\(\int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx\) [228]

Optimal result

Integrand size = 25, antiderivative size = 376 \[ \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=-\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{15 b^2 f}+\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{15 b^2 f \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}}-\frac {(a-8 b) (a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{15 b f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {(a+4 b) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{15 b f}+\frac {\sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{5 f} \] Output:

-1/15*(2*a^2-3*a*b-8*b^2)*sin(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^( 
1/2)/b^2/f+1/15*(2*a^2-3*a*b-8*b^2)*(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)/b^2/f/((a+b 
-a*sin(f*x+e)^2)/(a+b))^(1/2)-1/15*(a-8*b)*(a+b)*(cos(f*x+e)^2)^(1/2)*Elli 
pticF(sin(f*x+e),(a/(a+b))^(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)/b/f/(a+b-a*sin(f*x+e)^2)+1/15*(a+4*b) 
*sec(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)*tan(f*x+e)/b/f+1/5*s 
ec(f*x+e)^3*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)*tan(f*x+e)/f
 

Mathematica [F]

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

Integrate[Sec[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2],x]
 

Output:

Integrate[Sec[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2], x]
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4636, 2057, 2058, 314, 25, 402, 25, 402, 25, 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.

\(\displaystyle \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sec (e+f x)^5 \sqrt {a+b \sec (e+f x)^2}dx\)

\(\Big \downarrow \) 4636

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\left (1-\sin ^2(e+f x)\right )^{7/2}}d\sin (e+f x)}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 314

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}-\frac {1}{5} \int -\frac {4 (a+b)-3 a \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{5/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \int \frac {4 (a+b)-3 a \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{5/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {\int -\frac {a (a+4 b) \sin ^2(e+f x)+(a-8 b) (a+b)}{\left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{3 b}+\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {a (a+4 b) \sin ^2(e+f x)+(a-8 b) (a+b)}{\left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\int -\frac {a \left (2 (a-2 b) (a+b)-\left (2 a^2-3 b a-8 b^2\right ) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}+\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {\int \frac {a \left (2 (a-2 b) (a+b)-\left (2 a^2-3 b a-8 b^2\right ) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \int \frac {2 (a-2 b) (a+b)-\left (2 a^2-3 b a-8 b^2\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (2 a^2-3 a b-8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a-8 b) (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}\right )}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (2 a^2-3 a b-8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a-8 b) (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (2 a^2-3 a b-8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a-8 b) (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a-8 b) (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a-8 b) (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

Input:

Int[Sec[e + f*x]^5*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])/(5*(1 - Sin[e + f*x]^ 
2)^(5/2)) + (((a + 4*b)*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*b* 
(1 - Sin[e + f*x]^2)^(3/2)) - (((2*a^2 - 3*a*b - 8*b^2)*Sin[e + f*x]*Sqrt[ 
a + b - a*Sin[e + f*x]^2])/(b*Sqrt[1 - Sin[e + f*x]^2]) - (a*(((2*a^2 - 3* 
a*b - 8*b^2)*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 - 8*b)*b*(a + 
 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])))/b)/(3*b))/5))/(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 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] (verified)

Result contains complex when optimal does not.

Time = 36.37 (sec) , antiderivative size = 3376, normalized size of antiderivative = 8.98

method	result	size
default	\(\text {Expression too large to display}\)	\(3376\)

Input:

int(sec(f*x+e)^5*(a+b*sec(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/15/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)/b 
^2*(a+b*sec(f*x+e)^2)^(1/2)/(a*cos(f*x+e)^3+a*cos(f*x+e)^2+cos(f*x+e)*b+b) 
*((-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*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)^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)*a^3*b*Elli 
pticE(((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))*(-co 
s(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^2*b^2*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))*(12*cos(f*x+e)^3+24*cos(f*x+e)^2+12*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...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.36 \[ \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx =\text {Too large to display} \] Input:

integrate(sec(f*x+e)^5*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm="fricas")
 

Output:

1/30*((2*(-2*I*a^3 + 3*I*a^2*b + 8*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)* 
cos(f*x + e)^4 - (-2*I*a^3 - I*a^2*b + 14*I*a*b^2 + 16*I*b^3)*sqrt(a)*cos( 
f*x + e)^4)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(arcsi 
n(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^3 - 3*I*a^2*b - 8*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos( 
f*x + e)^4 - (2*I*a^3 + I*a^2*b - 14*I*a*b^2 - 16*I*b^3)*sqrt(a)*cos(f*x + 
 e)^4)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(arcsin(sqr 
t((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^2*b + 4*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + e)^4 + (I*a 
^3 + I*a^2*b - 4*I*a*b^2 - 4*I*b^3)*sqrt(a)*cos(f*x + e)^4)*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 
^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) - 4*((-I*a^2*b - 4*I*a*b^ 
2)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + e)^4 + (-I*a^3 - I*a^2*b + 4*I* 
a*b^2 + 4*I*b^3)*sqrt(a)*cos(f*x + e)^4)*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^2 + 4*(a^2 + 2*a*b 
)*sqrt((a*b + b^2)/a^2))/a^2) - 2*((2*a^3 - 3*a^2*b - 8*a*b^2)*cos(f*x ...
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate(sec(f*x+e)**5*(a+b*sec(f*x+e)**2)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(sec(f*x+e)^5*(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)^5, x)
 

Giac [F]

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

integrate(sec(f*x+e)^5*(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)^5, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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