 a^2+3 a b-2 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a b}}{3 a b}\right )}{f}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {(a+b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2}}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 \left (\frac {2 a}{b}-\frac {b}{a}+1\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{\sqrt {a+b \sin ^2(e+f x)}}-\frac {\frac {a (8 a-b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (8 a^2+3 a b-2 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a b}}{3 a b}\right )}{f}\)

Input:

Int[Cos[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(5/2),x]
 

Output:

(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*(((a + b)*Sin[e + f*x]*(1 - Sin[e + f*x 
]^2)^(3/2))/(3*a*b*(a + b*Sin[e + f*x]^2)^(3/2)) - ((2*(1 + (2*a)/b - b/a) 
*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/Sqrt[a + b*Sin[e + f*x]^2] - (-((( 
8*a^2 + 3*a*b - 2*b^2)*EllipticE[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*(8*a - b)*(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*b))/(3*a*b)))/f
 

Defintions of rubi rules used

rule 25

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

rule 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[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 401

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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

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. \(711\) vs. \(2(230)=460\).

Time = 6.96 (sec) , antiderivative size = 712, normalized size of antiderivative = 2.91

method	result	size
default	\(\frac {\left (5 a^{2} b^{2}+3 a \,b^{3}-2 b^{4}\right ) \cos \left (f x +e \right )^{4} \sin \left (f x +e \right )+\left (-4 a^{3} b -6 a^{2} b^{2}+2 b^{4}\right ) \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )-\sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, a b \left (8 \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+7 \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b -\operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-8 \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}-3 \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b +2 \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}\right ) \cos \left (f x +e \right )^{2}+8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{4}+15 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3} b +6 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b^{2}-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{3}-8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{4}-11 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3} b -\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b^{2}+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{3}}{3 a^{2} \left (a +b \sin \left (f x +e \right )^{2}\right )^{\frac {3}{2}} b^{3} \cos \left (f x +e \right ) f}\)	\(712\)

Input:

int(cos(f*x+e)^6/(a+b*sin(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/3*((5*a^2*b^2+3*a*b^3-2*b^4)*cos(f*x+e)^4*sin(f*x+e)+(-4*a^3*b-6*a^2*b^2 
+2*b^4)*cos(f*x+e)^2*sin(f*x+e)-(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*(cos(f*x 
+e)^2)^(1/2)*a*b*(8*EllipticF(sin(f*x+e),(-b/a)^(1/2))*a^2+7*EllipticF(sin 
(f*x+e),(-b/a)^(1/2))*a*b-EllipticF(sin(f*x+e),(-b/a)^(1/2))*b^2-8*Ellipti 
cE(sin(f*x+e),(-b/a)^(1/2))*a^2-3*EllipticE(sin(f*x+e),(-b/a)^(1/2))*a*b+2 
*EllipticE(sin(f*x+e),(-b/a)^(1/2))*b^2)*cos(f*x+e)^2+8*(cos(f*x+e)^2)^(1/ 
2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*a^ 
4+15*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin( 
f*x+e),(-b/a)^(1/2))*a^3*b+6*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b) 
/a)^(1/2)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*a^2*b^2-(cos(f*x+e)^2)^(1/2)* 
(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*a*b^3 
-8*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticE(sin(f* 
x+e),(-b/a)^(1/2))*a^4-11*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a) 
^(1/2)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*a^3*b-(cos(f*x+e)^2)^(1/2)*(-b/a 
*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*a^2*b^2+2* 
(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticE(sin(f*x+e 
),(-b/a)^(1/2))*a*b^3)/a^2/(a+b*sin(f*x+e)^2)^(3/2)/b^3/cos(f*x+e)/f
 

Fricas [F]

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

integrate(cos(f*x+e)^6/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**6/(a+b*sin(f*x+e)**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(cos(f*x+e)^6/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(cos(f*x + e)^6/(b*sin(f*x + e)^2 + a)^(5/2), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)^6/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int(cos(e + f*x)^6/(a + b*sin(e + f*x)^2)^(5/2),x)
 

Output:

int(cos(e + f*x)^6/(a + b*sin(e + f*x)^2)^(5/2), x)
 

Reduce [F]

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

int(cos(f*x+e)^6/(a+b*sin(f*x+e)^2)^(5/2),x)
 

Output:

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