\(\int \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx\) [230]

Optimal result

Integrand size = 23, antiderivative size = 222 \[ \int \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {\sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{f}-\frac {\sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{f \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}}+\frac {(a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{f \left (a+b-a \sin ^2(e+f x)\right )} \] Output:

sin(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/f-(cos(f*x+e)^2)^(1/2 
)*EllipticE(sin(f*x+e),(a/(a+b))^(1/2))*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2) 
)^(1/2)/f/((a+b-a*sin(f*x+e)^2)/(a+b))^(1/2)+(a+b)*(cos(f*x+e)^2)^(1/2)*El 
lipticF(sin(f*x+e),(a/(a+b))^(1/2))*((a+b-a*sin(f*x+e)^2)/(a+b))^(1/2)*(se 
c(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/f/(a+b-a*sin(f*x+e)^2)
 

Mathematica [F]

\[ \int \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx \] Input:

Integrate[Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]
 

Output:

Integrate[Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2], x]
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 4636, 2057, 2058, 314, 25, 27, 389, 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.

\(\displaystyle \int \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sec (e+f x) \sqrt {a+b \sec (e+f x)^2}dx\)

\(\Big \downarrow \) 4636

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 314

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 389

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

Int[Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]
 

Output:

(Sqrt[1 - Sin[e + f*x]^2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x 
]^2)]*((Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/Sqrt[1 - Sin[e + f*x] 
^2] + a*(-((EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[ 
e + f*x]^2])/(a*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])) + ((a + b)*Elliptic 
F[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]))))/(f*Sqrt[a + b - a*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 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 389

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

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] (verified)

Result contains complex when optimal does not.

Time = 7.98 (sec) , antiderivative size = 1831, normalized size of antiderivative = 8.25

method	result	size
default	\(\text {Expression too large to display}\)	\(1831\)

Input:

int(sec(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/f/(2*I*a^(1/2)*b^(1/2)-a+b)/((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(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+c 
os(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*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))*(-cos(f*x+e)^3-2*cos(f*x+e)^2- 
cos(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*b*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)^ 
3-4*cos(f*x+e)^2-2*cos(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)* 
b^2*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))*(-cos(f*x+e)^3-2*cos(f*x+e)^2-cos(f*x+e))+I*a^(3/2)*b^(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)*(-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)*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-b)/(...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 603, normalized size of antiderivative = 2.72 \[ \int \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=-\frac {4 i \, a^{\frac {3}{2}} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} \sqrt {\frac {a b + b^{2}}{a^{2}}} F(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) - 4 i \, a^{\frac {3}{2}} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} \sqrt {\frac {a b + b^{2}}{a^{2}}} F(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) + {\left (-2 i \, a^{\frac {3}{2}} \sqrt {\frac {a b + b^{2}}{a^{2}}} + \sqrt {a} {\left (i \, a + 2 i \, b\right )}\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} E(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) + {\left (2 i \, a^{\frac {3}{2}} \sqrt {\frac {a b + b^{2}}{a^{2}}} + \sqrt {a} {\left (-i \, a - 2 i \, b\right )}\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} E(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) - 2 \, a \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )}{2 \, a f} \] Input:

integrate(sec(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm="fricas")
 

Output:

-1/2*(4*I*a^(3/2)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*sqrt((a*b 
+ b^2)/a^2)*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^2 + 4*(a^2 + 2*a*b) 
*sqrt((a*b + b^2)/a^2))/a^2) - 4*I*a^(3/2)*sqrt((2*a*sqrt((a*b + b^2)/a^2) 
 - a - 2*b)/a)*sqrt((a*b + b^2)/a^2)*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^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) + (-2*I*a^(3/2)*sqrt 
((a*b + b^2)/a^2) + sqrt(a)*(I*a + 2*I*b))*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^(3/2)*sqrt((a*b + b^2)/a^2) + sqr 
t(a)*(-I*a - 2*I*b))*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*ellipti 
c_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*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/ 
(a*f)
 

Sympy [F]

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

integrate(sec(f*x+e)*(a+b*sec(f*x+e)**2)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(sec(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(sec(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(sec(e + f*x)**2*b + a)*sec(e + f*x),x)