\(\int \frac {\sec ^5(e+f x)}{(a+b \sec ^2(e+f x))^{5/2}} \, dx\) [283]

Optimal result

Integrand size = 25, antiderivative size = 325 \[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {2 a (a+2 b) \sin (e+f x)}{3 b^2 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {a \sin (e+f x)}{3 b (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {2 (a+2 b) E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \left (a+b-a \sin ^2(e+f x)\right )}{3 b^2 (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {\operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}}{3 b (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \] Output:

-2/3*a*(a+2*b)*sin(f*x+e)/b^2/(a+b)^2/f/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2) 
)^(1/2)-1/3*a*sin(f*x+e)/b/(a+b)/f/(a+b-a*sin(f*x+e)^2)/(sec(f*x+e)^2*(a+b 
-a*sin(f*x+e)^2))^(1/2)+2/3*(a+2*b)*EllipticE(sin(f*x+e),(a/(a+b))^(1/2))* 
(a+b-a*sin(f*x+e)^2)/b^2/(a+b)^2/f/(cos(f*x+e)^2)^(1/2)/((a+b-a*sin(f*x+e) 
^2)/(a+b))^(1/2)/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)-1/3*EllipticF(s 
in(f*x+e),(a/(a+b))^(1/2))*((a+b-a*sin(f*x+e)^2)/(a+b))^(1/2)/b/(a+b)/f/(c 
os(f*x+e)^2)^(1/2)/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)
 

Mathematica [A] (verified)

Time = 3.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.51 \[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \sec ^5(e+f x) \left (\sqrt {2} (a+b)^2 \left (\frac {a+2 b+a \cos (2 (e+f x))}{a+b}\right )^{3/2} \left (2 (a+2 b) E\left (e+f x\left |\frac {a}{a+b}\right .\right )-b \operatorname {EllipticF}\left (e+f x,\frac {a}{a+b}\right )\right )-2 a \left (a^2+5 a b+5 b^2+a (a+2 b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))\right )}{24 b^2 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{5/2}} \] Input:

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

Output:

((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^5*(Sqrt[2]*(a + b)^2*((a + 2* 
b + a*Cos[2*(e + f*x)])/(a + b))^(3/2)*(2*(a + 2*b)*EllipticE[e + f*x, a/( 
a + b)] - b*EllipticF[e + f*x, a/(a + b)]) - 2*a*(a^2 + 5*a*b + 5*b^2 + a* 
(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))/(24*b^2*(a + b)^2*f*(a + b* 
Sec[e + f*x]^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4636, 2057, 2058, 316, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4636

\(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right )^3 \left (a+\frac {b}{1-\sin ^2(e+f x)}\right )^{5/2}}d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 2057

\(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right )^3 \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^{5/2}}d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(Sqrt[a + b - a*Sin[e + f*x]^2]*(-1/3*(a*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x 
]^2])/(b*(a + b)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + ((-2*a*(a + 2*b)*Sin[ 
e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/(b*(a + b)*Sqrt[a + b - a*Sin[e + f*x]^ 
2]) + ((2*(a + 2*b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b 
- a*Sin[e + f*x]^2])/Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)] - (b*(a + b)*Ell 
ipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b 
)])/Sqrt[a + b - a*Sin[e + f*x]^2])/(b*(a + b)))/(3*b*(a + b))))/(f*Sqrt[1 
 - Sin[e + f*x]^2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x]^2)])
 

Defintions of rubi rules used

rule 25

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

rule 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[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 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 3042

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

rule 4636

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

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.74 (sec) , antiderivative size = 4473, normalized size of antiderivative = 13.76

method	result	size
default	\(\text {Expression too large to display}\)	\(4473\)

Input:

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

Output:

-1/3/f/(a^2+2*a*b+b^2)/(2*I*a^(1/2)*b^(1/2)-a+b)/((2*I*a^(1/2)*b^(1/2)+a-b 
)/(a+b))^(1/2)/b^2/(1+cos(f*x+e))/(a+b*sec(f*x+e)^2)^(5/2)*((-3*cos(f*x+e) 
^4+9*cos(f*x+e)^3+4*cos(f*x+e)^2-2*cos(f*x+e)+2)*((I*b^(1/2)+a^(1/2))^2/(a 
+b))^(1/2)*a^2*b^3*tan(f*x+e)*sec(f*x+e)^4+(-1/(a+b)*(I*a^(1/2)*b^(1/2)*co 
s(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)* 
(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x 
+e)))^(1/2)*b^5*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x 
+e)-cot(f*x+e)),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/ 
(a+b)^2)^(1/2))*(-6*sec(f*x+e)^3-12*sec(f*x+e)^4-6*sec(f*x+e)^5)+(-1/(a+b) 
*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f* 
x+e)))^(1/2)*(1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)+cos( 
f*x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*b^5*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b) 
/(a+b))^(1/2)*(csc(f*x+e)-cot(f*x+e)),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b 
^(3/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2))*(4*sec(f*x+e)^3+8*sec(f*x+e)^4+4*sec 
(f*x+e)^5)+4*I*a^(9/2)*b^(1/2)*((I*b^(1/2)+a^(1/2))^2/(a+b))^(1/2)*sin(f*x 
+e)+(-1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a 
-b)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2) 
*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*a^5*EllipticE(((2*I*a^(1/2) 
*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(f*x+e)),(-(4*I*a^(3/2)*b^(1/2)- 
4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2))*(2*cos(f*x+e)+4+2*se...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 1374, normalized size of antiderivative = 4.23 \[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*((2*(-I*a^3*b^2 - 2*I*a^2*b^3 + (-I*a^5 - 2*I*a^4*b)*cos(f*x + e)^4 - 
2*(I*a^4*b + 2*I*a^3*b^2)*cos(f*x + e)^2)*sqrt(a)*sqrt((a*b + b^2)/a^2) - 
(-I*a^3*b^2 - 4*I*a^2*b^3 - 4*I*a*b^4 + (-I*a^5 - 4*I*a^4*b - 4*I*a^3*b^2) 
*cos(f*x + e)^4 + 2*(-I*a^4*b - 4*I*a^3*b^2 - 4*I*a^2*b^3)*cos(f*x + e)^2) 
*sqrt(a))*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(arcsin( 
sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) + I*sin(f*x + 
e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) + 
 (2*(I*a^3*b^2 + 2*I*a^2*b^3 + (I*a^5 + 2*I*a^4*b)*cos(f*x + e)^4 - 2*(-I* 
a^4*b - 2*I*a^3*b^2)*cos(f*x + e)^2)*sqrt(a)*sqrt((a*b + b^2)/a^2) - (I*a^ 
3*b^2 + 4*I*a^2*b^3 + 4*I*a*b^4 + (I*a^5 + 4*I*a^4*b + 4*I*a^3*b^2)*cos(f* 
x + e)^4 + 2*(I*a^4*b + 4*I*a^3*b^2 + 4*I*a^2*b^3)*cos(f*x + e)^2)*sqrt(a) 
)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(arcsin(sqrt((2* 
a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) - I*sin(f*x + e))), (a 
^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) + (2*(-I* 
a^2*b^3 - 3*I*a*b^4 + (-I*a^4*b - 3*I*a^3*b^2)*cos(f*x + e)^4 - 2*(I*a^3*b 
^2 + 3*I*a^2*b^3)*cos(f*x + e)^2)*sqrt(a)*sqrt((a*b + b^2)/a^2) - (2*I*a^3 
*b^2 + 9*I*a^2*b^3 + 13*I*a*b^4 + 6*I*b^5 + (2*I*a^5 + 9*I*a^4*b + 13*I*a^ 
3*b^2 + 6*I*a^2*b^3)*cos(f*x + e)^4 + 2*(2*I*a^4*b + 9*I*a^3*b^2 + 13*I*a^ 
2*b^3 + 6*I*a*b^4)*cos(f*x + e)^2)*sqrt(a))*sqrt((2*a*sqrt((a*b + b^2)/a^2 
) - a - 2*b)/a)*elliptic_f(arcsin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a -...
 

Sympy [F]

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

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

Output:

Integral(sec(e + f*x)**5/(a + b*sec(e + f*x)**2)**(5/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\cos \left (e+f\,x\right )}^5\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \] Input:

int(1/(cos(e + f*x)^5*(a + b/cos(e + f*x)^2)^(5/2)),x)
 

Output:

int(1/(cos(e + f*x)^5*(a + b/cos(e + f*x)^2)^(5/2)), x)
 

Reduce [F]

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

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

Output:

int((sqrt(sec(e + f*x)**2*b + a)*sec(e + f*x)**5)/(sec(e + f*x)**6*b**3 + 
3*sec(e + f*x)**4*a*b**2 + 3*sec(e + f*x)**2*a**2*b + a**3),x)