\(\int \frac {\sec (e+f x)}{(a+b \sec ^2(e+f x))^{5/2}} \, dx\) [285]

Optimal result

Integrand size = 23, antiderivative size = 331 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 (2 a+b) \sin (e+f x)}{3 a (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {b \sin (e+f x)}{3 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {2 (2 a+b) E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \left (a+b-a \sin ^2(e+f x)\right )}{3 a^2 (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {(3 a+2 b) \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}}{3 a^2 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \] Output:

2/3*(2*a+b)*sin(f*x+e)/a/(a+b)^2/f/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/ 
2)-1/3*b*sin(f*x+e)/a/(a+b)/f/(a+b-a*sin(f*x+e)^2)/(sec(f*x+e)^2*(a+b-a*si 
n(f*x+e)^2))^(1/2)-2/3*(2*a+b)*EllipticE(sin(f*x+e),(a/(a+b))^(1/2))*(a+b- 
a*sin(f*x+e)^2)/a^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*(3*a+2*b)*Ellipt 
icF(sin(f*x+e),(a/(a+b))^(1/2))*((a+b-a*sin(f*x+e)^2)/(a+b))^(1/2)/a^2/(a+ 
b)/f/(cos(f*x+e)^2)^(1/2)/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 4636, 2057, 2058, 315, 25, 402, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4636

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(Sqrt[a + b - a*Sin[e + f*x]^2]*(-1/3*(b*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x 
]^2])/(a*(a + b)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + ((2*(2*a + b)*Sin[e + 
 f*x]*Sqrt[1 - Sin[e + f*x]^2])/((a + b)*Sqrt[a + b - a*Sin[e + f*x]^2]) - 
 ((2*(2*a + b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*S 
in[e + f*x]^2])/(a*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a + b)*(3*a + 
 2*b)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^ 
2)/(a + b)])/(a*Sqrt[a + b - a*Sin[e + f*x]^2]))/(a + b))/(3*a*(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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 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 = 8.36 (sec) , antiderivative size = 4473, normalized size of antiderivative = 13.51

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

Input:

int(sec(f*x+e)/(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)/a^2/(1+cos(f*x+e))/(a+b*sec(f*x+e)^2)^(5/2)*(-4*I*a^(1/2)*b^ 
(9/2)*((I*b^(1/2)+a^(1/2))^2/(a+b))^(1/2)*tan(f*x+e)*sec(f*x+e)^4-2*((I*b^ 
(1/2)+a^(1/2))^2/(a+b))^(1/2)*b^5*tan(f*x+e)*sec(f*x+e)^4+(-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^2*b^3*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a 
+b))^(1/2)*(cot(f*x+e)-csc(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)-4-18*sec(f*x+e)-32*sec(f 
*x+e)^2-26*sec(f*x+e)^3-20*sec(f*x+e)^4-10*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)*a^4*b*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b) 
)^(1/2)*(cot(f*x+e)-csc(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))*(12*cos(f*x+e)+24+24*sec(f*x+e)+24*sec(f*x 
+e)^2+12*sec(f*x+e)^3)+(-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)*c 
os(f*x+e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*a^4*b*El 
lipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(cot(f*x+e)-csc(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))...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 1293, normalized size of antiderivative = 3.91 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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